iGetIt! Music

HTML5 vs. Flash/Silverlight/JavaFX

2010-06-14T13:10:00.000-05:00

Given that I've invested many months in teaching myself Flex, and have a professional background in technology evangelism, I'm fascinated by Apple's decision not to support Flash/Silverlight/JavaFX/etc in the iPad.

Why did Apple ban Flash (among other things) from the iOS?

John Gruber's insightful blog post, Why Apple Changed Section 3.3.1, cuts right to the heart of the matter, to wit, that this is a tactic to help Apple establish the iOS API as a de facto standard for mobile apps.

This is a very dangerous tactic for Apple. Tactics that are perfectly legit for a non-monopolist (such as Microsoft's Windows pricing policy) are retroactively illegal for monopolists.

Apple's market share of the mobile handset market, at just 2.7%, is far from monopolistic. However, in 2009, 94% of mobile apps were sold into the iOS market. That is, if you want to profit from sales of your mobile app, then you must release it on iOS, and therefore you must develop it according to Apple's dictates. That is pretty much the definition of monopoly power. Once Apple is determined to have monopoly power, tactics such as Apple's banning Flash/etc. are likely to be found to be retroactively illegal. (I cannot over-stress the importance of the word "retroactively" in this discussion.) This is the market that will capture the attention of anti-trust regulators.

An anti-trust inquiry into Apple's tactics has already been launched in the US; I would not be at all surprised to see one follow rapidly in the EU.

Apple's actions impose two new costs on itself:

Direct costs: Anti-trust defense costs a fortune, in legal bills, fines, and―most importantly―executive focus.
Indirect costs: Acting in a monopolisitc manner, as Apple is now doing, makes it appear to be evil. Apple has profited enormously from its warm and cuddly reputation, which it has now started ruining. This indirect cost is very likely to far outweigh the direct costs of anti-trust defense.

If Apple's actions succeed in establishing it iOS API as the de facto mobile app standard, however, then these costs can be easily borne out of the immense profits that it will make from the resulting vendor lock-in. Apple would be far more powerful than Microsoft ever was, because it would control

the central processing unit (A4, like Intel's x86)
the hardware (iPhone/iPad/iWhatever, like Dell+HP+everyone else)
the OS (iOS, like Microsoft Windows)
the app distribution channel (AppStore, like everyplace you ever bought software), and
the software development tools (XCode, like Microsoft's Visual Studio and the open-source Eclipse tools platform).

Apple would have proprietary control over the entire value chain...except for mobile apps, from which it would nonetheless extract a 30% tax via the AppStore. It would have the most comprehensive computing monopoly since IBM's.

Here are some relevant links:

Gartner's Ray Valdes: HTML5 and the future of Adobe Flash, the best discussion of how Apple's SDK rules and the rise of HTML5 affect Flash. Qick answer: not much, for now.
Blogger David R Poindexter: Flash Apologists are Legion, whose comments are prototypically naïve.

Apple is attempting to cover two bases at once:

First, it is using the license restrictions on the iOS SDK (and related AppStore approval) to focus independent developers on the iOS API, thereby building a critical mass of support for it, and locking it in as the de facto standard API for mobile app software development, as described above.
Second, it is preparing for a future transition away from native (iOS or Android) apps to open-standards-based Web apps, in case its iOS-focused strategy either fails or dies a natural death due to the maturation of open standards such as CSS, HTML, etc. Hence, its aggressive support for those standards the open-source WebKit browser, on which Apple's Safari is based. WebKit is intended to ensure that if web apps emerge as the new de facto standard, no one else, outside of Apple, controls that API.

In short, Apple's "banning Flash" has absolutely nothing whatsoever with the technical merits of Flash; it is purely a platform play by Apple. All of Steve Jobs' comments regarding the technical merits of Flash are just smoke and mirrors, to distract attention from the core issue.

To further execute this strategy, I would expect Apple to take two steps:

Implement the iOS API on top of Android, allowing Apple to sell its developers an iOS-to-Android app-porting tool. If this tool were expensive enough, the profit from it would more than make up for any "sales of iPhones lost to Android phones due to the availability of iOS apps on Android," while further locking in the iOS API as the de facto mobile standard. Establishing the iOS API as the de facto API for Android apps would emasculate Android as a competitive threat, by ensuring that any new APIs developed for Android would be ignored by developers...until wrapped by new iOS APIs. If Apple could also extend its mobile advertising restrictions to iOS-based apps running on Android, then the porting of the iOS API to Android could be even more profitable. Alternatively (or in conjunction), Apple could license the iOS to other handset vendors, and license the iPhone as a reference design. This alternative is extremely unlikely, however, given Apple's corporate culture and business model.
Release, and feature, Apple-branded mobile apps in the best-selling and most-stable categories. Most casual users will favor the Apple-branded products simply because they are Apple-branded (so long as the apps don't suck). This will not only earn Apple additional revenue, but also give it a stable of in-house applications which can help drive future new iOS APIs to critical mass, and with which non-Apple apps will need to interoperate.

Both of these steps would increase the value of the iOS API standard to developers and consumers, thereby increasing its value to Apple and also locking it in even further.

So, what does all of this mean for Flash/Flex developers such as myself?

In the short run, Apple's having banned Adobe's Flash-to-iOS deployment tool is a pain. If I needed to sell my app to iOS users (which, thankfully, I do not), then I would have to use Apple's tools, framework, language, etc. for that version...which would not then be portable outside of Apple's universe. That is, I would be compelled to develop and maintain two codebases.
In the medium term, I would expect to see Adobe find workarounds (such as proxying Flash content onto the iOS as Opera Mini does), which will work just fine for many kinds of content (streaming video, for example), but not so well for some kinds (games and RIA's, for example).
In the long term, anti-trust action will almost certainly compel Apple to drop its development-tool restrictions.

And throughout this run, the only affected deployment platform is iOS―that is, Apple's mobile products. Flash continues to work great on Macintosh computers, and on all other computers; just not on Apple's mobile products.

Apple's tactic has slightly devalued Flash/Flex's cross-platform value proposition, but the demand for Flex developers has been rising so rapidly that Apple's tactic is a blip, not a barrier, to the monetization of Flex development skills.

Starbucks Via Ready Brew: taste-test

2010-06-07T22:19:00.000-05:00

Yesterday, I received a sample of Starbucks VIA(tm) Ready Brew Colombia in the liner of my Dallas Morning News—so I set up a blind taste test.

Bottom line: Starbucks VIA Ready Brew tastes like battery acid. We'll stick with Community Coffee Dark Roast.

The three contenders were:
- Our regular coffee, which is Community Coffee's Dark Roast (brewed in a Bunn BTX Thermo-Fresh Brewer)
- Nescafé Brasero, an instant coffee I bought on a recent consulting trip to Norway
- Starbucks VIA Colombia Ready Brew

The taste-testers: my wife Patti, her mother Gretta, and myself.

I knew which coffees were which, but Patti and Gretta knew them only as coffees A, B, and C.

First, we tried all three coffees black.
- Patti's preference order: C most preferred, then B, then A least preferred, with the comment that B was bitter, but A was worse.
- Gretta's preference order: C most preferred, then A, then B least preferred, with the comment that B was the most bitter.

Then, after adding sugar and half & half, they sipped again, were polled again, and gave the same preference orders.

They result: they liked their regular coffee (Community Coffee Dark Roast) best (coffee C), and were split on whether Starbucks (B) or Nescafé (A) was the worst.

So, we won't be switching anytime soon.

Scientific? Not even close. Fair? Absolutely. We were comparing the coffee we drank everyday to an alternative, which is exactly what Starbucks intended people do with its free samples.

If anything, our informal little taste-test made me look forward to trying the instant version of Community Coffee's Dark Roast. We already know that we like its basic flavor. The VIA packet contains 3.3g (0.1164oz) of instant coffee, and they cost about a dollar per packet in bulk online, so VIA costs about a dollar per cup. The instant Community Coffee Dark Roast, however, has a list price of $5.69 for a 7 oz jar, which (all else being equal) would make 60 cups, hence costing less that 10¢ per cup -- one-tenth the cost of Starbucks VIA. And, of course, Community Coffee's ground coffee costs even less per cup than that.

Apparently, Starbucks is continuing its business model of selling overpriced coffee to people with more money than sense (or taste).

Thanks for the sample, Starbucks! Now we know that our Community Coffee Dark Roast is not only less expensive, but also tastes better, than Starbucks.

Model, Conjecture, Hypothesis, Theory, Law

2010-06-02T15:49:00.000-05:00

In science, what is the difference between a conjecture, a hypothesis, a theory, and a law?

The best answer I was able to find, through Google, I found here, written by Jeffrey Glassman (who was apparently paraphrasing Michael Riordan). Here's the relevant section:

Science is all about models of the real world, whether natural (basic science) or manmade (applied science, or technology). These models are not discovered in nature, for nature has no numbers, no coordinate systems, no parameters, no equations, no logic, no predictions, neither linearity nor non-linearity, nor many of the other attributes of science. Models are man’s creations, written in the languages of science: natural language, logic, and mathematics. They are built upon the structure of a specified factual domain. The models are generally appreciated, if not actually graded, in four levels:

1. A conjecture is an incomplete model, or an analogy to another domain. Here are some examples of candidates for the designation:

“Ephedrine enhances fitness.”

“The cosmological red shift is cause by light losing energy as it travels through space.” (This is the “tired light conjecture.”)

“The laws of physics are constant in time and space throughout the universe.” (This one is known in geology as “uniformitarianism.”)

“Species evolve to superior states.”

“A carcinogen to one species will necessarily be carcinogenic to another.”

2. A hypothesis is a model based on all data in its specified domain, with no counterexample, and incorporating a novel prediction yet to be validated by facts. Candidates:

“Mental aging can be delayed by applying the ‘use it or lose it’ dictum.”

“The red shift of light is a Doppler shift.”

3. A theory is a hypothesis with at least one nontrivial validating datum. Candidates:

Relativity.

Big Bang cosmology.

Evolution.

4. A law is a theory that has received validation in all possible ramifications, and to known levels of accuracy. Candidates:

Newtonian mechanics.

Gravity.

Henry’s Law.

The laws of thermodynamics.

Each of these candidates can stir arguments worthy of a paper, if not a book, and no model is secure in its position. Weak scientists will strengthen their beliefs and stances by promoting their models while demoting the competition. Some familiar models fail even to be ranked because they are beyond science, usually for want of facts. Candidates:

Creation science or notions of “intelligent design.”

Astrology.

Parapsychology.

UFO-ology.

One of the positive outcomes of the extremely public debate about anthropogenic global warming (and evolution/creation, for all that) is that it has compelled thoughtful scientists to re-acquaint themselves with the Scientific Method. The Scientific Method is like democracy: the worst possible system, except for all of the others (paraphrasing Churchill).

I don't know whether Glassman (quoted above) is a credible scientist or not, but the whole point is that that's not the point: experimental results are the point, not the personality, pedigree, or popularity of the person who produced them (to be arrestingly alliterative).

JiMS proposes an alternative model for displaying, controlling, and understanding musical information. It is based on the Matrix's hypothesis that human cognition uses an isomorphic note-layout to classify and track tonal relationships over time.

The Matrix's hypothesis is, in turn, based on Sethares' theory that consonance arises from the alignment of tuning and timbre.

JiMS' hypothesis predicts that the cognitive map of tonal space observed by Janata, Krumhansl, etc. will prove to be tuning-independent across the syntonic tuning continuum—but to experiments have yet been performed to test/falsify/confirm this claim (which is why it's a hypothesis, not a theory).

I have every confidence that JiMS will prove to be the fastest path to deep musical understanding. But, what do I know? Until the courseware exists, no experiments using it can be conducted, so I have no supporting evidence...yet. Only time—and the application of the Scientific Method—will tell.

Whoa: Major and minor 4th and 5th

2010-05-30T20:13:00.000-05:00

(At the time of this posting, JiMS inline interactive courseware brings up this page when students press the "Whoa!" button for more information on the naming of the qualities of the diatonic 4th and 5th.)

In the fundamental scales of Western music (pentatonic, diatonic, chromatic, etc.) each generic interval occurs in exactly two qualities, called specific intervals.

This property is generally called Myhill's property after John Myhill, who co-discovered it with John Clough and Gerald Myerson, as described here.

Many of Westen music's most salient features have been shown to arise from this property.

Hence, it is extremely important to expose, through musical nomenclature,
• The difference between diatonic intervals that have
- one width, and
- two widths.

• The similarity among diatonic intervals that have
- width, and
- two widths.

The English language's traditional interval-naming scheme fails on both counts. Instead of dividing the diatonic intervals into two kinds (one-width and two-width), naming the kinds consistently, and naming the two qualities of the two-width kind consistently, it divides the diatonic intervals into four categories:
• One-width: perfect unison and perfect octave
• Two-width, smaller perfect: the 4th (smaller perfect; wider augmented)
• Two-width, larger perfect: the 5th (smaller diminished; wider perfect)
• Two-width, major and minor: the 2nd, 3rd, 6th, and 7th (smaller minor; wider major)

Furthermore, the English language's traditional interval-naming scheme fails to make an important distinction between
• diatonic qualities, and
• chromatic alterations.

For example, it uses
• "diminished" to refer to both
- the narrower diatonic 5th and
- the chromatically-narrowed cersions of the other diatonic intervals.
• "augmented" to refer to both
- the wider diatonic 4th and
- the chromatically-widened versions of the other diatonic intervals.

In summary, the English language's traditional interval-naming scheme
• Fails to distinguish among entities that are different:
- By using the term "perfect" to describe both one-width qualities and two-width qualities
- By using the term "augmented" to describe both diatonic qualities and chromatic alterations
- By using the term "diminished" to describe both diatonic qualities and chromatic alterations
• Makes false distinctions among entities that are the same:
- By using the different terms "perfect," "minor," and "diminished" to describe the same "narrower version of a two-width diatonic interval"
- By using the different terms "perfect," "major," and "augmented" to describe the same "wider version of a two-width diatonic interval"

JiMS does not break with tradition lightly. However, JiMS' goal of being the fastest path to deep understanding requires that it break with tradition in this case. By making a clean distinction between the one-width and two-width kinds of diatonic intervals (perfect and imperfect, respectively), and naming the imperfect intervals' narrower and wider qualities consistently (minor and major, respectively), JiMS not only exposes Myhill's Property—arguably the most fundamental pattern in all of Western music—but also reserves the names "diminished" and "augmented" to refer to chromatic alteration of those intervals.

JiMS-trained musicians are expected to (eventually) learn the traditional interval-naming scheme, in addition to the (more logical) one used by JiMS. However, because JiMS is expected to help students learn music's concepts at roughly three times the speed of traditional methods, JiMS-based students will have ample time to learn the traditional interval-naming scheme, too, and still be way ahead.

Lesson Feedback

2010-05-10T13:14:00.001-05:00

Catherine Schmidt-Jones, author of Understanding Basic Music Theory, was kind enough to offer her feedback on JiMS (lessons 1-6) in a recent email exchange.

-------------------
I suspect that what you have so far goes way too fast and too far without being grounded in what the student wants to do. In other words, the average person who can get through that much theory without getting to play anything and without losing interest has probably already learned traditional music theory. You introduce a lot of concepts very quickly, and this could get to be very intimidating for the beginner.

If you want to go for the people who want to make music but want an "easy way" to do it (and I agree that that is the ideal target population for you right now), you need to make sure that you show them as soon as possible that figuring out what you are talking about is going to "pay off" in terms of being able to reach their goals as musicians.

I would introduce the "you can play your computer keyboard" idea much sooner (as soon as is reasonable, actually), and have more exercises that let them get comfortable with doing that. If you can relate the exercises to the theory, so much the better. If you don't do this, then when they do get to the keyboard, that whole long string to play "Twinkle, twinkle" might also look very intimidating and hard to actually play. Don't scare them off; make it fun and engaging to hook them, and I believe they will sign up for more. Keep me posted!

-------------------

To which I responded (in part):

-------------------

I am toying with the idea of writing a Guitar Hero-like game using (a) the computer-based JiMS note-layout for a controller, and (b) piano-roll notation on the JiMS staff. With stickers on the appropriate keyboard buttons, color-coded (as in Guitar Hero) to match the corresponding staff-locations, students could start playing music pretty quickly -- without understanding any of it, of course, just as with Guitar Hero. Still, the game would allow me to introduce JiMS' piano-roll notation "under the radar," so to speak.

With such a game in place, the lessons could bounce back and forth between "lectures" (like the lessons I've already got) and "labs" in which one plays songs that illustrate the concepts presented in the lectures.

Writing such a game is a non-trivial programming challenge, so I've been putting it off until after my programming skills improve sufficiently. Your comments have ratcheted up my perception of its importance, though.

In short: everything you wrote was right on target, and I will change my priorities to take your feedback into account.

-------------------

Her response:

-------------------

I do think the guitar-hero-type playing-lab is a really good idea, even if the main musicianship goal is composition, since being able to play the "instrument" well makes the composition process go much more smoothly.

In fact, I believe that the reason composers tend to play piano or guitar is that being able to easily play and hear your own explorations of harmony theory is really important to developing the musical intuitions of a good composer, which suggests that your approach should work really well for composers. Improvisers too, if you can ever get somebody to build and sell an actual instrument. Maybe if your lessons develop a loyal following! Keep me posted, Kitty

I appreciate her candid feedback, and look forward to receiving more of it. Yours, too! ;-)

Lesson 7

2010-05-09T23:46:00.000-05:00

Here's Lesson 7, on interval nomenclature (kind, quality, degree, naming, and abbreviation).

The main file for this lesson includes over 2,000 lines of code, for a 10-minute lesson (200 lines/lesson-minute). That's more code than I would have expected. There must be an easier/faster way to generate these lessons.

Wiktionary defines "nomenclature" as "a name; a set of names or terms; [and/or] a set of rules used for forming the names or terms in a particular field of arts or sciences." While I've tried to keep the nomenclature of JiMS iGetIt! Music System (JiMS) as consistent with Western music's traditional nomenclature as possible, there are some cases in which improvements can recoup the cost of incompatiblity, with interest.

An example occurs in this lesson. In traditional Western musical nomenclature, intervals are assigned to the "perfect" or "imperfect" categories ("kinds") for no good reason that I can identify.

In JiMS, on the other hand, the differentiation between perfect and imperfect "kinds" of intervals is logical and meaningful: if an interval (of a given degree) occurs in the diatonic scale in one and only one size, it's perfect; if it occurs in two or more sizes, it's imperfect. Hence, the unison and octave are perfect, while all other diatonic intervals are imperfect...including the intervals traditionally named the "perfect fifth" and "perfect fourth."

This nomenclature is simple and logical. It also sets the stage for diatonic set theory, by making Myhill's property much easier to see, understand, and apply.

Is it reasonable for JiMS to use non-traditional names for these intervals? Well..."The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man." (George Bernard Shaw, 1903, Maxims for Revolutionists)

Lesson 6

2010-05-05T10:02:00.000-05:00

Here's Lesson 6, on modes and mode degrees.

I finally replaced the previous lessons' state-controlling button bar with a simple "Next" button, which only shows up at the appropriate time. This is much simpler and more intuitive, but restricts the user to a single path through the lesson, with no backing up. Good enough for now.

This lesson is, I think, an excellent example of the advantages that animations bring to online music education. The animation of the mode degree labels is impossible to replicate in a book, on a static web page, or even in a live lecture. (Compare it to this, for example.)

JiMS iGetIt! Music System (JiMS) doesn't use the traditional mode names -- Ionian, Dorian, Phrygian, etc. -- because those names are meaningless, and therefore must be memorized, which increases the cognitive load of the overall system.

Instead, JiMS names a mode after its scale and starting note-class -- "diatonic Do-mode," for example, or simply "Do-mode" if a diatonic context can be correctly assumed. This mode-naming system requires no memorization because it "says what it means." The name and the meaning are the same thing. Why require students to memorize the fact that "Phygian" means "diatonic Re-mode," when you can just call the mode "diatonic Re-mode" instead?

Furthermore, this same mode-naming pattern can be applied to any scale, whereas the traditional mode-naming scheme requires a unique name for every combination of scale and degree -- yet there is no standard for such unique names for non-diatonic scales.

A fundamental premise of JiMS is that using its non-standard nomenclature will increase students' learning-efficiency by a factor of at least three (and perhaps much more), producing a time-savings that will far exceed the time-cost of learning, in later lessons, to translate between JiMS and traditional nomenclature. As yet, of course, I have no hard data to support this premise, but the winds of cognitive science are at my back.

Next up: Diatonic intervals.

New address

2010-04-28T18:54:00.000-05:00

This blog was previously hosted on Blogger via FTP from www.iGetItMusic.com/blog, but Blogger discontinued its support for FTP-based blogs, so it had to move. Hence, the new address: http://blog.igetitmusic.com/.

I apologize for the inconvenience! :-(

--- Jim

This blog has moved

2010-04-28T13:20:00.001-05:00

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Flex 4's layouts, animations, and interpolators

2010-04-26T14:55:00.002-05:00

Below is an applet that demonstrates three new Flex features:
1. Custom layouts
2. Generalized animations
3. Custom interpolators

The applet, called AngleTest (source code here), shows the notes of the diatonic scale (Do, Re, Mi, etc.) placed around a circle:

Re is at the top to make the symmetry of the diatonic scale more obvious (because horizonal or vertical symmetry is much easier for human brains to recognize than symmetry along any other axis).

Initially, the scale is in Do-mode ("major," or "Ionian"). You can tell, because degree number 1—in green, to make it stand out—is next to Do.

If you click the "Next Mode" button (in the upper-left corner of the applet), the degree numbers change to reflect their positions in Re-mode ("Dorian"). Degree 1, in green, is now next to Re.

I don't know about you, but I find that sudden change of numbers, in place, to be almost impossible to follow. What changed, exactly, and how?

So, I added an "Animate" checkbox, also towards the upper-left corner. Click it, so that a checkmark appears in its check box, thereby enabling animation. Now, click the "Next Mode" button again.

Cool! The degrre numbers slide smoothly from one note-name to another, showing—with much greater clarity—the relationships of the modes to the underlying scale. That's due to the interaction of two new Flex features: (a) Flex's new custom layout feature, which my code uses to place each degree label at the angle specified by its "angle" property, and (b) Flex's new generalized Animation effect, which smoothly animates the angle property between values.

Er, hmmm, but...there's a problem. One of the degree numbers rotates in the wrong direction! Whichever degree label is on Mi goes all the way around the circle counter-clockwise to Fa, instead of going clockwise a short ways to Fa. WTF?

The problem is fairly simple, actually: Mi's angle is 330 degrees, and Fa's is 0 degrees. The natural way to interpolate the numbers between 330 and 0 is downward, through 300, 200, 100, and so on down to 0...and that's exactly what the Animation effect's default interpolator does. That's the only interval on the circle in which a degree label is moving from a large starting angle to a small ending angle, so it's the only interval for which the degree label goes the wrong way.

What we need here is a custom interpolator -- an *angle* interpolator -- which recognizes that angles fall on a circular continuum, rather than a linear continuum.

In the "Interpolator" panel, just below the "Animate" checkbox, are two radio buttons that let you select whether to use Animate's default "Number" interpolator or my custom "Angle" interpolator.

The "Angle" interpolator keeps all of the degree labels moving clockwise.

It isn't commercial-grade, though. I'm pretty sure that the increment and decrement methods aren't right, but my code doesn't need those functions so I haven't tested or revised them accordingly. (You've been warned!)

The layout of the note-names is a little off, too. I'm not sure why. I think that it's because the bounding boxes of the labels reserve space for descenders -- the bits of letters like 'y' and 'p' that stick down below the rest of the text. Not sure, and for now, don't care -- it's good enough for Version 1.0.

When I look at other people's sample Flex code, I am frequently stunned by the amazing effects they are able to get out of a tiny bit of code. My code's not like that. I seem to have to write reams of the stuff.

For example, the arcs that show the widths of the seconds around the circle—the red and blue arrows—are, well, *arcs*, and Flex's new FXG graphics code does not support arcs. Theoretically, one can program them using cubic Bezier curves, but I seem to have missed that post-graduate mathematics course. Instead, I kludged them up in Adobe Illustrator, ran Illustrator's output through Adobe's Flash Catalyst, and pasted the result into my app. The result looks fine, but involved reams of impenetrably-complex MXML/FXG that's almost certainly longer than necessary. But...hey! It works, and that counts for a lot.

Progressive Complexity

2010-04-06T12:39:00.000-05:00

In response to this comment on my post describing Lesson 005.1...

-----------------------

All of your criticisms above make perfectly valid points. I have what I *think* are good reasons for structuring JiMS the way I am, but in the absence of hard scientific evidence as to the efficacy of one structure vs. another, there is very ample room for differences of opinion. I value, very highly, such critical feedback, because it forces me to revisit my assumptions and see if they have any rational basis. That said...Many of the concepts that I'm teaching in JiMS with specifics, could be taught with greater generality. Here's why I'm starting with narrow, concrete specifics, such as the assumption of the syntonic temperament in 12-TET tuning. I could design JiMS such that it did not assume the syntonic temperament (i.e., alpha = P8, beta = tempered "perfect" fifth, with first comma tempered to unison) or 12-TET tuning (i.e., P8=1200 cents and P5=700 cents). Instead, I could teach that alpha and beta could be any interval. This would be a temperament-neutral approach. So, why am I basing JiMS firmly on the syntonic temperament...and with a fixed octave, to boot? Firstly, because JiMS is based on the philosophy of "progressive complexity," which states that "the simplest approximations of the truth should be presented first, with increasingly complex approximations added later, only as necessary." (I'm quoting myself here, but some prominent educator somewhere must have said something similar.) So, JiMS starts with the diatonic scale in the syntonic scale, in 12-TET tuning, and will progress through the chromatic scale also in 12-TET. It's only when JiMS needs to introduce the enharmonic scale -- which I've decided to do WAY later -- that the concept of "tuning" needs to be introduced. After that, JiMS can introduce Dynamic Tonality. Somewhere right before the introduction of Dynamic Tonality, I'll introduce the notion that the octave's width can be tempered in the syntonic temperament, too. After Dynamic Tonality, I can introduce non-syntonic temperaments. Remember the immediate goal of JiMS: to dramatically increase the efficiency with which non-musicians gain sufficient musical knowledge to write their own pop/rock music. To achieve this goal, students do not need to learn about non-syntonic temperaments. That being said, JiMS' *ultimate* goal is to establish a new universal paradigm of music which expands the frontiers of tonality. I believe that this ultimate goal cannot be achieved without first achieving JiMS immediate goal. Therefore, the immediate goal must take precedence in all of JiMS' design decisions. Specifically, I can't allow JiMS early lessons to be made more complex in order to facilitate the later introduction of non-syntonic temperaments. My recent re-writing of Lessons 4 and 5 is a manifestation of this design trade-off. I have a tendency to want to introduce non-12-TET and non-syntonic ideas sooner than necessary, because I think that JiMS' ability to support those concepts distinguishes it from traditional approaches, and because I think that these concepts are WAY COOL. However, it's simply not efficient to introduce those concepts too early. Secondly, the musical invariances on which JiMS is based -- transpositional invariance, tuning invariance, and cardinality invariance -- are all invariant only within a single given temperament. None of them are invariant across temperaments. I can't teach these invariances in the abstract; I can only teach them in the concrete context of a given temperament. Combined with the doctrine of progressive complexity, the temperament-dependency of musical invariances requires me to teach them within a given temperament (and tuning) first, and only then to generalize those teachings across (tunings and) temperaments. For example, consider the syntonic and schismatic temperaments. Both have the same generators -- P8 and P5 -- so they can share the same note-names, note-layouts, and staff notation. However, their comma sequence differs, and hence so does their mapping of partials to notes. For example, from a fundamental on Do, the syntonic temperament maps the fifth partial to Mi, while the schismatic maps it to Fe. As a result, the "shape" of the major triad (for example) is different, on JiMS keyboard and the JiMS staff, even across temperaments as closely-related as the syntonic and the schismatic. Combined with the doctrine of progressive complexity, this variance-across-temperaments requires JiMS to start by assuming the use of a single (tuning and) temperament, introducing other (tunings and) temperaments only when necessary (and it isn't necessary until after Dynamic Tonality has been introduced, using the syntonic temperament). Thirdly, the music of human cultures in the real world seems to be strongly biased towards temperaments generated by the P8 and P5, including the syntonic and schismatic. The syntonic temperament's tuning continuum includes nearly all of the tunings ever used by human cultures; the exceptions are arguably schismatic, such as Turkish and arguably some Indian music. This bias may arise from the human ear/brain/mind's apparent use, for the detection and tracking of tonal relationships, of a "map of the regions" generated by P8's and P5's (as exposed by Petr Janata's brain-scans). Such a map of the regions is the dual graph of the Wicki/JiMS note-layout, and hence is topologically identical to it. Combined with the doctrine of progressive complexity, this human cultural bias towards the syntonic temperament leads me to choose it as the basis of JiMS, with the added complexity of other temperaments being added to JiMS only much later in the lesson sequence. Please note that all of this is "just talk," however. I don't have any scientific evidence that proves that these design choices lead to the most efficient and accurate acquisition of musical knowledge by musical novices, or that graduates of such an educational program are able to advance the state of the art faster and/or more creatively. But then, no contrary evidence exists to refute these claims, either. The efficacy of alternative paradigms in achieving such objectives is an under-studied area. I hope that, as JiMS becomes available, such rigorous studies can be carried out. In the meantime, you're not wrong, and I'm not right. We just disagree as to when, in the sequence of ideas, non-12-TET tunings and non-syntonic temperaments should be introduced. On the other hand, we agree that these concepts SHOULD be introduced as soon as possible -- an agreement that differentiates us from the vast majority of music educators and music theorists, who neither understand nor care about these concepts. I hope that you will find this response to be in the cooperative, exploratory spirit of give-and-take in which it is intended.

Invitation to address UNT's Composition faculty

2010-04-05T21:05:00.000-05:00

On April 15th, I'll be addressing the faculty of the Division of Composition in the College of Music at the University of North Texas, just up the road in Denton (which is slightly north of Dallas; I'm in Mesquite, which is slightly south-east of Dallas).

UNT's Composition faculty is a computer-literate bunch; I look forward to meeting them.

This presentation is the result of my (a) emailing a number of the division's faculty members with information on how my work could facilitate theirs, and (b) a follow-up meeting with the division's Chair, Joseph Klein. That was a "flake filter" meeting; apparently, I passed was not filtered out.

My pitch to Dr. Klein was as follows:

If everyone were 100% certain that Idea X was about to emerge as their domain's new ruling paradigm, then every university in that domain would be racing to lead that emergence, in order to attract the best students, professors, research grants, etc., and to avoid obsolescence.
But by then, it would probably be too late to catch up with Idea X's early proponents.
The trick, then, is to identify ideas with high paradigm-shifting potential before it's too late to gain leadership.
There is a significant chance that the peer-reviewed findings of the Isomorphic Conspiracy could indeed lead to the emergence of a new paradigm in music, spanning the gamut of theory, composition, performance, education, technology, etc.
If UNT were to lead the way in establishing this new paradigm, then its leadership (all else being equal) would make it the top music school in the world. Just as Silicon Valley dominates computing tech, Detroit dominated auto manufacture (for a century, which ain't bad), and Vienna dominanted psychiatry, UNT could dominate the gamut of music under this new, far-reaching paradigm.
What if the Isomorphic Conspiracy's ideas don't pan out? To academics, it hardly matters. Consider the history of serialism. Within academia, serialism ruled for decades. One could hardly get one's foot in the door, let alone get tenure, without first establishing one's serialist bona fides. People based excellent, successful, and productive careers on serialism, despite its complete failure to make even the slightest impact on "real world" music-making. In short, even a failed attempt attempt to establish a new paradigm can be extremely successful for those who lead it.
But, again, if an attempted paradigm shift did prove to be successful, then the professors who led it would be like Watson and Crick in biology -- able to write their own tickets to any university in the world, now and forever.
...and here are our ideas; here's why they have the potential to establish a new paradigm in music; here's where UNT can add value; and here's how we can turn these ideas into a gold-mine of research funding.

UNT is very well-positioned to exploit this opportunity.

It has a high concentration of computer-savvy composers in its faculty.
It has two research programs that seem well-suited to explore and exploit it: CEMI and iARTA.
It is in a large metropolitan area (the Dallas-Fort Worth Metroplex, the fourth-largest economic region in the USA, with strong tech & finance industries), with a history of innovation and philanthropy.

My attempt to make a similar pitch to UT/Austin failed, in large part due to its music school's strangely backward-looking, technophobic culture.

So -- no pressure! -- if I am able to deliver a compelling presentation, this could be a milestone for both the Isomorphic Conspiracy and for UNT.

Lesson 005.1

2010-04-05T19:35:00.000-05:00

My latest lesson is Lesson 005.1 (source code here), which replaces Lesson 005:

Same crummy state-controlling button-bar at the bottom, for now. I really must fix that.

This lesson defines "scale" and "diatonic scale," and introduces JiMS keyboard -- i.e., the mapping of the Wicki/Thummer note-layout to the computer keyboard.

My lessons are starting to look a lot like PowerPoint presentations, except that their "graphics" are often interactive (e.g., JiMS keyboard). I've always liked PowerPoint, so the similarity is fine with me.

Although I was very strongly tempted to introduce other scales and even tunings at this point in the lessons. However, there is absolutely no advantage to the student in introducing those concepts now; they would just be a confusing distraction -- and the student's advantage must win all such design trade-offs. Hence, my decision to re-write Lessons 4 and 5, to provide a leaner, cleaner sequence of concepts.

I expect the next lesson (6) to introduce the term "mode," and discuss the modes of the diatonic scale. I think that I've got the components I need for that, but some of them haven't been used in a lesson before, and so will probably need to be tweaked...so don;t hold your breath for the next lesson. ;-)

Lesson 004.1

2010-04-01T19:39:00.000-05:00

My latest lesson is Lesson 004.1 (source code here):

Same crummy state-controlling button-bar at the bottom, for now. I really must fix that.

This lesson introduces a number of new terms. Each is clearly defined, and each definition is followed by a question to help cement understanding of the definition.

I now expect to revise Lesson 5 to focus on defining the term "scale." This may require introducing the notion of "tuning," but I don't want to go there yet, so I'll avoid it if I can.

Once the notion of "scale" is defined, then we can dive right into the Diatonic Scale, including its modes, intervals, chords, etc.

Rendering Notes into Text

2010-03-17T13:42:00.000-05:00

I'm a bit torn by a code architecture problem. A "Note" is a point in 2D tonal space, defined by the number of octaves and fifths that it is away from the origin note [0, 0] (which is also known as Re⁰). On the one hand, an abstract data type such as Note should not know anything about how it is rendered into graphic images or into text. The note [-2, 4] can be rendered in text as

the string "[-2, 4]"
the Text Layout Framework "TextFlow" data structure that represents "Fi⁰"
...and so on.

Likewise, the interval [-2, 4] can be rendered as

the string "[-2, 4]"
the string "major third"
the string "M3"
an arrow pointing from a given note to another note two notes rightward
...and so on.

The question is, what entity should render a given note (or interval) into a given text string?

The simple/stupid thing to do is have a method in the Note class, called something like toString(), that renders 'this' note into a canonical text form (e.g., "[-2, 4]"). Because strings are such a basic data type, having the Note class "know about" (and hence depend on) the String class is perhaps not a terrible violation of encapsulation.

But what about the rendering "Mi⁰"? To produce this, the Note class would have to know about (and hence depend on) the classes of Flex's Text Layout Framework, which is a pretty clear violation of Note's encapsulation.

Likewise, an Interval can be rendered into at least three different text strings ("[-2, 4]", "major third", "M3"). Should the Interval class implement methods for all of these different renderings? What if I want to add a new rendering? After doing so, I'd have to recompile all of the classes that used the Note class. Currently, my code base is small enough that this would be no big deal, but once I've got dozens of lessons, this kind of change could lead to a testing nightmare.

It seems to me that I've encountered a classic example of where the Factory design pattern is appropriate. I think I need a separate entity -- something that is not a Note, but which knows about Notes -- which renders Notes into...somethings.

There's be an abstract RenderNote class, with a method that would, given a Note, return an Object containing its rendering. RenderNote would have an abstract RenderNoteToString subclass, which would return a String for a given note. RenderNoteToString would have the subclass RenderNoteToStringVector, which would return the string "[-2, 4]" for the example note.

Then, RenderNote would also have a RenderNoteToTextFlow subclass, which would return a TextFlow (e.g., "Fi⁰") for a given note.

There would be a similar RenderInterval class hierarchy.

This all seems very complicated (as Factory deisgns usually are), but it would cleanly separate the text-rendering function out of Notes and Interval into separate classes. If, later on, I needed to render Note objects into a new kind of text representation, I could just add a new subclass to the RenderNote hierarchy; no recompilation or testing of the existing codebase would be necessary.

The result would be very much like Flex 4's use of "skins" for the graphical rendering of a UI control. The "RenderX" class could be seen as a "text skin," so to speak.

But...do I really want to mess with all of that complexity now? Or should I just slam a toTLFName() method into the Note class, and refactor it into a Factory if and only if it becomes a problem?

I think I'll do the latter.

Way behind

2010-03-17T13:03:00.000-05:00

I'm way behind on developing new lessons, for three reasons:
(1) I had a short consulting contract that occupied a couple of weeks of my time
(2) I need to correct some pedagogical deficiencies in earlier lessons (e.g., introducing the term "note" without defining it in Lesson 4) rather than focus on new lessons, and
(3) I invested a couple of days learning more about Adobe's Text Layout Framework (TLF).

I need to understand TLF better because JiMS makes heavy use of superscripts -- in note-names such as Re⁰ -- which, in Flex, require the use of TLF.

But, I'm back in the saddle now. A corrected version of Lesson 4 should be posted tomorrow (Lesson 004.01), with a modified version of Lesson 5 to follow a few days thereafter.

ExploreTuning1

2010-02-27T00:47:00.002-06:00

One of the cool things about JiMS iGetIt! note-layout (also used on the now-defunct Thummer) is that it has the same fingering in every tuning of the syntonic temperament.

This is kinda hard to explain, so I wrote a little Flash app to help. Here it is (source code here):

The slider on the left controls the frequency of Re⁰; all of the other notes' frequencies are determined by their geometric relationship to Re⁰, as a combination of octaves and fifths (as described here and here).

The slider on the right changes the width of the tempered major fifth (traditionally, "perfect fifth"), thereby changing the widths of all non-octave intervals -- that is, changing the tuning. A few notable tunings are labeled along the slider's track.

This chart shows what's happening:

(The colors in the chart do NOT correspond to the colors of the keyboard buttons in the applet above.)

On the keyboard app above,
1. Every note in a given note-class (such as all of the Re's) has the same color.
2. Two dfferent note-classes' notes have the same color if their frequencies, in the chart above, intersect in the current tuning.

For example, in 7-tet, a given diatonic note and all of its chromatic variations (a) control the same frequency, and hence (b) are drawn with the same color. Example: Ra, Re, and Ri are all red in 7-tet. Hence, there are only 7 "frequency classes" in 7-tet. That is, only 7 frequencies, and their octaves, occur in it.

BUT THERE ARE STLL 19 NOTES PER OCTAVE. Many of them just share the same frequency-classes. For example, Ra, Re, and Ri are still different NOTES; they just happen to control the same frequencies when tuned to 7-tet.

Likewise, if one moves the right-hand slider all the way down to 5-tet, then only the 5 notes of the pentatonic scale have unique frequency-classes, all of the diatonic, chromatic, and enharmonic notes (i.e., all of the notes of well-formed scales of cardinality higher than the pentatonic) share/duplicate these pentatonic notes' frequency-classes.

If one slides the slider up to 12-tet, only the chromatic notes have unique frequency-classes; the enharmonic notes (that is, the notes of those well-formed scale with cardinality higher than the chromatic) share/duplicate these chromatic frequency-classes.

In 19-tet, or 31-tet, or in most other tunings, each note-class of the enharmonic scale controls a different frequency-class.

(One of the strangest tunings is 17-tet, in which the pairs De-Li and Se-My are enharmonic. Set the slider to 17-tet, and play Se⁰ and My⁰, in the upper-left and lower-right corners of the keyboard, respectively. Different notes, same frequencies.)

This makes me wonder about the relationship between "scales" (that is, subsets of the enharmonic scale's note-classes) and "tunings" (is the pentatonic scale "really" the pentatonic scale all across the tuning range? Why or why not? How about the diatonic scale...in 5-tet?).

Now, the tunings that are far from 12-tet sound like crap when played using harmonic timbres (try it!), such as the timbre produced by the keyboard applet above. That's because the applet is only tempering the tuning, not the timbre, too. Tunings sound best when played using a "related" timbre -- that is, a timbre in which the partials align with the tuning's notes. Indonesian gamelan orchestras, playing in slendro's 5-tet scale, are playing instruments that emit timbres that (when crossed with a harmonic timbre) fit 5-tet. Tradtitional Thai and African music, played in 7-tet, is played on instruments that emit timbres that fit 7-tet...just as Western timbres fit the tunings near 12-tet.

With electronic sound synthesis, one can temper the timbres to match the tuning in real time -- by shoving a timbres' partials around -- so that voila! You get to have (or choose not to have) consonance in any tuning.

Which bring us to Dynamic Tonality.

Here's a simple example of dynamic tonality, using the above keyboard applet:
1. Slide the tuning to 19-tet (using the tuning slider at the right).
2. Play the ReFiLa triad. Very nice; very restful.
3. Slide the tuning to 5-tet (at the top of the slider).
3. Play the ReFiLa triad again. Too much tension! Must release!
4. Slider the tuning back to 19-tet, and play the ReFiLa triad again. Aha...sweet relief.

What you're experiencing is a novel means of creating tension and relief -- that is, of controlling emotional affect -- in tonal music.
A. In 19-tet, the ReFiLa triad is your basic major triad, which fits well with the harmonic series, and sounds restful.
B. Widening the fifth from 19-tet to 5-tet widens the triad's major third (Re-Fi) by so much that it begins to sound like a sus4 instead. That's one form of tension.
C. Also, widening the fifth from 19-tet to 5-tet pulls the tuning's notes out of alignment with the timbre's (harmonic) partials, creating another form of tension. The notes are "out of timbre."
D. Tuning back to 19-tet relieves the tension of the pseudo-sus4, and also brings the notes back "into timbre."

If one can temper one's timbres in addition to tempering one's tunings, then one can introduce "out of timbre" tension to any triad, including the tonic major triad.

The above experiment would be more compelling if the underlying synth could alter the frequency of a note being played after it started playing (i.e., pitch bend), but, alas, it cannot (so far as I can tell).

You can explore Dynamic Tonality more deeply with the Max/MPS-based TransFormSynth, described here.

P.S.: Why the ReFiLa triad, instead of the DoMiSo triad? Because Re⁰ -- being the center of symmetry (more or less) of all well-formed scales -- is the "origin note" from which the frequencies of all all other notes are determined. As such, Re's frequency doesn't change when the tuning changes, but the frequencies of all other notes do change. Clearly, the applet need to be extended to support the ability to specify a "tonic note-class," which would make the tonic note-class' members (e.g., Do) stable instead of Re. Always more work to do. ;-)

Guthman Musical Instrument Competition

2010-02-24T18:33:00.000-06:00

The Thummer is a contestant in Georgia Tech's Guthman Musical Instrument Competition, being held later this week.

Dr. Monty Cole, a high school friend of mine, happens to work just down the road at Mercer University, and has kindly offered to present the Thummer there on my behalf.

Unfortunately, we haven't been able to get one of the (rapidly aging) Thummer prototypes working, so the presentation will rely on videos of other people performing on it, rather than a live performance.

Here's the presentation, as a compressed PowerPoint file: Guthman.zip

It's super-short, relying primarily on three videos. I'm not sure that the PowerPoint file will be able to locate the videos properly after one downloads, unzips, and moves them to some other file folder. One may need to open the presentation in PowerPoint, go to the slides that contain the videos, double-click on the video graphics, and update the video-link to reference the video files' new location.

Why do the videos have a "Wondershare" logo across their upper-left corner? Because I used a trial version of Wondershare's Video Converter for Mac to convert the video files from WMV to MOV format.

Marek Zabka: Let's Talk

2010-02-23T16:11:00.000-06:00

Marek Zabka, a Lecturer at Slovakia's Comenius University, is hot on our heels.

His paper Generalized Tonnetz and Well-Formed GTS: A Scale Theory Inspired by the Neo-Riemannians shows that he's investigating the same generalized approach to music theory that Andy Milne, Bill Sethares, and myself are pursuing (our references here), on which JiMS iGetIt! Music System (JiMS) is based.

Interestingly, Dr. Zabka does not cite any of our papers, which I presume means that he's unuaware of them.

He has not yet connected his approach to isomorphic keyboards or -- more importantly -- to a generalization of timbre, so we're still ahead of the pack.

Clearly, the foundations of our mutual approach are "in the air," much as infinitesimal calculus was in the 1660's and natural selection was in the 1850's.

I don't have Dr. Zabka's contact information, and can't find it on the web. If you, kind Reader, know how to contact him, or can forward this to him, I would welcome the opportunity to welcome him to into our growing collaboration.

Lesson 005.0

2010-02-20T01:02:00.001-06:00

Here's my first draft of Lesson 5 in JiMS iGetIt! Music System (source code here):

Same crummy state-controlling button-bar at the bottom, for now. I really must fix that.

This lesson is 640x480, rather than the much smaller dimensions of the previous lessons. The larger size doesn't fit this blog very well, but it makes the lesson's text easier to read -- especially the note-button labels.

In this lesson, we build the "Fundamental Scales" -- that is, music's "well-formed scales." I'm not using the "well-formed scale" phrase yet, because to do so, I also need to introduce Myhill's property, and we're still a few lessons away from that.

In Lesson 6, I expect to introduce the notion of tuning, to show how the world's different musical cultures are related, and to establish the argument that to learn music using JiMS is to use a very general approach -- not limited to traditional Western music, for example. I had hoped to put that into Lesson 5, but it was just too much information. It needed its own lesson.

As of this lesson, my courseware has not just drifted, but positively galloped away from mainstream approaches to music education. Yet one can see that the concepts it introduces are quite simple, when shown using JiMS isomorphic keyboard and on-screen animations.

This lesson is late because I spent a week doing the final packing, cleaning, etc. to get our Austin house on the market. That's done; the coast is clear. More lessons! (More cowbell!)

Cardinality invariance

2010-02-07T01:23:00.000-06:00

All isomorphic note-layouts, by definition, have the property of transpositional invariance: the same fingering in every key.

Non-trivial isomorphic keyboards also have the property of tuning invariance: the same fingering in every tuning (of those temperaments with the same generators as the note-layout).

I've blogged before about the fact that the Wicki note-layout has another invariant property, not yet named: its fingering patterns are the same for well-formed scales of any cardinality (again, assuming that the layout and temperament use the same generators). However, that property has not yet been assigned a name.

I hereby define cardinality invariance as "the same fingering in every well-formed scale, regardless of cardinality" (for a given generator-pair).

JIMS' (Wicki) note-layout has this property. The Wesley note-layout has it, too. Most other isomorphic note-layouts don't have it. I don't yet know what mathematical characteristics confer it. But now, at least, it has a name: cardinality invariance.

Languages, Frameworks, and Idioms

2010-02-06T23:01:00.001-06:00

I've recently realized that I'm not "re-learning how to do computer programming," as I had intended. I'm just learning
- new languages (ActionScript, [M]XML),
- a new framework (Flex),
- new development tools (Eclipse), and
- their relevant idioms.

The data structures, algorithms, and fundamental abstractions are all pretty much the same as they were 17 years ago. There are a few new concepts, but basically, it's old wine in new bottles.

It could be, that a more recently-experienced programmer would look at my code and ask "why aren't you using glorpization here?" -- and the answer would be, I haven't a clue what glorpization is, because its use became widespread during my programming hiatus, so I never encountered it before.

One example of this, I suspect, is iterators. The use of iterators for traversing collections had only recently come into vogue went I left programming in the early 1990's. So when I returned to programming, using Adobe's Flex, I tended to traverse its ArrayCollections using indexing (that is, "for i=0 to foo.length") rather than iterators, whether implicit ("for each element in foo") or explicit ("cursor = foo.getCursor; result = cursor.findAny(key)...").

Another thing I missed was the introduction of associative arrays as an underlying mechanism of dynamic programming. "Static typing be damned -- just slam another property on that instance!" This was anathema, back then. Now, you can hardly turn around without bumping into associative arrays. XML appears to one big nested associative array. Flex depends on them.

It's quite odd to see a practice that was formerly considered to be a hanging offense -- "Egad! Self-modifying code! Run away, run away!" -- find its way into the core of modern programming.

I am still finding XML to be impenetrable. It's so simple that I can't understand it, because its (assumed) simplicity is shrouded in impenetrable layers of complexity.

My current confusion on the topic reminds me very much of how confused I was when I was first learning object-oriented programming (OOP). Everyone kept describing OOP in anthropomorphic/magical terms that clouded the issue. They'd say "message passing" when what they really meant was a function call. They'd say "overriding a method" when what they meant was a function call. Only after I discovered C++'s vtables did OOP become clear to me. At its heart, OOP was just a little bit of compiler (and runtime) code that maintained tables of pointers to functions. It was simple. Why did everyone insist on hiding this simplicity with layers of complexity?

I'm pretty sure that XML is the same -- a very simple idea layered in complex crap.

For example, my online music education lessons rely heavily on the use of the computer keyboard as a musical input device. I am quite aware that much of the world does not use the English language's "standard" QWERTY keyboard layout, using other layouts like QWERTZ, AZERTY, etc. instead. I would like to write my code such that I can add new layouts without recompiling all of my lessons.

I've got all of the relevant data structures working just fine in my code, as in-line ArrayCollections of Key objects (which define a name, code, and physical position for each key) -- but I can't figure out how to implement them in XML. It ought to be brain-dead simple. It probably IS brain-dead simple. But I just wasted an entire day on this issue without making any progress whatsoever, as far as I can tell. Too many layers of complexity around XML's simple core.

So, screw it. For now, I will continue to keep XML in the "too hard" file, along with fly fishing, calculus, preparing beef Wellington, and serialism.

Lesson 004.0

2010-02-04T23:48:00.000-06:00

Here's my first draft of Lesson 4 (source code here):

It uses the same crummy state-controlling button-bar at the bottom, for now. I'm going to write a new control for progressing-through-the-lesson-control soon, probably this weekend.

Moodle
I spent some time earlier this week looking at Moodle, the open-source learning management system. I'd like to package my lessons in a Moodle wrapper, because it has excellent support for all sorts of things (like gradebook databases) that I don't want to have to think about. Unfortunately, Moodle's Flash/Flex support is seriously deficient. Fortunately, an effort appears to be underway to address this deficiency. Therefore, I will proceed as if Moodle will have excellent support for Flash within the near-enough future.

I met the Moodle guys when I was living near their home base in Perth, Western Australia. I knew at the time that they had a good chance of beating their commerical competition. The signs were there, even then. Moodle hasn't been gaining market share as rapidly as I had expected, though. It needs some professional help with its evangelism, I suspect, to accelerate its rate of growth. If Moodle doesn't pick up the pace, it could be the next MySpace. It's "do or die" time.

Music & Pedagogy
Lesson 4 is the first lesson to introduce JIMS keyboard. The keyboard is introduced by deriving the pentatonic scale from an octave-reduced stack of (tempered) (major) fifths. Notice that the lesson never qualifies the term "fifth" -- that is, it doesn't call it a "perfect" fifth or a "major" fifth. I don't want to, or need to, open that can of worms quite yet. All in good time.

The next lesson, Lesson 5, will state that JIMS' unique approach gives its students the power and flexibility to understand and describe the music of many cultures. It will suppor this statement by extending the Stack of Fifths to produce the diatonic, chromatic, and enharmonic scales, and by showing that -- using a tuning slider -- the student can change the tuning to match that of many different non-Western cultures and Western eras, while retaining the simplicity and consistency of JIMS' keyboard's pattern.

I think that it's important to make this point early on, because immediately after making it, the lessons will shift their focus to the diatonic scale, and spend a LOT of time in the diatonic world thereafter. If the flexibility of the JIMS keyboard isn't demonstrated early on, a knowledgeable music teacher, reviewing JIMS' early lessons, might reasonably conclude that JIMS teaches concepts that are applicable only to the diatonic scale. I need to plant the seed of JIMS' power early on, even if I don't water it until much later.

Programming
I'm becoming more comfortable with the architecture that I'm using for these lessons, in which the lesson's content is implemented in the transitions between Flex's application states. If I choose the states wisely, then the architecture works well -- even if this architecture is, as I suspect, an unanticipated application of Flex's "state" feature.

Flex's 4 Beta 2's implementation of stateGroups seems to be a bit buggy (as one might expect from a new feature in a beta-version framework). It seems to clobber properties that aren't set by the relevant states. For example, you'll notice that in Lesson 4 above, the note's octave numbers disappear partway through the lesson. That appears to be a manifestation of the stateGroup bug. I spent a couple of hours trying to work around it, before deciding that the lack of octave numbers, in those states, was not a big enough bug to worry about. Also, don't use stateGroups to affect the setting of the properties of a slider, because the max/min/value will be set to NaN under conditions that I haven't spent the time to rigorously quantify.

Now, if I were a really serious beta-tester, I'd dig into Adobe's online bug reports and open-source nightly builds of Flex, to track down the bug and try to identify a fix. However, I'm confident that the bug is severe enough that others will have done this work, so it will be fixed in the release version. Although my use of the state feature in my application's architecture is, as I've suggested above, likely to be unusual, the use of stateGroups is not, so other people should be encountering this bug. If it persists in the next release, I will become more actively concerned.

Schedule
I've decided to try to post a new lesson each week. That would give me fifty lessons in a year. Assuming that I'll make the first dozen free, on a "try before you buy" basis, then those who subscribe to the paid lessons will get an additional 38 lessons (because 50 - 12 = 38). Thirty-eight is more than three dozen, and hence is three times the number of free lessons -- which ought to make the paying customer feel like they are getting enough to make the $29.95 purchase worthwhile. Of course, I'll be adding new lessons constantly thereafter, but with 50, I ought to have enough to "go live" and start selling subscriptions.

I'll have to pause my output while writing a "notation" component, but since JIMS' sequencer-like notation is so much simpler than traditional notation, it shouldn't slow me down by too much.

Lookin' good. ;-)

Lesson 003.0

2010-01-29T20:00:00.001-06:00

Here's my first draft of Lesson 3 (source code here):

Like the first two lessons, it uses a gross-looking and non-intuitive button-bar (along the bottom) to move from state to state. I need to replace that with a simpler/better "Next" button that only appears when one can proceed, and along with "Quit" and "Previous" buttons. The button bar is better for my development purposes, though, because it allows me to jump around non-sequentially.

Musically, this lesson shows JIMS starting to diverge from traditional representations of musical information. There is no international standard way of indicating the octave to which a note belongs. Some musicians indicate the octave of the piano keyboard; some musicians use MIDI numbers; some musicians use apostrophes -- it varies across the globe. So one more variation can't hurt, and might help.

In JIMS, octaves are numbered relative to the octave of the "origin note." In Lesson 3, Fred takes the note Bob sings as his origin, and numbers all octaves from it. Octaves are numbered along a number line, with higher octaves being positive and lower octaves being negative, as described in Lesson 3.

This system is entirely relative. The note Xx⁰ is in the same octave as the origin note, irrespective of the frequency associated with the origin note. As my father used to say, "Everything is relative (but relatives aren't everything)."

Develping Lesson 3 took much longer than it should have, in part because I spent a week (or more) rewriting my QWERTY/Wicki keyboard code to Flex 4...which I then decided not to use in this lesson after all. I'll use it soon enough, though.

StateGroup bug in Flex 4 Beta 2

2010-01-29T19:09:00.000-06:00

I hit an annoying bug in Flex 4 Beta 2 today: using stateGroups to set the min/value/max of a slider control SOMETIMES sets those values to NaN instead of the specified values. It can be worked around by not using state groups...which of course makes state groups considerably less useful.

Here's a Flex app that reproduces the bug 100% of the time (source code here):

I've edited the code and walked Flex's source code in the debugger, trying to track down the cause of the failure, and also tried to find a mention of the bug in Adobe's bug database...but it's already eaten up my whole day, and I've got lessons to finish, so I'll just stop using stateGroups. That's life in beta-land.

Bummer. :-(

wordWrapping a Flex Panel's title?

2010-01-10T19:12:00.000-06:00

How does one turn on word wrap in the title of a Flex Panel?

I'm using Panels to present questions, and the question statements can occupy more than one line. However, the Panel clips them to just one line. There must be a way to make the Panel use as many lines as necessary to display the title's text in full, preferably from MXML.

How?

I suspect that the answer involves using "Cascading Style Sheets," which is on my steadily-growing "must-learn" list. That list seems to be growing from the bottom faster than I can scratch newly-learned things off the top, which is a bit worrisome. The problem isn't that cascading style sheets are rocket science; I'm sure that they are not. It's just one more thing on the list.

Lesson 2

2010-01-09T16:50:00.000-06:00

Here's my first cut at Lesson 2 in JiMS iGetIt! Music System (source code here):

No radical departures from mainstream theory or pedagogy, so far. I'm not super-happy with the state-based architecture that I'm using, and there are some bugs (unimplemented events, actually) in the Flex 4 beta that I had to work around, but...so far, so good.

Interval width changes across the syntonic tuning continuum

2010-01-03T13:03:00.002-06:00

If we stack nine tempered major fifths (traditionally called "perfect fifths") above Re, and nine below it, we get the following generated collection:
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
De-Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li-My

Plotting these intervals' relationships across the syntonic temperament's tuning continuum produces this chart:

(You might want to open this chart into its own window, so that you can look at it, without scrolling, while reading the text below.)

This chart follows the following JIMS conventions:
- Interval names are traditional, except for
+ 4ths and 5ths: wider is "major," narrower is "minor"
+ (that way, 4ths and 5ths follow the same naming-pattern as all of the other non-octave intervals)

- All intervals follow the standard JIMS color-code:
+ major intervals in blue
+ augmented intervals in cyan (an "extreme blue")
+ minor intervals in red
+ diminished intervals in magenta (an "extreme red")

- All chromatic variations of a given diatonic interval share the same note-line symbol. For example,
+ All the unisons (Ra, Re, Ri) are marked with x's.
+ All of the seconds (Me, Mi, My) are marked with squares.
+ All of the thirds (Fa, Fi) are marked with vertical lines.
+ etc.

The legend, at the right of the chart, displays the generated collection of notes, in the same order (bottom to top) as they appear in the list at the top of this blog post. Each note's name is followed, after a colon (':'), by its interval-from-Re. Observe that the follow a pattern: augmented intervals at the top, then major intervals, then unison (Re), then minor intervals, then diminished intervals at the bottom of the list.

The vertical scale, on the left, indicates the width of a given note from Re.

The horizontal scale, on the bottom, indicates the width of the tempered major fifth (M5), that is, of the generator of the generated collection. The scale includes the valid tuning range of the syntonic temperament, which can be thought of an an extended meantone tuning system.

The widths of the intervals between Re and every other (non-octave) note is controlled by the width of the generator, M5. As the width of the M5 increases, from left to right across the chart, the widths of all of the non-octave intervals change:
- The intervals below Re in the legend, representing minor and diminished intervals, slope downwards as the M5 increases, indicating that they narrow.
- The intervals above Re in the legend, representing major and augmented intervals, slope upwards as M5 increases, indicating that they widen.
- The farther a note is from Re in the legend, the steeper its slope.

Consider, for example, the widths of the unisons. As the generator (M5) increases in width:
- Re (unison) is unchanged at 0, because it is the basis from which all other intervals are measured. Its note-line is shown at the very bottom of the chart area, as a series of black x's.
- Ra (diminished unison, d1), shown with magenta x's, decreases in width. It's note-line drops from 0 cents below Re (i.e., 1200 cents above Re), on the left edge of the chart, to 240 cents below Re (i.e., 960 cents above Re) at the right edge.
- Ri (augmented unison, A1), shown width cyan x's, increases in width, from 0 cents above Re on the left to 240 cents above Re on the right.

All of the unisons start, on the left, at 0, and separate as the width of the generator increases.

Likewise, consider the widths of the seconds-from-Re:
- Me (minor second, m2) drops rapidly from 171 cents to 0.
- Mi (major second, M2) rises slowly from 171 cents to 240.
- My (augmented second, A2) rises sharply from 171 cents to 480.

Just as with the unisons, all of the seconds start together (at 171 cents) and separate as the width of the generator increases. Generally, all of the chromatic variations of a given diatonic degree start at the same point on the left-hand edge of the chart, and diverge as the M5's width increases rightwards across the chart. (Note that 1200 and 0 are the same octave-reduced interval, so that Ra, which intersects the left edge at 1200, intersects it at the same interval as Re and Ri, which intersect it at 0.)

7-edo
The seven left-edge-intersection-points divide the octave into 7 equally-wide intervals, forming a 7-note "equal division of the octave," abbreviated "7-edo."

(The phrase "N-tone equal temperament" and its abbreviation "N-TET," used in Wikipedia and elsewhere, is avoided in JIMS, because it confuses the important distinction between tunings and temperaments...an explanation of which is beyond the scope of this blog post.)

5-edo
Likewise, the right-hand edge of the chart, at M5=720, shows that a completely different combination of notes intersect to divide the octave into five equally-wide intervals: 5-edo. (Again, note that 1200 and 0 are the same octave-reduced interval, so Di, intersecting the right edge at 1200, and Me, intersecting the right edge at 0, are intersecting it at the same interval.)

12-edo
Near the middle of the chart, at M5=700, you can see that seven pairs of note-lines cross. From top to bottom, the crossing pairs are:
1100 - Ra and Di (d1 and M7)
900 - De and Ti (d7 and M6)
800 - Te and Li (m6 and A5)
600 - Le and Si (m5 and M4, traditionally named d5 and A4)
400 - Se and Fi (d4 and M3)
300 - Fa and Mi (m3 and A2)
100 - Me and Ri (m2 and A1)

The notes in the crossing pair are always 12 notes apart in the 19-note stack of M5's (check for yourself, using the chart's legend).

The crossing note-pairs are said to be "enharmonic" (i.e., have the same pitch) in 12-edo. This is the "equal temperament" tuning familiar to most modern musicians -- so familiar, in fact, that many such musicians do not realize that other tunings exist, or that there is such a thing as a tuning (let alone a temperament).

17-edo
Slightly to the right of 12-edo, at M5-706 cents, two other note-lines cross:
352 - Se and Mi (d4 and A2)
847 - De and Li (d7 and A5)

All of the note-lines intersect the vertical line labeled "17-edo" at 17 equally-spaced intervals, so M5=706 is 17-edo tuning.

In 17-edo, the major second is subdivided into three equally-wide intervals by the augmented second and minor second. For example, see how the gap between Re (black x's, at the bottom) and Mi (blue squares, near the 200 cent horizontal line) is evenly divided by Ri (A1, cyan x's) an Me (m2, red squares). Note that at this point along the horizontal axis (M5=706), Me is closer to Re (i.e., lower in pitch) than Ri is.

In 17-edo -- and indeed everywhere rightward of 12-edo -- minor/diminished intervals are lower in pitch than the augmented/major intervals with which they are enharmonic in 12-edo.

19-edo
Likewise, the vertical line labeled "19-edo" marks the spot, at M5=695, where the note-lines subdivide the octave into 19 equally-wide intervals: 19-edo tuning. At this tuning, a major second (for example, Re-Mi) is divided into three equally-wide intervals by and augmented unison (Ri) and a minor second (Me).

In 19-edo -- and indeed everywhere leftward of 12-edo -- minor/diminished intervals are higher in pitch than the augmented/major intervals with which they are enharmonic in 12-edo.

Dynamic Tonality
Despite the changes among the relationships between intervals across the syntonic temperament's tuning continuum, the sound of tonal harmony's basic structure survives, as shown in this video (with over-the-top narration, for which I apologize):

This dynamic flexibility of tuning, combined with the consistent fingering of the Wicki/JIMS keyboard, can be used to create musical effects that are truly new, such as the tuning progression in this piece, C to Shining Sea, by William Sethares. We call the result Dynamic Tonality.

Well-formed scales beyond the chromatic

2010-01-02T18:21:00.044-06:00

First, let's review the construction of the chromatic scale.

Stacking 13 tempered perfect fifths (P5's) one atop the other, centered on Re, produces the following 13-note generated collection:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si

The Le-to-Di and Me-to-Si 12-note subsets of this generated collection are both just transposition of each other, so either can be used to represent a 12-note contiguous subset of the above 13-note generated collection. In the following discussion, the Me-to-Si subset will be used.

The Me-Si generated collection's notes can be adjusted so that they all fall within a single octave. We will arbitrarily define the octave to start on Do. The result is a "well-formed scale," in this case the chromatic scale.

The chromatic scale has the following note sequence and interval sequence:
note sequence: Do-Di-Re-Me-Mi-Fa-Fi-So-Si-La-Te-Ti-[Do2]
interval sequence: A1-m2-m2-A1-m2-A1-m2-A1-m2-m2-A1-m2

...where:
A1: augmented unison
m2: minor second

In the syntnonic temperament's valid tuning range -- that is, when the width of the P5 is anywhere between 686 and 720 cents wide -- the m2 is wider than the A1. Hence, in the syntonic temperament, the chromatic scale has the following width sequence:
width sequence: S L L S L S L S L L S L

That's 7 L's and 5 S's.

With that review, we can now go...

Beyond the Chromatic
In the syntonic temperament, then, the well-formed scale with the next-highest cardinality after the chromatic's 12 will have the cardinality:
Cardinality' = 2L + S = (2 * 7) + 5 = (14) + 5 = 19.

Stacking 19 tempered P5's one atop the other, centered on Re, produces the following generated set:
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
De-Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li-My

...with the extra notes (relative to the chromatic scale) appearing the ends and shown in boldface.

Octave-reducing this generated set, and arbitrarily defining the octave to being on Do, gives the following 19-note note sequence and interval sequence:
Do-Di-Ra-Re-Ri-Me-Mi-My-Fa-Fi-Se-So-Si-La-Li-Te-Ti-De-[Do2]
A1-d2-A1-A1-d2-A1-A1-d2-A1-d2-A1-A1-d2-A1-d2-A1-d2-A1

...where:
A1: augmented unison
d2: diminished second

Clearly, as we sub-divide the octave into more pieces (i.e., into higher-cardinality scales), those pieces must get smaller.
Scale Cardinality Large Small
Pentatonic 5 m3 M2
Diatonic 7 M2 m2
Chromatic 12 m2 A1
Enharmonic_19 19 A1 d2

At each successively-higher cardinality, the formerly-small interval width becomes the new large width, and a new small width is introduced.

On a 19-note-per-octave Wicki/JIMS note-layout, and played in 19-tone equal temperament (P5=695, at which the A1 and d2 are both 1200/19=63.16 cents wide), this scale looks/sounds like this (source code here):

Now, let's explore the alternative cardinality-successor to the chromatic scale.
Stacking 17 tempered P5's one atop the other, centered on Re, produces the following generated set:
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li

...with the extra notes, relative to the chromatic, added to either end, and shown in boldface.

Octave-reducing this generated set, and arbitrarily starting defining the octave to being on Do, gives the following 17-note note sequence and interval sequence:
Do-Ra-Di-Re-Me-Ri-Mi-Fa-Se-Fi-So-Le-Si-La-Te-Li-Ti-[Do2]
m2-d2-m2-m2-d2-m2-m2-m2-d2-m2-m2-d2-m2-m2-d2-m2-m2

On a 17-note-per-octave Wicki/JIMS note-layout, played in 17-tone equal temperament (P5=706), this scale looks/sounds like this (source code here):

And there you have it: the next-higher-cardinality scales after the chromatic are 17 and 19.

Syntonic and Mavila

2010-01-02T13:34:00.003-06:00

In an earlier post, I calculated the cardinalities of successive well-formed scales -- pentatonic (5), diatonic (7), and chromatic (12) -- and animated their interval-patterns on the Wicki/JIMS note-layout.

What we saw -- with some interpretative help from Andy Milne -- was that:

each successive well-formed scale came in two versions: one with X large intervals and Y small intervals, and one that was vice versa (Y large and X small); and that
the sequence of intervals that defined both versions was the same; the only difference between the two versions was the tuning (that is, the width of the tempered perfect fifth, since that is the generator of the generated set that defines a well-formed scale).

For example, the diatonic "generated set" is Fa-Do-So-Re-La-Mi-Ti, which produces the note-sequence (in Do-mode) Do-Re-Mi-Fa-So-La-Ti-[Do2], which has the inter-note interval sequence M2-M2-m2-M2-M2-M2-m2.

In the syntonic temperament's valid tuning range (P5=(686, 720)), the M2 is wider than the m2, so this sequence can be written as the width sequence L-L-S-L-L-L-S, which is 5 large (L) and 2 small (S) intervals.

However, as P5's width shrinks towards 686, the m2 widens and the M2 shrinks, such that they become equal at around P5=686 cents, producing 7-tone equal temperament tuning.

If one narrows the P5 even further, one leaves the syntonic temperament and enters what Erv Wilson called the Mavila temperament, in which the m2 is wider than the M2. There, this same pattern (note sequence: Do-Re-Mi-Fa-So-La-Ti-[Do2] == interval sequence: M2-M2-m2-M2-M2-M2-m2 ) has the width sequence S-S-L-S-S-S-L, because in the Mavila temperament's valid tuning range, m2 > M2.

Alternatively put, the diatonic note note sequence and (hence) interval sequence are unchanged from syntonic to Mavila; the only thing that's changed is the relationships among the interval-widths, in that syntonic's m2 < M2 becomes Mavila's m2 > M2.

The same meta-pattern applies to the chromatic scale (all from Do):
note sequence: Do-Di-Re-Me-Mi-Fa-Fi-So-Si-La-Te-Ti-[Do2].
interval sequence: A1-m2-m2-A1-m2-A1-m2-A1-m2-m2-A1-m2

Within the syntonic temperament's valid tuning range, the m2 is wider than the A1 (i.e., m2 > A1), so the above chromatic note/interval sequence produces the following width sequence:
width sequence: S L L S L S L S L L S L

However, if the P5's width is narrowed so that it crosses out of the syntonic temperament's valid range into the Mavila temeprament's valid tuning range, then the width-relationship of the m2 and A1 is reversed, such that m2 < A1 -- producing a chromatic width sequence in Mavila that's the opposite of that in the syntonic:

width sequence: L S S L S L S L S S L S

Apparently, Andy's algorithm for calculating the sequence of cardinalities for successive well-formed scales, and the count of large & small intervals in each, produces a single scale, of which there is a syntonic variant and a Mavila variant. Let's see, in my next post, if that pattern continues, as we explore well-formed scales beyond the chromatic.

Kodály, Wicki, and iSlate

2009-12-31T13:43:00.003-06:00

The Kodály music education Methodstarts young students with pentatonic songs, then slowly introduces them to the "extra" diatonic intervals, and then eventually to the "extra" chromatic intervals.

Now, imagine that the youngest students were presented with a "pentatonic keyboard" in the Wicki/JIMS note-layout, like the one shown in the (non-interactive) image at right.

For the pentatonic songs used initially by the Kodály Method, this pentatonic-only keyboard would be ideal. It would contain only the notes (intervals) that the students were currently learning. (Other keyboard controls, not shown, would be used to indicate the tonic and define its pitch.) Also, it would give students a visual, tangible metaphor for tonal space, hence (potentially) accelerating their development of audiation skills.

Then, when they were introduced to the "extra" intervals of the diatonic scale, they could get a new diatonic keyboard.

As you can see, it's the same as the pentatonic keyboard, with the addition of Fa and Ti along the left and right edges, respectively. In effect, the diatonic keyboard's extra notes expand the "tonal space" to which the student is exposed.

Again, by containing only the notes in the scale currently being studied, such a keyboard has the potential to sharpen student's focus.

Later, as the student progressed to learning about chords, they could be presented with a two-handed diatonic keyboard, suitable for self-accompaniment. (The note-layouts are mirrored for cognitive convenience, and angled for ergonomic convenience.)

...which would, in turn, be superseded by a two-handed chromatic keyboard:

...and eventually, a two-handed enharmonic keyboard, featuring all 19 intervals of the enharmonic scale:

The latter keyboard looks rather overwhelming, and it probably would be, if it were the first keyboard a student encountered. However, after starting with the simple pentatonic keyboard and working progressively up through the diatonic an chromatic keyboards, the enharmonic keyboard wouldn't seem like such a big deal. It just adds a few extra notes at the outer edges of the keyboard, leaving its pentatonic/diatonic/chromatic core unchanged.

The main advantage of this approach is that the student always uses a keyboard that has precisely enough note-controlling buttons to achieve the required pedagogical goals, thus encouraging proper focus and minimizing distraction/confusion. Of all of the isomorphic note-layouts, the Wicki note-layout is best for this purpose. Each successively-wider Wicki keyboard enables the student to see farther into tonal space, literally expanding their tonal horizons.

The main disadvantage is that the student must trade-up keyboards rather frequently.

Perhaps this disadvantage could be ameliorated by using a virtual multi-touch keyboard, such as the much-rumored Apple iSlate (see article here):

Such a multi-touch sensitive display would perhaps lack the tactile feedback needed in a true performance instrument...but that's not the point. The Kodály Method stresses the use of one's voice as one's performance instrument. Hence, in a Kodály context, the Wicki note-layout keyboard would be used not for performance (absent the Thummer [sigh]), but rather for pedagogy -- i.e., in helping students apply additional senses (sight, touch) to the development of proper audiation skills.

Using a virtual keyboard would enable new intervals to be introduced not just one scale at a time, but one note at a time -- first just So and Mi, then also Do, then Re, then La, etc. -- following the standard sequence of the Kodály Method.

Apple's iSlate is likely to be to expensive for K-12 music instruction. Give it 10 years, however -- maybe less -- and iSlate clones will be cheaper than traditional band instruments.

Cardinality of well-formed scales

2009-12-31T06:24:00.004-06:00

Let’s apply Andy's next-highest-cardinality-MOS-scale-calculating algorithm, starting with the pentatonic scale.

First, here's an animation of the pentatonic scale on a Wicki note-layout (source code here):

As you can see, the pentatonic scale has two large steps (m3’s) and three small steps (M2's).
Name         L   S   Cardinality
Pentatonic   2   3       5

So far, so good.

Remember, a well-formed scale is drawn from a stack of tempered perfect fifths; the stack has the same number of notes in it as the scale does: the cardinality of the scale. The pentatonic scale's stack of P5's (laid on its side) looks like this:
-2 -1 0 1 2
Do-So-Re-La-Mi

Now, let’s apply Andy's algorithm to these values of L and S to get the L and S of the next-higher-cardinality MOS scale, i.e., L’ and S’ respectively.

Cardinality’ = 2L + S = (2 * 2) + 3 = 4 + 3 = 7, which agrees with our expectations for the diatonic scale, which is a god sign. ;-)

X = L + S = 2 + 3 = 5
Y = L = 2

Therefore, the next-higher-cardinality-than-pentatonic MOS scale will have either 5 large steps and 2 small steps, or vice versa.

The diatonic scale has five large steps and two small steps, with cardinality 7, so that seems to be the “right choice.”

However, I am confused. Does this mean that there ALSO exists some “Bizarro-Diatonic” scale of cardinality 7 which has five small steps and two large ones? If so, what is that scale? If not, why not?

This process gives us the following result:

Name              L   S   Cardinality
Diatonic          5   2        7
Bizarro-Diatonic 2   5        7

Hmmmm, it’s a little weird to have the Bizarro-Diatonic scale as a result of this algorithm, but what the heck, let’s press on.

[Edit: Andy Milne was kind enough to point out that the Bizarro-Diatonic scale is more properly named the Mavila scale, following Erv Wilson. I haven't been able to find out much about it on the web. When I understand it better, I'll put up an appropriate animation of its interval pattern.]

The diatonic scale described above looks like this (source code here):

The diatonic scale's stack of tempered P5's is just like the pentatonic's, but it has one additional note at each end (Fa and Ti):
-3 -2 -1 0 1 2 3
Fa-Do-So-Re-La-Mi-Ti

Now, let’s apply the stated algorithm to the diatonic scale’s L and S values, to find the next-higher-cardinality MOS scale (which OUGHT to be the chromatic scale, if all goes well).

Taking the values L' = 2 and S' = 5 from the diatonic scale...

Cardinality’’ = 2L’ + S’ = (2 * 5) + 2 = 10 + 2 = 12, which is the cardinality of the chromatic scale, which is encouraging.

X’ = L’ + S’ = 5 + 2 = 7
Y’ = L = 5

Therefore, the next-higher-than-diatonic MOS scale will have either 7 large steps and 5 small steps, or vice versa.

Hmmmm, that's odd. Can there be two different versions of the chromatic scale?

Yes, it turns out that there can, and the existence of the two different versions answers a question that's been puzzling me for the last couple of weeks.

Here's an animation of the first version of the chromatic scale (source here):

There are 12 intervals in the chromatic scale, so any drawing of them is going to look complicated, and this animation's drawing is no exception. But if you look closely, you can see a lot of structure in its pattern of intervals.

Firstly, the scale has only two interval sizes, as predicted: minor seconds (m2's, in red) and augmented unisons (A1's, in pink).

Second, there are five A1's (smaller intervals) and seven m2's (larger intervals). In 12-tone equal temperament ("12-tet"), the A1 and m2 happen to be equally wide (at 100 cents), but they are still different intervals, so they have different shapes on the Wicki note-layout.

Third, all the interval lines are parallel to each other. Among them, they outline a chromatic staff (well, kinda sorta).

One of the note-buttons, Le, isn't used in the animation above, however -- none of the interval-arrows ever reach it. You might well ask, "why did Jim include Le in the animation, then?"

To answer this leading question, let's look at an animation of the other version of the chromatic scale (source here):

It looks very much the same, as you would expect. The only difference is that, after the scale goes up an m2 from Fi to So, it goes up another m2 from So to Le, rather than turning back towards Si with an A1 as the previous version of the chromatic scale did.

We can call the first version "Chromatic Si," and the second version "Chromatic Le." In 12-tone equal temperament, there is no difference between them, but there would be a difference in (say) 1/4-comma meantone tuning, which had a leading role in Western music for many centuries (and which appears to have been used in the tuning of ancient Chinese bells).

In the stack of tempered perfect fifths that forms the chromatic scale, Le and Si are on opposite ends:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si

...which is the same as the diatonic stack, but with extra notes added at the ends (in boldface).

This chromatic stack has 13 notes, but only 12 can be included in the chromatic scale. One must choose whether to include Le or Si; you can't have both, because then you'd have a scale of cardinality 13, not 12.

If you choose to include Le, you get the Chromatic_Le scale.
If you choose to include Si, you get the Chromatic_Si scale.

[Edit: I got all excited when I first saw this, because, after about 4am, my counting skills decline precipitously -- so I thought that Chromatic_Le had 5 small and 7 large intervals, and Chromatic_Si, vice versa. They are just transpositions of each other, and nothing to get excited about. One should never trust (or at least, never post) late-night epiphanies.

As further evidence of the decline in my late-night counting abilities, I also declared the down-and-right-pointing intervals (e.g., from Do to Di) to be "diminished seconds," when they were clearly augmented unisons (how "clearly"? In both chromatic animations, the A1 arrows ALWAYS connect notes that begin with the same initial consonant -- that is, chromatically-altered versions of the same note. Hence, the interval MUST be a variation on unison. What an idjit I am!). I have relabeled them accordingly, and updated this blog post's text accordingly. Thanks to Andy Milne for giving me the heads-up on these errors; see his comment below.

I realy should have changed the color of the A1 arrows to be cyan, while I was relabelling them, to follow my convention that all augmented intervals are colored cyan...but I forgot, and now I'm too tired again. Later, perhaps.]

I've been up all night working on this blog posting (and its animations), so now, at 6am, I'm off to bed. Soon, I'll put up another post that continues walking up the cardinality chain to 17-tone scales, 19-tone scales, and beyond.

Cardinality sequence of MOS scales

2009-12-30T19:59:00.001-06:00

It is well-known (among mathematically-inclined music theorists) that there are well-formed scales with cardinality 5 (pentatonic), 7 (diatonic), and 12 (chromatic). ("Cardinality" is the number of notes in the scale.)

But, why these cardinalities, and not others? Why are there not well-formed scales of cardinality 6, 8, 9, etc.? Also, what cardinalities come after 12, that are well-formed?

I asked this of Andrew Milne, who gave a great answer, which I have appended below (with hyperlinks and occasional [editorial comments] added for your convenience...).

The quick answer is that the next couple of well-formed scales fter the chromatic have cardinalities 17 and 19.
------------------------------------------------------------

From: Andrew Milne [mailto:andymilne@tonalcentre.org]
Sent: Wednesday, December 30, 2009 11:10 AM
To: 'Jim Plamondon'; 'Bill Sethares'
Subject: RE: well formed scales, cardinality: next after 12?

Hi Jim and Bill

A lot of these concepts – well-formedness, Myhill’s, etc. are mathematically related and their definitions are hard to separate. But this is the method I use to determine the cardinality and tuning range of MOS/well-formed scales.

First you need a Farey sequence of order N. Then any Farey triple (three consecutive terms of the sequence), whose middle member has a denominator of N (i.e., j/k, M/N, p/q) gives the tuning range for an MOS scale of N tones. For example, Farey(12) has the triple 4/7, 7/12, 3/5, which tells us that there is a 12-tone MOS (the chromatic scale) which occurs for tunings where 4/7 < beta/alpha < 3/5. We know this method works, but Bill and I still haven’t worked out quite why.

By generating successively higher-order Farey sequences, you can find triples with higher denominators, and hence the tuning ranges of higher cardinality MOS scales. Note that as the cardinality goes up, the tuning range gets smaller. (This is all illustrated in the MOS labyrinth picture [for which, see Figure 5 in this paper]).

For your example, the next well-formed scale after the 12-tone chromatic is 17 if 7/12 < beta/alpha < 3/5, and 19 if 4/7 < beta/alpha < 7/12. You can either calculate this yourself or, more easily, just read it directly off the MOS labyrinth diagram.

Actually, I’ve just remembered, there’s another easy way to do this. If your MOS has L large steps, and S small, the next higher MOS has cardinality 2L + S with L’ large steps and S’ small where EITHER

L’=L+S large steps and S’=L small OR
S’=L+S small steps and L’=L large; and so on.

For example, the 7-tone diatonic has L=5, S=2; the next higher MOS has 2L + S = 12 tones.

If this 12-tone MOS has L’ = L+S and S’ = L, then the next higher MOS is 2L’+S’ = 2(L+S)+L = 3L+2S = 19 tones.
If this 12-tone MOS has S’ = L+S and L’ = L, then the next higher MOS is 2L’+S’=2L+L+S = 3L+S = 17; and so on.

I presume that this rule is a direct result of well-known properties of the Farey sequence.

Every MOS/well-formed scale has a tuning range over which it is also proper (Rothenberg’s term, or “coherent” in Balzano’s terminology). There is a method to find this using the Farey sequence of a higher order (or Stern Brocot tree – which is the same thing but arranged into layers), which was explained by Thomas Noll, in one of the emails he sent when reviewing our JMM paper [see draft here]. I can’t quite remember what he said (I do have a copy, so I can check), but it amounted to something like stepping up to the next higher MOS scale and using that tuning range, or similar. For example, the diatonic scale has a tuning range of 4/7 to 3/5, but is proper only over 4/7 to 7/12.

Presumably there are also ways to generalise these things for 3-D tunings producing “pairwise well-formed” scales, and higher-D tunings. This would certainly form the basis of a groundbreaking paper...

Andy

------------------------------------
More to follow.

Lesson 001.0

2009-12-21T22:00:00.002-06:00

Here's the first lesson in JiMS iGetIt! Music System (source code here):

It's not terribly impressive, of course, but one must start somewhere, both as a student of music and as a student of coursware development.

Well, it doesn't display correctly in IE/Windows, just as my last couple of test projects didn't. Works fine on Safari/Mac. Clearly, I can't continue to ignore this IE/Windows problem.

States in Flex 4 (entry 3)

2009-12-16T09:33:00.000-06:00

Here's a minor iteration to my recent state-controlling button-bar (source code here):

The main difference is that, in this version, whenever the application's state changes, the StateButtonBar's currently-selected state is updated to reflect that change.

Say, for example, that the user presses the StateButtonBar's State3 button, thus setting the application's currentState to "State3". Then, the user presses the stand-alone foo button, beneath the StateButtonBar. This sets the application's currentState to "foo", and because the StateButtonBar now listens for application state changes, it is notified of the state change and can update its own state (by selecting the ButtonBarButton labeled "foo") accordingly.

US Patent up for review

2009-12-14T12:56:00.000-06:00

The US version of one of my WIPO patent applications, Musical Button-Field Layout for Alphanumeric Keyboards, has recently been assigned for examination.

I'm too broke to pay a patent attorney to "prosecute" (drive forward) this patent application -- heck, Thumtronics already owes patent attorneys more than it (or I) can repay. However, if I can prosecute it myself for a couple of hundred bucks and a couple of dozen hours, I can probably do that, on behalf of iGetIt! Music.

The twist claimed by this patent application is not just the Wicki-to-QWERTY mapping, but rather, mapping each note of tonic solfa (in the Wicki pattern) to the QWERTY keyboard, and relying on the QWERTY keyboard's underlying computational power to map these intervals to the correct pitches, given a common reference pitch. Hence, "La" always maps to the same button on the QWERTY keyboard, but that button does not always map to the same pitch. It's a "movable Do keyboard," right there on your computer.

If I'd realized, before starting the Thummer project, that I could map Wicki's note-layout to an alphanumeric keyboard as described in this patent application, then I might have focused on a software-only product from the outset, and saved myself a fortune in hardware-development costs.

Based on my experience with Thumtronics, here's my advice to budding inventors:

One language, one market. If you're a native speaker of English, file only in the USA (although perhaps via WIPO's international process, because it gives you more time). If Japanese, in Japan only, and if Chinese, in China only. If you're European, file in the USA only, until Europe adopts a single unified patent system that enables prosecuting (i.e., "driving forward") a single pan-Euopean patent in a single language.
Don't hire a patent attorney. Instead, (a) carefully read dozens of patents related to the one you expect to file, paying particular attention to the specialized language used, (b) carefully compose your own patent description, claims, and drawings, based on what you learned from that study, and finally (c) buy the latest version of Patent It Yourself, and follow its instructions to the letter.

Thumtronics broke both of these rules, and the resulting expenses were an unsustainable drain on its finances. According to LegalZoom, "thousands of inventions are patented each year but only a minuscule amount actually generate substantial, if any, profits." Too true.

With such low odds of returning any value, the cost of patenting must be equally low for patents to be worth filing at all.

All of the patent attorneys in the world will tell you the same lie: that they cannot possibly estimate the cost of prosecuting your patent; that every patent is different; and that the best thing to do is to go ahead and get the process started. This is complete crap. Given all of the patents that have been filed, it is surely possible to find the average cost (and standard deviation) of patent filing by number of claims, number of descriptive pages, number of drawings, number of countries, number of languages, etc. This is basic data-mining.

However, if starry-eyed inventors knew in advance what the process might cost, they would be less likely to invest in it. Hence, it is in the patent attorney's interest to keep the costs of the process as opaque as possible, while leaving its early stages inexpensive. It's a scam, made worse by the high barriers to entry erected to limit entry to patent attorney's profession, thereby limiting competition and increasing the margins that patent attorneys can charge.

As with so many other aspects of modern global capitalism, the patent system has been skewed away from "defending the rights of the little guy" towards being a tool for their oppression. The big guys can afford to work the legislative/industrial system, in which legislators sell -- for "campaign contributions" and other graft -- the right to expropriate money from the little guys. The little guys just get screwed.

So, why bother to invent at all? Well, because -- as Canada Bill Jones is quoted as saying -- "Of course the game is rigged! But it's the only game in town, and if you don't bet, you can't win."

Rather than spending a fortune on patent attorneys, I submit that one should file one's own patents -- keeping the cost as low as possible. Then, once you've produced a product based on the patent that is generating a profit stream, buy patent insurance to defend those profits. A million-dollar policy currently costs around US$25K/year.

The most carefully-crafted patent is worthless if you can't afford to defend it, whereas even a poorly-crafted patent, backed up by a million-dollar defense fund, will deter infringers.

As always, while justice whispers, money talks.

States in Flex 4 (entry 2)

2009-12-12T18:19:00.000-06:00

This app looks the same as the last one I posted, but it implements the transitions between states using Flex's 'transitions" mechanism, rather than just executing onEnterState event handling code (source code here):

Other differences include:
1) on entry to State1, the label for the current state is spun at the same time as the narration is played, and
2) on entry to State2, a second sound is played after "test narration two" completes.

It shows the same clipping problem in IE on Windows. I'm going to have to look into that.

States in Flex 4

2009-12-11T16:15:00.001-06:00

One way to architect the code for a "lesson" is as a series of states. Flex 4 (now in beta) has improved its support for states, so this seems like a reasonable approach to explore.

Here's a program that demonstrates a trivial use of states in Flex 4 Beta 2 (source code here):

A ButtonBar at the bottom is initialized with the names of each of three states, and a label reflects the name of the currently-selected state. When the ButtonBar is clicked, the state is changed. Whenever a new state is entered, a short narration is played.

To use this approach in my courseware, I'll need to break each lesson up into a series of states, and define the actions to be taken whenever a new state is entered. Flex 4's new-and-more-general animation model should make it easier to define these actions than was previously the case.

I have no idea whether this is the "best practice" way to architect interactive courseware in Flex. I'm kinda flying blind here. Suggestions welcome.

[Postscript: All three ButtonBar-buttons display properly in Safari on my Mac, but the rightmost part of the StateTest1 widget, including most of the third ButtonBar button, is clipped off in Internet Explorer on my Windows machine. I don't know why. Welcome to the joys of cross-browser incompatibility, I guess.]

Sound synthesis solution...for now

2009-12-10T17:55:00.001-06:00

Here's a new version of SoundTest that sounds much better (source code here):

The difference is that this new version calls sound-generating code from a beta version of the SonoFlash library. Assuming that this library will be offered at a reasonable price once it is released in final form, I expect to use it for most of my courseware's lessons.

Adobe has a list of requested features that one can vote for, and MIDI is on the list, but apparently it has never made the cut. Perhaps Adobe should acquire SonoFlash, and use its team (and cross-platform sound engine) to implement MIDI in Flash.

Sound synthesis problem

2009-12-08T14:24:00.000-06:00

I spent the last month packing and moving, but I'm now back online (at last!).

Why doesn't this code work?

Here's what works:

Click either "Play" button; a note will sound, and the button's name will change to "Stop".
Click the same button again to silence the note; the button's name will change back to "Play."
Click the other "Play" button; a different note will sound (an octave higher than the first note); click it again to silence it.

So far, so good. :-)

Here's what doesn't work:

Click on either "Play" button, sounding a first note.
Click on the other "Play" button, sounding a second note while the first button is still sounding.

Bad things now happen. The sounds do not blend (as one would expect octaves to do), but instead grate against each other in a nasty alternating, overlapping grind. Also, the application's performance slows to a crawl. Click on either button, to silence its note, and you'll see that it takes forever for the button to respond to the click.

I'm not sure how to fix this. The code seems to be working as intended. The flaw seems to be in how I'm using Flex/Flash's sound architecture -- that is, my tactics are correct, but my strategy's wrong.

Comments welcome. :-)

Superfreakonomics, global warming, and music

2009-11-13T11:33:00.000-06:00

A contrarian chapter on global warming in Superfreakonomics has sparked intense controvery. I am partcularly intriqued by the "shape" of the debate -- the kinds of arguments that are being used in both directions.

I found the following paragraph, from this Bloomberg blog post, to be particularly interesting:

Dubner wonders why everyone is so angry. In part, it’s because the book’s blithe remedies – “We could end this debate and be done with it, and move on to problems that are harder to solve,” Levitt told the U.K. Guardian newspaper – are an insult to the thousands of scientists who have devoted their careers to this crisis.

Hmmm. Let's paraphrase that paragraph as follows.

Mr. Inventor wonders why everyone is so angry. In part, it’s because his invention's blithe remedy is an insult to the thousands of people who have devoted their careers to this crisis.

People who have devoted their careers to Paradigm A can become very angry when you make a case for Paradigm B. Their anger spills over from attacks on Paradigm B (which are an intrinsic part of the scientific process) to personal attacks on those who dare to back it (which are an intrinsic part of human nature).

If JIMS Isomorphic Music System starts to gain momentum, I can reasonably expect to have such anger directed at me, for daring to challenge established music theoretical and music education orthodoxy. In a way, this would be a sign of progress. Such anger would prove that the Establishment was on the defensive, which is a big step forward for something that is currently beneath the Establishment's notice.

The Importance of a Good Notation

2009-11-09T15:35:00.001-06:00

In his 1911 book An introduction to mathematics, Alfred North Whitehead wrote (p. 59):

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the [human] race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties.

Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. The consequential extension of the notation to decimal fractions was not accomplished till the seventeenth century. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.

----------------

Great quote, isn't it?

It clarifies the two essential benefits of a good notation, to wit, that it:
1. Enables domain specialists to advance the state of the art; and it
2. Enables a higher percentage of non-specialists to master the domain's fundamentals.

That's a pretty powerful combination, which explains why notational improvements have been the key to so many of humanity's great leaps forward.

Likewise, JIMS Isomorphic Music System (JIMS)
1. Enables creative artists to advance the state of the art (through such novel effects as Dynamic Tonality), and
2. Enables a higher percentage of non-musicians to master the musical domain's fundamentals.

Or, at least, that's my claim. Time will tell. ;-)

In the meantime, today's music education establishment will continue to argue -- as Greek mathematicians did in their day -- that their domain's high failure rate is due to the inherent difficulty of their domain, not due to the imperfection of their notation (and instrumentation). Perhaps JIMS, too, will astonish domain experts by doing the impossible.

JMTP paper rejected

2009-10-24T14:17:00.003-05:00

Today I received notification that my recent submission to the Journal of Music Theory Pedagogy (JMTP) was rejected. The rejection letter is appended below.

The reasons for the rejection were many, but all boil down to this one, from Reviewer #2:

We have precious little time to teach students as it is now; if in their theory and/or aural skills classes they are dealing with a new notational system, I believe that will not support their progress in reading and performing traditional music.

One might paraphrase this as "Our patients are dying left and right, despite our bleeding them with the best leeches available. Your proposed 'germ theory' does nothing to improve the efficiency of our leeches, and hence has no place in the practice of modern medicine."

This submission/review/rejection process proved to me that academia, per se, will reject JIMS reflexively. This proof validates my decision to target my JIMS-based online courseware at "musically-inclined but not-formally-trained individual consumers" (using rock music rather than the Common Practice classics), thereby following a path very similar to that of Rosetta Stone's breakthrough language-learning courseware.

Once the first version of JIMS courseware is on a self-sustaining trajectory, I can produce separate version that is dumbed down (by using traditional note-names and staff notation), thereby meeting the needs of modern academia.

In the meantime, illigitimi non carborundum, and back to developing the courseware!

-----------------------------------------

From: Steve Laitz [mailto:***]
Sent: Saturday, October 24, 2009 9:00 AM
To: Jim Plamondon; Andrew Milne; William Sethares
Subject: Re: JMTP & Paths to Musicianship

Dear Professors Plamondon, Milne, and Sethares,

I write to inform you that your article “Sightreading Music Theory” will not be accepted for publication by the Journal of Music Theory Pedagogy. Unfortunately, the reviewers found the paper to have little to do with its title: "Sight-Reading Music Theory." They were somewhat confused in that the paper appears to be a rewrite of The ThumMusic System.

The reviewers felt that the article is not relevant to pedagogy in general and it does not demonstrate direct application to the teaching of theory. One issue each reviewer voiced was a frustration concerning constant references that were “beyond the scope of this paper” and that these references were often directed to your own website for explanation. They felt that anything that is not common knowledge in the theory community needs to be explained within the paper.

The reviewers felt that the non-theorist who is teaching theory at the collegiate level—and there are plenty of these folks who read JMTP--will not find the paper particularly helpful in pedagogical matters. Further, even the professional music theorist will reap little benefit from the paper given that issues are explored but not clearly explained. For example, the acronym JIMS is used throughout the paper, but there is never an explanation for what the letters represent, except in the abstract, and even there, the letter “J” is not defined. Other examples include the “pitch buttons” and references to “playing pitches,” but without clear definition at that point in the article. Regarding the Thummer instrument and the related ThumMusic system, both of which you mention late in the article, this all requires far more explanation and, more importantly, direct focus on the applicability of the JIMS System and Thummer instrument to theory teaching.

Below are detailed comments from each of the three readers that I hope will help you, should you decide to revise the paper and submit it to another journal.

-------------------------------

Reviewer #1:

Page 2, “scenario”: how can a HS band student have no background whatsoever and still play an instrument? How much does the author’s “etc.” include? Students would have to at least read music in a single clef in order to play, and presumably would know a piano keyboard. And that is one of my issues throughout this article—any pedagogy should help student musicians with the music they will encounter in any setting outside a theory classroom: in performance (printed scores); in analyzing scores in other classes, etc. Music is not going to be rewritten to accommodate a new technology.

Reviewer #2

1. Page 2, #3, Musical Isomorphisms, term “chromatic staff”: is he referring to a standard staff (capable of showing any pitch) or a specialized staff. I think this needs to be made clear.

2. Page 3: last sentence before section 3.2: re music-control interfaces: if he means piano layout, fretboard layout, etc, I think he should say that. I’m not sure most of us refer to the layout of notes on different instruments as music-control interfaces.

3. Page 4, the sentence that says the “extra” notes will become clear later – I’m not sure they do. Or at least, not without a lot of work on the reader’s part – see my note #21

4. Page 4, 2nd paragraph under Figure 3: “To play in C major . . . beyond the scope . . . ” – I think he needs to provide an explanation. If I understand correctly, this is somewhat equivalent to a transposing instrument – regardless of the key, the fingering always remains the same. Rather than changing horns, for example, one sets G as tonic, or A as tonic, etc.

5. Page 5, section 3.3: “Chromatic staff”: this 2nd reference suggests the chromatic staff is not our standard staff, yet I’m not sure how many people would know this reference. I am not familiar with it as far as I recall. So again, I think some explanation is in order.

6. Page 5, re Fig. 4: I think some examples would be in order to show what he is talking about.

7. Page 5, next paragraph, beginning “At the far left”: end of paragraph -- again, I think an example is in order.

8. Page 5, next paragraph, beginning: “All other symbols.” He cites his website for a detailed description of JIMS staff notation. Well, once again, if this article is supposed to explain “Sight-Reading Music Theory,” and the author is wishing to encourage support for this alternative system of notation, then I think fundamental information concerning these things is not at all beyond the scope of this paper, but belongs squarely in it.

9. Page 6: Are the terms “double harmonic major” and “double harmonic minor” common terms? They aren’t as far as I am aware, so again, should be explained, at least in terms of where these scales are found or employed.

10. Page 6: small point: Fig. 5: Make the outline of the white honeycomb cells darker – they are hard to see. But, if the scale dots should be white, how are they to be seen against the while honeycomb?

11. Page 6, Fig. 7 and 8 confuse me. I ultimately can see them the way the author intends, but it took some time and effort. Perhaps it would be better to use 2 diagrams to show first major, then minor triads. It took me some time to figure out what he was doing in this figure, which is duplicating each note (Do, do), to show how a single pitch fits into either a major or minor triad. If his system is supposed to make things simpler, this absolutely does not do that.

12. Page 7, Section 3.4.1: he says that we should minimize memorization load: Great! I’m all for using as few terms as possible, and not creating new ones, especially if they conflict with terms already in use for the same concept (one of my problems with Edwin Gordon’s writings). But he then introduces yet more terms to be memorized. Thought experiment or not, one cannot ignore the fact that students will be learning terms like Plagal and Authentic (and should!!) in music history classes, and or encountering them in various other classes and musical environments. In my view, music notation is not going to change. We have precious little time to teach students as it is now; if in their theory and/or aural skills classes they are dealing with a new notational system, I believe that will not support their progress in reading and performing traditional music. If a student has a gig, he or she needs to be completely conversant with standard music notation. Using the proposed set of terms would make it difficult for students trained that way to converse and play with other musicians, as I don’t think this will ever gain world-wide or even country-wide adoption.

13. Page 7: Similarly, bulleted items no. 3 and 4 strike me as confusing, and would add to what a student would have to memorize. (I admit that I am only vaguely aware of Nashville numbers, so I don’t know if this corresponds to them in some way, and I can see how this corresponds somewhat to how one would read a jazz chart. But it is still not really the same, so creates the necessity of learning yet another system on top of what students will need to learn in order to perform any standard notated music, or music from a lead sheet.)

14. Page 8, secondary dominants: maybe I’m slow, but I don’t get how his system makes this “entirely clear,” since in my experience, secondary dominants are never entirely clear to any but a very few students. So once again, I think he is asking too much of the reader to have to go to his website to understand how this would work (“beyond the scope of this paper”), since this explanation would seem to be at the core of what his title suggests. (Oh, wait – is he trying to drum up business on his website???)

15. Page 8, last paragraph, first sentence: “Likewise every occurrence . . .”— I’d like an example to see how exactly every Do-Fa-Sol-Do chord progression will look like every other.

16. Page 9, last bulleted item before section 4.2: I suppose this is correct, but how would it correspond to what students are seeing when they read actual music?

17. Page 9, section 4.2.1, Creative Power:
a. Again, one must go to his website to understand – very problematic

b. Syntonic temperament – again, one is forced to go to his website to try to learn what he is talking about. I used to know quite a bit about tuning and temperament, but I’m not sure from the article what he is talking about. Did he mean this was a Just system? That didn’t seem to be right. And how, btw, can a syntonic system be equal tempered? That is not my understanding of syntonic at all. Perhaps that is my own ignorance, but I’m guessing other readers would have a problem with this concept as well. So I did go to his cited publication to see what he meant. It cleared it up for me, but I think he needs to explain what he means in the article. I suspect that would not be clear to very many readers at all.

c. I think this could be pared down by saying something like “this system is capable of producing any number of tuning systems by simply setting the system via the controller.”

d. Next paragraph: very small point, but in the first line, the word “retaining” should be “retain.”

18. Page 10, fig. 10 – this seems to be getting beyond the scope of this paper, plus I’m not sure I completely understand his figure. I understand cents, I understand commas, I understand equal and non-equal tunings, but I don’t quite get what his figure is showing. And in fact, this whole section on tuning seems slightly out of place with what the first part of the article seems to be about. If the article is about pedagogical efficiency, and “sight-reading” theory (I don’t really think that is the appropriate title for this article, either), then this digression into tuning seems to me a bit out of place I believe this would require more explanation, but that would truly be beyond the scope of this paper.

19. Page 11, 2nd paragraph: “three full 8vas, of 19 buttons” – I think this needs more explanation: coming from a section on tuning, and tunings with many divisions of the 8va, this becomes confusing. I think he needs to specify that his 19 notes include enharmonic equivalents, including “De” and “My.” He says earlier on p. 4, below Table 1, that the need for these “extra” notes will become clear later in the paper, but he never again addresses that. This would appear to be the place to do so.

20. Page 11: Under Fig. 11, I have no idea what he means by 10 degrees of freedom.

21. Page 12, section 5, Metrics: umm, sorry, but I am once again confused. What does he mean that before Guido invented sight-singing his singers could sing but didn’t know any songs?? Of course they did. They learned them by rote and memorized them, just as any child learns the ABC song, Happy Birthday, etc, without ever learning to read music.

22. Page 12, section 5, second paragraph, 1st sentence: “ . . . quality of a music theorist” – I thought this was about teaching students, not music theorists?? Following sentence: “On the one hand”: this seems to be getting at a separate agenda.

23. Page 12, section 5: Small item towards end of 2nd paragraph: “To identify of key centers” – obviously, delete the work “of.”

24. Page 12, section 5, end of 2nd paragraph: “to recognize modulation . . . ” – he never demonstrated this earlier on, when he should have.

25. Page 12, Section 6, Previous Work, end of 1st paragraph, “most viewers absorbed the basics quickly.” As a reader, I would like to know what those basics included.

26. Page 13, top: Huh?? “this paper cannot and does not propose that JIMS be used today in music theory pedagogy.” I thought that this was what the article was supposed to be about?

Reviewer #3:

Page column line comment

1 2 15 On which page in Einstein’s article does this quote appear?

2 1 13 You should use a gender-neutral reference for the students. I would suggest you use “his/her” rather than “her.

4 Table 1 The fifth scale step in Tonic solfa is “Sol” and not “so.”

4 2 5 You write, “to play in C Major, one must indicate . . . that Do should sound the pitch C.” How is a student to determine what the tonic note is?

What are the criteria and where in this paper have you established this?

How does the student know if the piece is in major or minor?

5 2 1 You need a musical example to illustrate Ri as an “upward-pointing note-head” and Me as “a downward-pointing note-head.”

6 2 7 You need to have a page reference for Euler (is page 6 in Cohn’s article?).

7 2 5 I don’t agree with your assertion: “the ‘major scale’ and ‘minor scales’ are not scales at all.” A scale is a collection of notes that span the octave; the mode is the specific pattern of steps and half steps that encompass the octave. In his harmony book (Harmony, rev. ed. New York: W.W. Norton, 1948), Piston opines, “tonality is synonymous with key, modality with scale” (29).

7 2 20 Here I take exception to your premise that you are not “defining new terms for specialized uses.” Moreover, if a student is to be conversant and literate, s/he must know what “authentic” and “plagal” mean. Finally, for those of us who use do-based minor, reading la-re-fa-ti-mi-la does not allow me to audiate i-iv-VI-ii-V-I (aside from the infrequent progression of iv-VI!). Am I to assume mi means (in a minor) E-G#-B?

8 1 5 “3Do” reminds me of Percy Goetschius’s nomenclature in The Material Used in Musical Composition (New York: G. Schirmer, 1889). The difference resides in how it is written: in Goetschius IV2 = IV@. I doubt that the literate musician will know that “5So7” means V$.

8 2 14 How do I know that Re7 “is the dominant of the dominant” in a diatonic D-mode rather than the diatonic ii‡?

9 1 26 I do not believe that the ability “to transpose notation among clefs and keys; to identify key centers and key relationships; to recognize modulation to closely related keys; and so on” is irrelevant. A French horn player in band must know how to perform at sight an Eß part on his/her F horn.

13 1 1 This says it all: “Clearly, this paper cannot and does not propose that JIMS be used today in music theory pedagogy.” Moreover, nothing in this paper convinces me that JIMS will “improve the efficiency of theorist-training.”

References. By this I assume you mean a bibliography. However, some of the documentation is inaccurate or missing. For example, the ISBN for d’Arezzo is 1-896926-186 (not 978-1896926186). For Cohen, where was the book published?

------------------------------

Steve Laitz
Editor, Journal of Music Theory Pedagogy

Isomorphism & diatonic set theory

2009-10-20T18:11:00.003-05:00

There are lots of isomorphic note-layouts -- for example, the Bosanquet, Fokker, Janko, Wesley, Chromatic Button Accordion (B-system and C-system), and Wicki.

JIMS uses the Wicki note-layout for a variety of reasons that are beyond the scope of this post.

The Wicki note-layout is proving to have some interesting mathematical properties. For example, consider any well-formed scale constructed by stacking N tempered perfect fifths and subtracting octaves (an "alpha-reduced beta-chain," where alpha is the octave and beta is the tempered perfect fifth), and N is the "cardinality" of the scale (that is, the number of notes in the scale).

The Wicki note-layout appears to be unique in that such well-formed scales are always tightly packed together on the keyboard, with no "holes" between the notes of the scale.

For example, consider the well-formed scale of cardinality 5 (pentatonic). It's notes [Do Re Mi So La] form a single tight group that (a) has no "holes" in it, and (b) is symmetrical around Re.

The well-formed scale of cardinality 7 (diatonic) is likewise tightly grouped and centered.

So is the well-formed scale of cardinality 12 (chromatic). Notice that both Le and Si are included, which is redundant; they represent the same note in the 12-tone well-formed scale, whether in 12-tone equal temperament tuning or not. I've just included both in the drawing for symmetry. The chromatic scale is the only well-formed scale with even cardinality (well, among those scales with cardinality less than or equal to 19, anyway), which is kinda messing with my head a bit.

And so on, for the well-formed syntonic scale of cardinality 17:

...and 19:

...and 21:

...and so on, ad infinitum.

To put it another way, the Wicki note-layout appears to be unique in that, to increase the cardinality of the syntonic scales playable on a Wicki note-layout, all one needs to do is add more notes to the left & right edges of the note-layout.

The other isomorphic note-layouts do not share this property. Their design intermingles scale notes and non-scale notes. As a result, they do not present the same pattern of notes for well-formed scales of all cardinalities.

By way of comparison, consider the Chromatic Button Accordion's C system note-layout (CBA-C), shown at right.

The CBA-C layout works fine for the chromatic scale, but if you wanted to use it exclusively for the pentatonic or diatonic scales, the note-layout would be full of holes. Alternatively put, neither the pentatonic nor diatonic note-sets map to compact, contiguous button-sets in the CBA-C note-layout.

Likewise, look at the line of "semi-tones" running up-and-rightwardly from C on the CBA-C note-layout. If one wanted to put the Db and C# on separate buttons there's no room. There's only one button-space between C and D; if has to serve for both Db and C#. The CBA-C note-layout does not have a clean "edge" to which the Gb could be added, as the Wicki note-layout does. As a rule of thumb, any note-layout with a contiguous line of "semitone"-controlling buttons has the chromatic scale "baked in," because the "semitone" is only a meaningful concept in chromatic scale (i.e., in the well-formed scale of cardinality 12). In scales of cardinality higher than 12, there is no "semitone." There are augmented unisons and there are minor seconds, but there are no semitones.

Now, look back at the patterns that well-formed scales make on the Wicki note-layout. These patterns all share three characteristics:

(a) They have no "holes" between the notes of a scale of given cardinality.

(b) They are symmetrical around Re.

(c) All of their notes fall on adjacent rows, with one row being one button/note wider than the other (including the chromatic/12, because I included both Le and Si, which is cheating, just a little).

On the other hand, one can see (using the scale chooser on the interactive keyboard below) that non-well-formed scales, such as the Neapolitan, Melodic, Harmonic Major, Marmonic Minor, and Double Harmonic Minor, do not share all of these characteristics.

This suggests that there is some common element that is shared by (a) the definition of well-formedness and (b) the definition of the Wicki note-layout. I do not yet know what that common element is, but it's pretty obvious that it's in there somewhere. (I think that it has something to do with the fact that on the Wicki note-layout, the "beta-stack" corresponds directly to one hexagonal line of note-controlling buttons, and the "alpha-stack" corresponds directly to a second, semi-perpendicular line. But I'm not sure.)

If you can shed any light on this common element, please don't hesitate to let me know. :-)

Separation of concerns in music education

2009-10-17T16:23:00.000-05:00

Computer science has an important concept called separation of concerns, first described by Dijkstra in his 1974 paper On the role of scientific thought as follows:

Let me try to explain to you, what to my taste is characteristic for all intelligent thinking. It is, that one is willing to study in depth an aspect of one's subject matter in isolation for the sake of its own consistency, all the time knowing that one is occupying oneself only with one of the aspects. [...]

It is what I sometimes have called "the separation of concerns", which, even if not perfectly possible, is yet the only available technique for effective ordering of one's thoughts, that I know of. This is what I mean by "focusing one's attention upon some aspect": it does not mean ignoring the other aspects, it is just doing justice to the fact that from this aspect's point of view, the other is irrelevant.

JIMS is "concerned" with one thing, and one thing only: the efficiency with which the concepts of tonal music can be learned.

This does not mean that I consider other musical "concerns" (such as performance skills, music history, ehtnomusicology, etc.) to be unimportant. Quite the contrary! Rather, it is only by studying each concern in isolation that its unique characteristics can be understood with sufficient clarity to enable the concern to be re-integrated with other related concerns.

In this video, starting about 8:50m in, Dijkstra compares writing software to writing music, and discusses the impact of higher-level programming languages—that is, languages that express programming concepts at a higher level of abstraction. This discussion is apt because one could consider JIMS to be a "higher level language" for music, compared to traditional notational and gestural languages.

In the above video at 17:25m, there's a quote about elegance from EWD 1284 which reads,

In some ways programs are among the most complicated artefacts mankind ever trued to design, and personally I find it fascinating to see that reasoning about them is so much aided by simple, elegant devices such as predicate calculus and lattice theory. After more than 45 years in the field, I am still convinced that in computing, elegance is not a dispensable luxury but a quality that decides between success and failure; in this connection I gratefully quote from The Concise Oxford Dictionary a definition of "elegant", viz. "ingeniously simple and effective". Amen.

The use of "simple and effective devices" (such as isomorphic note-layouts, solfege, and the tonnetz) is precisely how JIMS intends to help students "reason about music."

Diatonic Set Theory

2009-10-14T12:15:00.000-05:00

I've been reading up on diatonic set theory, using Timothy Johnson's Foundations of Diatonic Set Theory and various scholarly papers (thank God for Google!). It all seems to be based firmly on the syntonic temperament (that is, on stacks of tempered perfect fifths, in which the syntonic comma is tempered to unison).

This is absolutely the right simplifying assumption to make initially. Now, however, it seems reasonable to explore the application of its findings to other temperaments (such as Magic). Presumably, it will be discovered that some the "global" rules apply across a well-defined subset of all possible temperaments, and that each temperament has its own "local" rules.

Knowing which rules are global, and which local rules exist in any given temperament (such as Magic), could go a long way towards defining the intrinsic music theories of these alternative temperaments -- temperaments that now have, for the first time ever, the possibility of local consonance and dynamic tonality.

Circle of Nths

2009-10-13T22:02:00.006-05:00

Here's my latest Flex control:

The component shows the "Circle of Nths" for a given N and a given scale.

Hopefully, the component's controls are fairly self-explanatory.

Musical Issues
The component uses the interval-naming scheme discussed here. In brief,
- There are two interval categories: perfect and imperfect.
- The only perfect interval is unison (and its octaves); all other intervals are imperfect.
- Each interval also has a quality: diminished, minor, perfect, major, or augmented.
- All imperfect intervals of diatonic Fa-mode (Lydian) have the quality "major" (including its fourth, the "major fourth").
- All imperfect intervals of diatonic Ti-mode (Locrian) have the quality "minor" (including its fifth, the "minor fifth").
- All perfect or major diatonic intervals, when chromatically widened, have the quality "augmented."
- All perfect of minor diatonic intervals, when chromatically narrowed, have the quality "diminished."
- The resulting interval-names are used to name the widths of the intervals of all scales, whether diatonic or not. (Specifically, one does NOT generate interval names for non-diatonic scales by applying to them the name-generation algorithm described above; instead, one just names a non-diatonic scale's intervals using the corresponding interval-names generated for the diatonic scale.) Actually, it may be that these names only make sense within the syntonic temperament; other temperaments, such as Magic, may require different interval-names. I haven't looked into these other temperaments enough yet to know for sure.

This use of color matches this animation of the relationships among the diatonic modes' intervals as one moves from mode to mode along the major-minor axis (which is also the Circle of Fifths).

Using this naming scheme makes it easy to see that, within the diatonic scale, all non-octave intervals occur in exactly two sizes (major and minor). This is Myhill's property, and it is the essential characteristic from which the other properties of the diatonic scale emerge (e.g., maximal evenness, cardinality equals variety, structure implies multiplicity, and being a well formed generated collection). It is also the property from which Dynamic Tonality arises. It is also easy to see that this property is not shared by any of the Prime Scales (i.e., those shown in the scale-selection combo box).
In his book Foundations of Diatonic Set Theory, Timothy Johnson uses a single-octave note-circle for all Circles of Nths. His Circle of Fifths, shown on Page 82, is one example. Using a single-octave circle shows the relationships among the notes clearly, whereas using (N-1)-octave circles shows the relationships among the intervals clearly.

Programming Issues
The sliders don't have tick marks or labels because I can't figure out how to make Spark sliders show these things. Halo sliders had a property, tickInterval, that I could set for this purpose, but Spark sliders don't. I spent a couple of hours searching the documentation and source code (always the best documentation), but couldn't find anything that looked right.

If you know how to decorate a Spark slider with tick marks and bounds labels, please let me know.

The component's interval-arrows are also drawn in a stupid manner -- by simply drawing a sequence of connected straight line segments. I'd rather use an elliptical Path, a la Degrafa/SVG, but Flex 4's FXG stuff -- despite being otherwise quite spiffy -- does not support elliptical paths (why not?).

This component is NOT a good example of how to use Flex 4's Spark architecture, because it doesn't. It is a very Halo-like component, making no use whatsoever of Spark's skinning or layout enhancements.

Now that I've got the basic control working to my satisfaction, I'll see if I can break it up according to the proper Spark-style architecture (components, skins, layouts).

The component is also a fairly egregious example of ravioli code (i.e., "encapsulated spagetti code"). The one component is doing way too much; its source file is nearly a thousand Lines Of Code long (1 K-LOC). That's nothing to compare to Flex 4's 12 K-LOC UIComponent, which is the Mother of All Ravilolis (of necessity) -- but it's still a signal that my component probably should be broken up.

I also need to learn how to bring MXML data into a library-based component. If anyone can tell me how to do that, I'd welcome the instruction.

Once I've re-architected the component to use Spark's new architecture, I should be able to change its superclass to Slider, and presto change-o, enable dragging a thumb around the "clock" to change its mode. Being able to interactively change the Circle on Nth's mode will make it easier for for a student to see the relationship between an interval's width and its degree in a given mode. (The width of every 4th in the diatonic Circle of 4ths, for example, corresponds to the width of the 4th in the diatonic mode of the interval's starting-note.)

I'd also like to explore smooth animation of the component's state-changes. That way, when the control is changed from (say) being a Circle of 2nds to a Circle of 3rds, the note-labels can move around and change size slowly enough for the eye to follow, hopefully making the transition itself easier to understand.

Hence, this control is providing me with lots of opportunity to explore Flex 4's new architecture and "learn by doing."

Myhill's Property and Interval Names

2009-10-09T00:38:00.000-05:00

The characteristic of having two versions of each simple interval is known in diatonic set theory as Myhill's property, and it is the source of many other musically-significant characteristics.

It seems to me that this property is more-easily exposed and explained if the two versions of each simple interval are named consistently, e.g., major and minor, rather than calling some perfect, some augmented, and some diminished.

The Major-Minor Axis

2009-10-03T15:27:00.001-05:00

I've written a little Flash application to show how the modes of the diatonic scale relate to one another, using JIMS' Keyboard. It's not intended to be courseware, but rather to answer some questions that have come up in a different forum (which is why I didn't clean up its bugs).

To run the app, click here.

There are two button on the lower-right corner of the screen:
- Minor-ward: Moves the mode one degree towards the minor end of the major-minor axis (Ti).
- Major-ward: Moves the mode one degree towards the major end of the major-minor axis (Fa).

Initially, the app just shows JIMS Keyboard.

Click once on the "Minor-ward" button. The simple diatonic intervals (henceforth, "intervals") of Fa-mode (Lydian) will be revealed. All of its intervals are major, because Fa-mode is the major endpoint of the major-minor axis. Note that this discussion uses the renaming of "perfect" intervals discussed here.
Click again on Minor-Ward button. Fa-mode's pattern of (major) intervals will be slid up-and-right to Do, the next note minor-ward along the diatonic major-minor axis. Fa-mode's pattern of intervals fits Do-mode just fine, except for one interval: the major fourth. In the previous mode, it ended on Ti, but in this mode, it ends off the diatonic scale, on Fi. Therefore, we must replace the previous mode's major 4th with a minor 4th. By shifting the interval's endpoint to Fa, we get Do-mode's minor 4th.
Click on Minor-ward again. Do-mode's pattern of intervals, including the m4, will be slid along the major-minor axis to So. Do-mode's pattern of intervals fits So-mode just fine, except for one interval: the major 7th. In the previous mode, it ended on Ti, but in this mode, it ends off the diatonic scale, on Fi. Therefore, we must replace the previous mode's major 7th with a minor 7th. By shifting the interval's endpoint to Fa, we get So-mode's minor 7th.
Click on Minor-ward again, shifting the previous mode's pattern of intervals to Re-mode. Again, the previous mode's intervals all fit Re-mode just fine, except for the major third, which ends off the diatonic scale, on Fi. Replacing the major 3rd (ending on Fi) with a major third (ending on Fa), we get the intervals of Re-mode (half major, half minor).

By now, the pattern should be clear: at each step minor-ward along the major-minor axis, the only interval changed in width is the (major) interval ending on Ti, which is replaced by a (minor) interval ending on Fa.

Click on Minor-ward again to see the Ti-ending major 6th change to a Fa-ending minor 6th.
Click on Minor-ward again to see the Ti-ending major 2nd change to a Fa-ending minor 2nd.
Click on Minor-ward again to see the Ti-ending major 5th change to a Fa-ending minor 5th.

Now, we've arrived at Ti-mode, at the minor end of the major-minor axis. All of its intervals are minor.

To go back down the axis in the other direction, click on Major-ward. All of Ti-mode's intervals will fit Mi-mode just fine, except for Ti-mode's minor 5th. In Ti-mode, this 5th ended on Fa, but now it falls off the diatonic scale onto Te -- so it must be switched to end on Ti, instead, giving Mi-mode its major 5th.
Before clicking on Major-ward again, identify the interval that ends on Fa. It's Mi-mode's minor 2nd. That's the interval that will be changed when moving down the major-minor axis to La-mode.
Click on Major-ward, and watch Mi-mode's Fa-ending minor 2nd be replaced by a Ti-ending major 2nd in La-mode. Which interval will be replaced next? The one that ends on Fa. Which one is that? La-mode's minor 6th.
Click on Major-ward again to see La-mode's Fa-ending minor 6th be replaced with Re-mode's Ti-ending major 6th.
Click on Major-ward again to se Re-mode's Fa-ending minor 3rd be replaced by So-mode's Ti-ending major 3rd.
Again, and So-mode's Fa-ending minor 7th is replaced by Do-mode's Ti-ending major 7th.
Again, and Do-mode's Fa-ending minor 4th is replaced by Fa-mode's Ti-ending major 4th.

Unfortunately, code bugs prevent you from going back up the axis, or from reversing course mid-way along the axis.

Nonetheless, this simple app usefully exposes some of music's patterns:

Fa-mode (Lydian) is "the most major" mode, and Ti-mode (Locrian) the "most minor," each being at extreme ends of the major-minor axis, which runs along an axis of major fifths.
Moving up the axis towards minor, the (major) interval ending on Ti will be swapped for the (minor) interval ending on Fa.
Moving down the axis towards major, the (minor) interval ending on Fa will be swapped for the (major) interval ending on Ti.
Re-mode (Dorian) is half-major and half-minor, giving it a uniquely-ambiguous position along the axis.
Stepping from Do-mode to So-mode changes an odd-numbered degree (7th), and so does the adjacent step from So-mode to Re-mode (3rd). These are the ONLY two adjacent steps along the major-minor axis which both change odd-numbered intervals. This is significant, because tonal harmony is based on stacking odd-numbered degrees (that is, 3rds) in the mode of a given chord's root. (Similarly, the steps Re-to-La-to-Mi change the 6th and 2nd degrees, which might matter more to stack-of-4ths [quartian] harmony, as found in some jazz, than to stack-of-thirds [tertian] harmony).
The traditional names for the 4ths and 5ths obscure the consistency of these patterns. These intervals' names should follow the same pattern as the other two-value diatonic intervals, that is, the larger size (traditionally "augmented 4th" and "perfect 5th") should both be called "major," and the smaller size (traditionally "perfect 4th" and "diminished 5th") should be called "minor." The only intervals that should be called "perfect" are unison and its octaves, because they alone are distinguished by having only one size in the diatonic scale.
The The traditional names for the 4ths and 5ths also obscure the potential consistency of the "diminished" and "augmented" names. Once the names of the 4ths and 5ths are regularized, then "augmented" and "diminished" intervals can be recognized as referring consistently to chromatic alterations of diatonic intervals.

Much more betterish. ;-)

Singing through the syntonic comma

2009-09-30T15:40:00.000-05:00

On a Thummer-like keyboard, from C, count up four perfect fifths (from F through C, G, and D, to A). In Just Intonation tuning—which perfectly aligns notes with the harmonic series’ partials—the perfect fifth is 701.955 cents wide, so that’s (4*701.955=) 2807.82 cents. Subtract a couple of octaves (2*1200=2400 cents) from that and you get a remainder of 407.82 cents.

Now, another way to get from F to A on a Thummer-like keyboard is to got rightward by a single major third. In Just intonation, a major third is 386.31 cents wide. Subtracting this major third from your octave-reduced stack of four perfect fifths, you get (407.82-386.31=) 21.51 cents, which is the syntonic comma, which is the difference between a 10/9 major second (Re) and a 9/8 major second (also Re). The ratio of 10/9 over 9/8 is (9*9)/(10*8) = 81/80, which works out to this exact same 21.51 cents. (I don't want to go into more of the math here.)

21.5 cents is a LOT – more than one-fifth of a semi-tone. It is very audible, even to untrained musicians. If two singers are out of tune by a syntonic comma, you and everyone else in the audience WILL hear the difference, as a strong beating between the two singer’s voices.

This is a significant problem for vocal groups (if they actually want to sound good), especially when singing in close harmonies a capella, as (for example) barbershop quartets do.

If you sing the I-vi-ii-V-I chord progression in Just Intonation, for example, the I chord that you end on will be a syntonic comma lower than the I chord that you started on. This is called “commatic drift.” To avoid this drift, singers must learn to distinguish between two different Re’s: the 10/9 Re at the root of the ii chord and the 9/8 Re in the 5th of the V chord. This requires singers to be able to sing the two different notes correctly, and–at least as importantly–know when to sing each one and not the other.

I don't see how the use of Do-based minor either helps or hinders a singers ability to correctly choose and sing the right Re. I would welcome having someone explain this to me. I can't find much discussion of this issue on the Web, which suggests that I my misunderstanding of the issues is so deep that I can't even choose the right search terms. Either that, or writing about singing is like dancing about architecture, so none of the relevant discussion is written down.

One of the great advantages of the syntonic temperament is that it tempers out the syntonic comma (hence its name), so chord progressions like the one above “work” without either two Re’s or commatic drift. The cost of this tempering is that the notes of such a temperament are not perfectly aligned with the partials of the Harmonic Series.

There are two ways to address this. One is to adapt the pitches of the notes as they are played to align them with the proper JI intervals. There has been a ton of work on this kind of adaptive tuning. It presumes that the only timbre that’s interesting is the harmonic series, and that all tuning should be adjusted to align with harmonic partials.

Dynamic Tonality supports this. You can use the Tonality Diamond in the TransFormSynth to choose a “major JI” tuning or a “minor JI” tuning, at either vertical end of the tonality diamond. This keeps the tuning at a 5-limit JI, and adjust the 5th partial to align with the 10/9ths Re (in major) or 9/8ths Re (in minor). Of course, if you do this, then moving the tuning slider has no effect, because your use of the tonality diamond has indicated that you want to use a JI tuning, not a tempered tuning.

On the other hand, Dynamic Tonality can also address the problem by tempering the timbre to match the current tuning. Just keep the tonality diamond’s dot a near the vertical center of the tonality diamond, along the axis from “fully harmonic” timbres to “fully tempered” timbres, and you can adjust the tuning slider to your heart’s content. This option was not available to previous generations of theorists, because they didn’t have the necessary computing power. But it’s available to us. Even singers, singing into microphones, can have their amplified timbres adjusted in real time to fit the current syntonic tuning (such as 12-tet).

(I have this recurring nightmare that the only popular use of Dynamic Tonality will be to make 12-tet more consonant, thereby locking it in as the de facto standard forever.)

Point being, that Dynamic Tonality makes the problem of the syntonic comma completely disappear by tempering it out of the harmonic series, thus eliminating the syntonic comma at its source.

La-based minor revisited

2009-09-30T14:13:00.000-05:00

I have gotten a lot of feedback on my earlier post, Why La-based Minor?

In brief,

my understanding of Do-based minor was wrong.
I am now considering a different scheme, which seems to me to combine the best of both La-based and Do-based minor.

My new thinking, which is still somewhat half-baked, goes like this.

Do-based minor emphasizes the parallel minor, while La-based minor emphasizes the relative minor. Why emphasize one over the other? Why not clearly distinguish between the two?

Case 1: Starting in Major
Let’s assume that you’re going to notate/play a piece starts in major (diatonic Do-mode), so you transpose the pitches under the keyboard to move the desired tonic pitch to Do. Let’s say that you’re playing in C Major, so Do is C, La is A, Mi is E, and Me is Eb.

When the piece begins, in Do-mode, all is well. The tonic is on Do (C), and that mode’s third degree is Mi (E), which is where we pegged the keyboard, so those notes have those pitches.

1a) If the piece wanders from Do-mode into its parallel minor (C minor), all is well. The tonic stays on Do (C), and Do-mode’s third degree is Me (Eb), but no new transposition of the keyboard/notation is necessary, because those notes already have those pitches.

1b) If the piece wanders from Do-mode into its relative minor (A minor), all is well. The tonic moves from Do (C) to La (A), and La-mode’s third degree is Do (C), but no new transposition of the keyboard/notation is necessary, because those notes already have those pitches.

The next question is, “where do the scale dots and tonic indicator go?”

We're starting in diatonic Do-major, so the scale dots are on the usual diatonic notes, Do, Re, Mi, Fa, So, La, and Ti, with the tonic indicator on Do.

1a) Moving from Do-major to Do-minor, the scale dots change to Do, Re, Me, Fa, So, Le, and Te, with the tonic indicator staying on Do.

1b) Moving from Do-major to La-minor, the scale dots stay the same (i.e., on Do, Re, Mi, Fa, So, La, and Ti), but the tonic indicator moves to La.

The interplay of the scale dots and the tonic indicator show, in JIMS staff notation, what is happening.

If the scale dots change, but the tonic indicator stays on the same note, then the music has moved to a parallel mode.
If the scale dots stay the same, but the tonic indicator moves to a different note, then the music has moved to a relative mode.

This is not a factoid to be memorized, per se, but rather something which can be observed from the note-patterns on JIMS keyboard as one plays.

This system usefully distinguishes Do-major's relative minor (La-minor) from its parallel minor (Do-minor). The two would be notated using different notes on JIMS staff, and played using different buttons on JIMS keyboard.

When used strictly by “La-based minor” singers, the notation and keyboard could be transposed such that both the parallel and relative minors always used La as their tonic. However, that would NOT be the general case. the main difference between my previous proposal and this one.

Case 2: Starting in Minor
Now, let’s assume that you’re going to notate/play a piece which starts, and remains primarily, in A minor, so we map A to La.

2a) If the piece wanders from minor into its relative major (C Major), all is well. The tonic moves from La (A) to Do (C), and Do-mode’s third degree is Mi (E), but no new transposition of the keyboard is necessary, because those notes already have those pitches.

2b) If the piece wanders from minor into its parallel major (A Major), all is well. The tonic stays on La (A), and La-mode’s third degree is Do (C)…wait a minute. That’s not right. We’re talking La-MAJOR now, not La-MINOR.

Case 2b above is (I suspect) at the heart of the conflict between the La-based minorists and the Do-based minorists. Who would ever expect to find a major scale with La as its tonic?

If Case 2b's minor-to-parallel-major-and-back movement dominated a given piece, then it would make sense to start the piece in Do-minor. Then, when the mode changed from the primary minor key to its parallel major, the major mode’s tonic would be Do, as one would normally expect. Using D-minor in this way would emphasize that the core relationship in this piece was parallel, not relative.

Let's look at the scale dots and tonic indicator in Case 2.

2a) Starting in La-minor, the scale dots are on the usual diatonic Do, Re, Mi, Fa, So, La, and Ti, with the tonic indicator on La. Moving to the relative major, the scale dots remain the same, but the tonic indicator moves to Do.

2b) Starting in Do-minor, the scale dots are on Do, Re, Me, Fa, So, Le, and Te, with the tonic indicator on Do. Moving to the parallel minor, the scale dots change to Do, Re, Me, Fa, So, La, and Ti, with the tonic remaining on Do.

If the music moves farther than one relative or parallel step away from the tonic, in either direction, then it has modulated (hasn't it?). One could either keep shifting the stack of scale dots to reflect the changing set notes in the current diatonic scale (much like introducing more sharps and flats into a key signature, with all of the disadvantages thereof), or simply transpose JIMS staff and keyboard (using a transposition indicator and user-interface gesture, respectively).

Limitations
If a piece's tonal center is ambiguous, then no tonally-focused notation (like JIMS) is going to offer significant advantages over less-tonally-focused notations (like traditional notation).

However, most music played and listened to by the majority of the people in the First World is strongly tonal (and even strongly modal, if one considers the major scale to be the Ionian mode), which plays to JIMS' strengths, so I don't think that this limitation counts for much.

Advantages
The above-described approach seems to me to combine the best of both Do-based minor and La-based minor, by distinguishing unambiguously between the relative and parallel relationships. The person notating a song would need to do a significant amount of work to analyze what’s happening in a given piece, in order to notate it correctly -- but that's a GOOD thing, because once this analysis is done by the notator, it is very easily accessible by the student and/or performer.

Having a clear distinction, in JIMS notation, between parallel and relative intervals helps distinguish between notes that have the same name but are a comma or two apart (in Just Intonation), thereby helping singers it the right notes. Nonetheless, for purely vocal music in the La-based minor tradition, one could transpose JIMS notation (and perhaps an accompanying JIMS keyboard) to keep the scale dots constant, i.e., to use a La-based minor whether that minor was relative or parallel. But this would not be *required* under the above-proposed revisions to the JIMS system.

I think that this refinement is a considerable improvement to JIMS. It exposes a meaningful difference—the difference between relative and parallel keys—in a clear and unambiguous manner. This is in line with JIMS' neo-Riemannian roots.

Perhaps this refinement is sufficient to make JIMS useful to those who teach music using a Do-based minor system. I hope so. In my wildest dream, I imagine that JIMS, with this refinement, might be sufficient to heal the centuries-long rift between the La-minorists and the Do-minorists.

Comments and corrections welcome! :-)

Meantone temperament

2009-09-29T08:42:00.000-05:00

I spent most of yesterday editing Wikipedia's article on the meantone temperament.

By mid-2006, it had become clear to Bill Sethares, Andy Milne and I that our music theoretical work was focused on what have historically been called "extended meantone temperaments." We considered using that name in our own work, but we rapidly learned that the tuning community would have none of it. The term "meantone" was loaded down with an oppressive weight of historical baggage. For us to redefine the term, even slightly, or to broaden the scope of its usage, was anathema. Hence, we called our thingy the "syntonic temperament," and its valid tuning range the "syntonic tuning continuum," and so on, to avoid violating sacred historical precedent.

But it's really just a continuum of extended meantone tuning, in which we're tempering timbres in addition to notes, and thereby retaining the option of consonance across the entire tuning continuum.

The jargon of traditional tuning theory seems to me to be quite seriously muddled, especially in its failure to distinguish cleanly between a "temperament" (a set of rules, defined by a comma sequence, for mapping partials to notes) and a "tuning" (a combination of generator widths). This lack of distinction probably arises from tuning theory's obsession with the Harmonic Series (which is perfectly understandable, given the dominance of harmonic timbres in the history of Western music). If one assumes that the only timbres that matter to music are harmonic timbres, then the "mapping of partials to notes" is an irrelevant step, so tunings and temperaments become essentially the same thing...as the historical jargon-muddle reflects.

All of which made re-writing Wikipedia's Meantone temperament article harder.

NOw, the fun part will be seeing how long my edits last. Will the tuning community's Old Guard simply revert them away, hence defending tuning theory's status quo? Our will the edits be accepted, albeit perhaps with slight revisions?

Now that our theory is backed up by a slew of peer-reviewed scientific papers, it would be rather difficult to justify simply reverting the edits out of existence. Passions can run quite high in such tiny communities, however, so I am loathe to predict the outcome.

CircularSlider

2009-09-27T15:34:00.005-05:00

Here's my first attempt at a Spark-based Flex control, using Flex's new skinning architecture. It's a circular slider. This (uncopyrighted) file, posted by Jacob Goldstein, was a big help. Thanks, Jacob! :-)

There's some seriously-weird stuff happening in this control.
- Firstly, it attempts to draw a horizontal slider's track, for reasons that I haven't found yet. This "ghost track" appears to be intercepting some of the slider's events, which causes the thumb to jump around erratically during a thumb-drag.
- Secondly, it resists resizing. Apparently, I need to locate methods that faciliate resizing Spark controls and override them.

It's a start. ;-)

Custom Keyboard skins?

2009-09-25T21:50:00.000-05:00

These days, books can be printed "on demand" ...can keyboard skins?

Keyboard skins are the ultra-thin latex sheets that fit over the keys on a computer keyboard, like the one at right.

I'd like to be able to offer a keyboard skin pre-printed with the note-pattern for JIMS Keyboard, but I can't afford to order them up front...and who knows how well they'd sell? They'd have to be printed "on demand."

If you know of any firm that offers "on demand" keyboard skins, please let me know.

Back to coding!

2009-09-17T19:41:00.002-05:00

I've spent the last couple of months storyboarding my first batch of lessons, on Musical Sounds, the Harmonic Series, the Diatonic Scale, Modes of the Diatonic Scale, Diatonic Intervals, and the Major-Minor Axis.

Now, I'm going to start coding them up as interactive lessons, using Flex/Flash and some video. This is going to take me a while, as my coding skills are still pretty rusty. Adobe's about to release Flex 4, which I should probably use instead of Flex 3, so that my de-rustified coding skills can be as up-to-date as possible.

Once my first batch of lessons is online and gathering feedback, I expect to start storyboarding the next batch, covering Diatonic Triads and Modal Harmony, making extensive use of JIMS™ Tonnetz, which is of course aligned with JIMS™ Keyboard (see www.igetitmusic.com/papers/Perception.pdf).

The tonnetz is a great tool or exposing the relationships among triads. Consider this depiction of the neo-Riemannian PLR relationships between the C minor triad, labeled Q, and its three neighbors on a tonnetz:

This graphic shows that performing, on Q, the

Relative operation produces Q's R-major triad (Eb-G-Bb);
Parallel operation produces Q's P-major triad (C-E-G);
Leading-tone exchange operation prodices Q's L-major triad (Ab-C-Eb).

Obviously, just because I'm using a tonnetz doesn't mean that I have to emphasize a neo-Riemannian approach to harmony. I can use a more neo-Rameau-ian(?), root-movement-oriented approach instead. The point is that using a tonnetz enables me to go either way, or to mix and match as appropriate.

The best thing about this is that JIMS Tonnetz is not some abstract representation of tonal space, but is, instead, a concrete aspect of JIMS Keyboard, as implemented on a computer's standard QWERTY keyboard:

Being able to relate JIMS Tonnetz directly to the sound-controlling JIMS Keyboard should make it possible for me to SHOW people how chords relate to each other, rather than trying to EXPLAIN it.

Also, the tonnetz is the dual graph of Schoenberg's chart of the regions, which is rather handy.

However, for now, I must stop thinking about harmony and start thinking about coding up the first batch of lessons.

Once upon a time...

2009-09-11T15:32:00.003-05:00

Below is a copy of another post recently made to a discussion on Daniel Levitin's Facebook page. It's a re-statement of the content in this older post, but I like the clarity of this new restatement.

In brief, my argument is that (a) there's a difference between "musical talent" and "the ability to handle arbitrarily high UI loads," and that (b) reducing the UI load in a given domain in can increase the success-rates that novices enjoy in that domain.

----------------------------------

Darin wrote:

> If someone has innate talent, then as that person
> practices and progresses, he or she will recognize
> the progress, will be recognized by others for the
> progress, and as a result, will develop real passion
> for the pursuit. If somone does not have innate
> talent, such person will practice, not make much
> progress, will see the lack of progress, and be told
> about it by others, and the passion will not take
> hold, but will wither, and the person will move on
> to some other endeavor.

>
> People with real talent are few.

I respectfully disagree. "Talent" has absolutely nothing to do with it. Please let me explain by example.

1. Once upon a time, the Cherokee were illiterate. The English alphabet was a poor fit with the Cherokee language, so efforts to spread literacy among the Cherokee failed. Then one Cherokee invented a writing system that fit Cherokee perfectly, enabling literacy to sweep the Cherokee almost overnight. Did the Cherokee suddenly gain a "talent" for literacy?

(http://en.wikipedia.org/wiki/Cherokee_syllabary)

2. Once upon a time, the Koreans were illiterate. The Chinese ideographic script was a poor fit with the Korean language, so the efforts to spread literacy among the Koreans failed. Then the Koreans invented Hangul, which fit Korean perfectly; now, "a bright child can become literate in a day, and a dull child in ten." Did the Koreans suddenly gain a "talent" for literacy?

(http://en.wikipedia.org/wiki/Hangul)

3. Once upon a time, physicists couldn't puzzle out the interactions of quantum mechanics, nor could students learn about them efficiently. Then, Feynman invented "Feynman diagrams," and students could understand such interactions in less than a semester. Did physics students suddenly develop a "talent" for quantum mechanics?

(http://en.wikipedia.org/wiki/Feynman_diagram)

4. Once upon a time, European mathematicians could not conceive of "x to the power of y," because Roman numerals could not notate the concept, and the Roman abacus could not calculate it. Then Fibonacci explained how to use Arabic (actually Hindu) numerals and algorithms, and the scope of European mathematical thought widened dramatically. Did Europeans suddenly develop a "talent" for mathematics?

(http://en.wikipedia.org/wiki/Liber_Abaci)

5. Once upon a time, the "value" of a church singer dependend as much on "how many songs he had memorized" as on how well he could sing them, because all songs had to be memorized by rote. Then Guido d'Arezzo invented staff notation and solfeggio, enabling novices to become valuable church singers much more rapidly. Did such novices suddenly gain a "talent" for singing?

(http://en.wikipedia.org/wiki/Guido_of_Arezzo)

6. Once upon a time, all mathematical calculations had to be executed longhand, making them expensive and error-prone. Then logarithms were invented, and many calculations could be accelerated by looking them up in tables of pre-calculated logarithms. Did this suddenly increase people's "talent" for calculation? Did the invention of the slide rule? Of the pocket calculator?
(http://en.wikipedia.org/wiki/Logarithms#History) (http://en.wikipedia.org/wiki/Calculator#Pocket_calculators)

7. Once upon a time, learning and practicing chemistry was extraordinarily difficult, with the properties of each element having to be learning individually, and its guiding principles (e.g., phlogiston) being fundamentally incorrect. Hence, few gained mastery over chemistry. Then Lavoisier discovered the combustion principle, Mendeleev invented the Periodic Table of the Elements, and Bohr deduced the planetary model of the atom, all of which reduced the investment of time necessary to master chemistry, thereby dramatically increasing the percentage of the human population that could afford to master chemistry. Did students suddenly gain a "talent" for chemistry?
(http://en.wikipedia.org/wiki/History_of_the_periodic_table)
(http://en.wikipedia.org/wiki/Phlogiston#History)
(http://en.wikipedia.org/wiki/Bohr_model)

YESTERDAY, learning and practicing music-making was extraordinarily difficult, with the patterns of each key, clef, scale, mode, tuning, instrument, timbre, etc., having to be learning individually, and its guiding principles (e.g., 12-tone equal temperament) being fundamentally incorrect. Hence, few gained mastery over music-making. Then [insert here a list of scientific discoveries and technological inventions that, arguably, have not yet been made], all of which reduced the investment of time necessary to master music-making, thereby dramatically increasing the percentage of the human population that could afford to master music-making. Did students suddenly gain a "talent" for music-making?

Of course not.

In all of the above examples, the problem was a lack of technology, not of "talent." The traditional technology of music-making—staff notation, instruments, and theory—is the problem. As with all of the above examples, fixing the technology will fix the problem.

Until we fix the technology of music-making, it hardly seems fair to blame the victims—music students—for their "lack of talent." (Why beholdest thou the mote that is in thy brother's eye, but perceivest not the beam that is in thine own eye?
(http://www.godrules.net/para/luk/parallelluk6-41.htm)

To argue otherwise is to argue that either

all of the above examples are wrong, or that
"music is different."

I would welcome the opportunity to dismember either argument. ;-)

Respectfully,

Jim Plamondon
Unaffiliated Musical Heretic

The (Isomorphic) Cortical Topography of Tonal Structures

2009-09-10T19:05:00.000-05:00

At the request of Daniel Levitin, I added this post to his Facebook page's discussion board, which I will also paste below.

-------------------------

Gentlepersons,

Most work in music cognition assumes 12-tone equal temperament, which is a perfectly reasonable starting point. However, I suspect that the findings thereof can be easily generalized to alternative tunings, using some recent discoveries.

The first discovery is the two-dimensional syntonic temperament. Its tuning continuum includes nearly all of the tunings ever used by humankind in the real world, from the from the 7-tet ("7-tone equal temprament," hence "7-tet") tunings related to the timbres of the Thai ranat and African balafon to the 5-tet tunings related to the timbres of the Indonesian gamelan, with 17-tet, Pythagorean, 12-tet, the meantones, and an infinite number of other tunings in between.

The second discovery is the relationship between two-dimensional temperaments, such as the syntonic temperament, and two-dimensional isomorphic keyboards.

If the pattern of notes on a two-dimensional keyboard is generated by the same two intervals that generate a two-dimensional temperament (such as the syntonic temperament), then the keyboard will be "isomorphic" with that temperament. What this means is that any given interval in that temperament will have the "same shape" in every tuning of that temperament. Therefore, any given combination or sequence of intervals also has the "same shape" everywhere on an isomorphic keyboard, in every tuning of that temperament. This is tuning invariance.

For a demonstration of tuning invariance on an isomorphic keyboard (with embarrassingly-over-the-top commentary), please see this video.

This tuning invariance applies to all syntonic tunings, including tunings that are equal and non-equal, regular and irregular (such as “well-temperaments”), and also "rank-2, 5-limit Just Intonation" tunings (see proofs here or here).

Alternatively put, syntonic tunings include Western (Pythagorean, 12-tet, 1/4-comma meantone, 31-tet, “circulating”) and non-Western (Indonesian, Thai, Mandinka African) tunings, and the JI tunings used both in the West and in non-Western cultures (which rarely exceed 5-limit; the blues is, arguably, 7-limit, but that case is also well-handled by an isomorphic note-layout).

The one non-syntonic temperament which I can find to have been used by humankind in the real world is the (Turkish) schismatic temperament. Because its generators are the same as those of the syntonic temperament, it is compatible with the syntonic temperament's isomorphic keyboards, and hence with the conclusions of this posting—but it is a special case, beyond the scope of this posting, so I won't mention it again.

The third discovery—at least, we haven't been able to locate any prior art yet—is that such isomorphic keyboards include within their pattern of notes a tonnetz, as described by Euler/Oettingen/Riemann etc. (see Figures 7 and 8 in this paper.) An important point is that such a tonnetz is tempered; that is, it is not based on "ratios of small whole numbers" (i.e., Just Intonation), but rather on a mapping from these "just" intervals to intervals of the syntonic temperament.

Such a "tempered" tonnetz has the same tuning invariance as the isomorphic keyboard from which it is drawn. Hence, the relationships among the notes on such a tonnetz are tuning invariant, too.

The tonnetz is (I believe) well-known to be the dual graph of the "chart of the regions" described by Schoenberg and others (see this book, p. 105). Hence, any such tempered "chart of the regions" is likewise tuning invariant.

The map of perceptual tonal space described by Krumhansl, Janata, and other cognitive psychologists, is precisely such a tempered "chart of the regions." Hence, this map *ought* to be tuning invariant, too.

The hard-wiring of a tuning invariant map of perceptual tonal space could help explain both

The diversity of real-world tunings, in that an infinity of syntonic tunings are compatible with such a perceptual space, and
The limitations on that diversity, in that

only the tunings of the syntonic (and perhaps schismatic) temperament fit this perceptual space, and
a culture’s dominant instruments must produce a timbre that is closely "related" to such a tuning (wherein "related" has the meaning described here).

The latter point must not be overlooked in any related experiments. For example, using harmonic timbres for all tunings will produce invalid results.

If perceptual tonal space were indeed found to be tuning invariant, then this could would be an important scientific step towards a truly universal theory of music.

Neither I nor my collaborators have the skills or knowledge of musical cognition sufficient to execute the kinds of experiments needed to explore this issue further. We would be delighted to help, though. Ping me at jim@iGetItMusic.com.

Thanks!

Jim Plamondon
Unaffiliated Musical Heretic

The Cortical Topography of Tonal Structures

2009-09-09T15:54:00.000-05:00

Here's an interesting experimental result, from Per Janata's article The Cortical Topography of Tonal Structures Underlying Western Music in the December 2002 edition of Science:

In contrast to distributed cortical representations of classes of complex visual objects that appear to be topographically invariant (26), we found that the mapping of specific keys to specific neural populations in the rostromedial prefrontal cortex is relative rather than absolute.

Within a reliably recruited network, the populations of neurons that represent different regions of the tonality surface are dynamically allocated from one occasion to the next. This type of dynamic topography may be explained by the properties of tonality structures. In contrast to categories of common visual objects that differ in their spatial features, musical keys are abstract constructs that share core properties. The internal relationships among the pitches defining a key are the same in each key, thereby facilitating the transposition of musical themes from one key to another.

Two observations about this:

The brain may recognize individual pitches using "fixed Do," but it recognizes tonal relationships using "movable Do with a La-based minor."
The neural topography revealed by this experiment is compatible with an isomorphic note-layout, and therefore it ought to be tuning invariant. This observation could enable the experiment's results to be generalized beyond 12-tet to include not only 12-tet, but nearly all of the pre-modern and non-Western tunings ever used by humankind.

Triad names

2009-09-07T14:29:00.004-05:00

Aha!

Consider the traditional naming of triads (built by stacking thirds from the root upwards):
- m3, m3: diminished triad
- m3, M3: minor triad
- M3, m3: major triad
- M3, M3: augmented triad

What is "diminished" about a diminished triad? Its diminished fifth, according to the traditional interval-naming scheme.

What is "augmented" about an augmented triad? Its augmented fifth, according to the traditional interval-naming scheme.

That is, the traditional naming-rule for triads is:
- If both of the triad's thirds are the same width, name the triad after the width of the fifth.
- Else, name the triad after the width of its bottom third interval (i.e., the one between the chord's mode's 1st and 3rd degrees).

If JIMS re-names the narrow fifth "minor" and the width fifth "major," as discussed below, then the traditional triad-naming rule doesn't make sense within JIMS. That means that JIMS either needs to (a) not rename the fifths, (b) rename the triads, or (c) redefine the triad-naming rule.

I like the latter option best (re-defining the triad-naming rule). With this approach, JIMS' rule for naming triads would be:
- If both thirds are minor, the triad is "diminished."
- If the thirds differ, name the triad after the width of its bottom third.
- If both thirds are major, then the triad is "augmented."

This rule produces the same triad names as the traditional rule, but without relying on the name of the triad's fifth.

I like this approach because it recognizes that the triad is what is diminished or augmented, not the triad's fifth, which is as independently major or minor as the thirds are.

I'll have to think about the naming of seventh chords, too. There seems to be a lot of variation in seventh-chord naming anyway, between classical and jazz traditions, so a little more variation wouldn't be shocking.

Simplified solfege

2009-09-05T15:42:00.006-05:00

One simplification which JIMS could adopt, but has not (so far), is a simplification of the solfége naming rules for sharpened and flattened notes.

The most common current rules are these:
- The vowel 'e' indicates intervals that are "flattened from the diatonic." There's a conflict with "Re," so "Ra" is used to indicate the flattened form of Re. The 'a' in "Ra" the matches the 'a' in Fa and La, of course, which confuses the rule further.
- The vowel 'i' indicates intervals that are "sharpened from the diatonic." There are two conflicts, with Mi and Ti. Instead of providing unique names for the sharpened versions of these notes, Fa and Do — which are enharmonic (in 12-tet) with the missing note-names — are used.

This traditional solfége system is not well-thought-out in two ways:
- The use of the vowels 'e' for flattened intervals and 'i' for sharpened intervals conflicts with the occurrence of those vowels in the diatonic scale, leading to exceptions and loss of obvious meaning.
- The absence of names for the sharpened versions of Mi and Ti restricts its use to 12-tone equal temperament. All other temperaments have sharpened versions of Mi and Ti that are distinct from Fa and Do, respectively. (Besides, the difference between Mi and its sharpened version is an augmented unison, not a minor second — a difference which should be exposed, not hidden.)

They followning rules would be much simpler:
- The vowel 'u' (pronounced as in moo, coo, and new) would mean "flattened from the diatonic" (i.e., diminished, using the terminology proposed here).
- The vowel 'y' (pronounced as in tie, lie, and my) would mean "sharpened from the diatonic" (i.e., augmented).

With these proposed rules, the naming of chromatic variations of diatonic notes would be entirely consistent within JIMS, and would also be consistent with the naming of diminished and augmented intervals within JIMS. Diminished intervals end in 'u'; augmented intervals end in 'y', every time.

Another reason to like the above rules is that they make the 'i' endings of Mi and Ti stand out. The only notes ending in 'i' are at the lower ends of the only two minor seconds in the diatonic scale. Unfortunately, the rest of the diatonic scale's notes' letters do not convey similar structural meanings. I've played with a lot of alternatives to the standard Do Re Mi names, and none of them are significantly better than the standard names, especially given how familiar these names are, even to non-musicians.

As to the "singability" of these vowels...

Singing 'u' is no problem. When pronounced as indicated, it's a "pure vowel" (monophthong, meaning "one vowel").

(How is it that the Greek word "phthong," meaning "vowel," has six consonants, but only one vowels? I would expect the word for "vowel" to be an anagram of the vowels — something like "youwaei." This approach does not work terribly well for consonants, I'll admit. "Bcdfghjklmnpqrstvxz" is rather hard to pronounce.)

Singing 'y' could be a problem, because it is a diphthong (i.e., two vowel sounds smooshed together, as in "eye," "you," "boy," and "cow"). To pronounce a diphthong correctly, one must slide the tongue from one location to another while speaking it. If you're singing the vowel, it's not obvious exactly when to do this tongue-slide, which makes it harder to sing (and learn to sing) diphthongs than pure vowels.

However, the diatonic scale's traditional solfa names already include diphthongs (notably Do, So, and Re). This suggests that singing 'y' would not be a major problem.

So, why doesn't JIMS use the more-complex traditional solfa-naming rules, instead of these simpler rules?

Because one of the groups that is most under-served by traditional notation is the vocal music community. Singers can transpose their voices on the fly, but transposing their notation is much harder, making JIMS attractive. Internationally-popular vocal instruction methods such as Kodály are based on "movable Do with a La-based minor." These methods would be much better-served by JIMS than by traditional notation.

If JIMS uses the solfa note-names that Kodály users are familiar with, then JIMS can slide right into their established practices, and simply work better for them than traditional notation does today.

On the other hand, if JIMS uses different solfa note-names than the Kodály standard, however, then this would be a barrier to their use of it.

So far, the benefits of being compatible with the Kodály method have, in my opinion, outweighed the advantages of shifting to the simpler system. However, Kodály is used almost exclusively in schools, and I'm not targeting schools with JIMS, because they are so incredibly resistant to change.

We'll see.

Unison and octave?

2009-09-05T12:59:00.002-05:00

Well...darn.

In an earlier post, I explored the possibility of eliminating the "perfect/imperfect" interval class distinction. When considering the "perfect" class, I only considered the fourths and fifths, kinda sorta overlooking the (rather important) unison and octave.

Oops. My bad.

I can't see how one could have minor and major octaves or unisons. They don't have two different sizes within the diatonic scale; they only have one. Each can be augmented and diminished, but those are chromatic operations, not diatonic operations.

So, I think we're stuck with two interval classes.

Let's define "perfect" to mean "has only one interval size in the diatonic scale." That definition fits unison and octave, but no other diatonic intervals -- specifically, not the fourths or fifths.

"Imperfect," then, would mean "has two sizes in the diatonic scale." This is true for all intervals that are not unison or its octaves (i.e., 2nds, 3rds, 4ths, 5ths, 6ths, 7ths, [not 8ths], 9ths, 10ths, etc., ad infinitum). This definition of "imperfect" moves the fourths and fifths out of their traditional place in the "perfect" class, and into the "imperfect" class.

"Perfect" and "imperfect" are lousy names for these interval classes, because they are meaningless. That is, the distinguishing feature of each interval class is not encoded in the class names. Better names would be "one-width" and "two-width," for example, because these names "say what they mean."

One could still call a diatonic octave a "perfect" octave, even though its interval-class name was "one-width." The class name does not have to match the interval-name. The major third is not called the "imperfect third," after all.

Tying this all together....

There are two interval-classes:
- One-width: unison and its octaves.
- Two-width: all other diatonic intervals.

Each interval-class has its own interval-naming rule:
- One-width: diminished/perfect/augmented
- Two-width: diminished/minor/major/augmented

In the above interval-naming rules, the '/' symbol corresponds to the "augmented unison." Therefore, when the rule is read from left to right, each name denotes a note that is an augmented unison higher than the previous name. In the syntonic temperament, the augmented unison is the vector [-4, 7] (down four octaves, then up seven tempered major fifths).

The augmented unison should not be confused with the minor second, which, in the syntonic temperament, is the vector [3, -5] (up three octaves, then down five tempered major fifths). Augmented unisons separate Se, So, and Si (all of which share the same leading consonant, to show that they are all related to the same note of the diatonic scale), whereas a minor seconds separates Mi from Fa (which have different leading consonants, to show that they are different notes of the diatonic scale).

BTW, to reverse an interval-vector's direction, one reverses its signs. For example, to turn the "augmented unison" vector [-4, 7] into the "diminished unison" vector, one changes the signs of the numbers in the vector, getting [4, 7]. Likewise, to turn the "minor second up" vector [3, -5] into the "minor second down" vector, one changes the signs of the numbers in the vector, getting [-3, 5]. Same magnitude, opposite direction.

In summary, the proposed interval-naming changes are:
1. To move the fourths and fifths out of the perfect interval-class into the imperfect interval-class.
2. To rename the perfect and imperfect interval classes to "one-width" and "two-width," respectively, so that the class names "say what they mean."

I am still debating whether or not to incorporate this change in JIMS. On the one hand, this change would make interval-naming much easier to teach, and would expose many other meaningful relationships. However, these changes would also be the first in JIMS to break compatibility with the vocal music instruction methods based on "movable Do with a La-based minor," such as Kodály.

Maintaining compatibility with the Kodály method could significantly increase JIMS' rate of adoption, because its users are particularly under-served by traditional notation (and accompaniment instruments).

Nobody's "perfect"

2009-09-04T10:36:00.010-05:00

In my previous blog post, What is a "perfect" interval, really?, I described the traditional "interval classes" as:
- Perfect: diminished/ perfect /augmented
- Imperfect: diminished/ minor/major /augmented

Then I asked, "is this distinction meaningful? And if so, how?"

The short answer is "no, it's not meaningful."

The long answer, provided by Andy Milne, is that all Moment of Symmetry (MOS) scales (also called well-formed scales), including the diatonic scale, have two interval sizes. The wider interval could always be called the "major" interval, and the narrower one, the "minor" interval. That's the interval-naming rule associated with the "imperfect" interval class.

In short, the traditional distinction between "perfect" vs. "imperfect" intervals is a distinction without a difference. One could classify all intervals as being "imperfect," in any MOS scale, without loss of information. Because all tonal intervals would therefore fall into a single class, the need for any such classification would be removed.

An Alternative to Perfection
For example, the diatonic scale includes two fourths, one smaller (e.g., F to B) and one larger (e.g., C to F). Traditionally, fourths are members of the "perfect" interval class, with the narrower interval being named the perfect fourth and the wider interval being named the augmented fourth. However, these intervals could instead be named the minor fourth and major fourth, respectively.

Likewise, the diatonic scale includes two sizes of fifths, one smaller (e.g., B to F) and one larger (e.g., C to G). Traditionally, fifths are members of the perfect class, with the narrower interval being named the diminished fifth and the wider interval being named the perfect fifth. However, these intervals could instead be named the minor fifth and major fifth, respectively.

Using only the imperfect interval class's interval-naming rule would result in the following interval names (with changed names in italics):

There are two reasons why using two naming rules, when one will suffice, is bad.

Error of the First Type
The first reason why its bad to make a distinction between perfect and imperfect intervals is that this distinction implies that there is a difference between them, when no such difference exists. The diligent student, on learning of the distinction, will naturally attempt to discover what the difference is. When the student fails to find such a difference (because there is no difference to find), this failure will tend to reinforce the student's feeling that music is Kafkaesque — that is, "marked by a senseless, disorienting, often menacing complexity." Alternatively, the student may internalize the failure, concluding that "I'm just too stupid to understand music."

Either way, this is bad pedagogy.

There's a simple rule to follow, while designing systems, that can help one avoid Kafkaesquery: Occam's Razor, which states that "entities should not be multiplied unnecessarily." In this case, the interval-naming rules of the perfect and imperfect interval classes are the "entities." If only one rule is needed, then Occam's razor says, don't create more rules.

In short, the first reason why it is bad to have two interval classes, when one will suffice, is that having two classes implies the existence of meaningful patterns that do not, in fact, exist. Exposing meaningless patterns is a false positive error (also known as a "type I error").

Error of the Second Type
The second reason why it is bad to have two interval classes, when one will suffice, is that having two classes hides the existence of meaningful patterns. Hiding meaningful patterns is a false negative error (also known as a "type II error").

Consider the relationships of interval-names to modes. Using only the one "imperfect" interval class, the relationship between interval-names and modes reveals the major-minor axis through the modes. The table below shows the diatonic modes in circle-of-fifths order, from Fa to Ti (using movable Do with a La-based minor, of course):

This table (and hence the one-class interval-naming system) exposes a number of clear patterns.
- Fa-mode (Lydian) is "the most major" mode; all of its intervals are major.
- Ti-mode (Locrian) is "the most minor" mode; all of its intervals are minor.
- As one moves down the table from Fa-mode to Ti mode (and hence around the circle of fifths), each mode introduces a new minor interval, making it "more minor" than the previous mode.
- The "new minor interval" of each mode is always the interval from that mode's tonic to Fa.

It's not that these patterns were impossible to detect when using two interval-naming rules; it's just that using a single rule makes these patterns easier to detect. Increased ease-of-detection reduces the chance of false negative errors. Alternatively put, it increases the chance that students will detect (and therefore have a chance to understand) these meaningful patterns.

Fossilized Tradition
The music establishment uses this overly-complex interval-naming system for one reason, and one reason only: because the music establishment uses this overly-complex interval-naming system. It's done because it's done. A few centuries ago, some intervals were renamed in accordance with the major/minor system, but some of the old names stuck.

Consonance
I am aware that the "perfect" moniker is applied only to "the most consonant" intervals, i.e., the unison, octave, wide fifth, and narrow fourth. At most, that's a reason
- to name the narrow fourth "perfect" while calling the wide fourth "major," and
- to name the wide fifth "perfect" while calling the narrow fifth "minor."

Diatonic and Chromatic Intervals
Another meaningful distinction that is blurred by the traditional interval-naming scheme is the difference between diatonic intervals and chromatic intervals. By calling the FaTi interval an "augmented fourth" and the TiFa interval a "diminished fifth," no distinction is made, in the naming of intervals, as to whether they are diatonic or chromatic.

Using only the "imperfect" interval-naming rule makes a clear distinction between these two different kinds of intervals: minor and major intervals occur in the diatonic scale, whereas diminished and augmented intervals do not. This is a meaningful difference, easily detected.

Communication Compatibility
Inspection of the table of interval names above shows that the likelihood of confusion, between people using the old names and those using the new, is quite low.

It's OK to have many different names that refer to the same thing, as in "Bob," "my brother Bob," "my father Bob," "Mr. Bob Johnson," etc. That's a many:1 mapping. No problem. So when a traditionalist hears someone say "minor fifth" for the first time, he'll need to learn a new name, but there's no ambiguity. Four of the five changed names fall into this many:1 category (minor fourth, major fourth, minor fifth, major fifth).

Two changed names, however, use existing names differently that they were used before (augmented fourth and diminished fifth). When a traditionalist hears someone say "diminished fifth," he'll have to wonder...is that an "old style" diminished fifth, or a "new style" diminished fifth?

That would be a problem...if it were to ever come up. However, it won't, because the "doubly altered" intervals with which new-style diminished fifth and augmented fourth may be confused are exceedingly rare. That is, someone using the new-style interval names would almost never say "diminished fifth" or "augmented fourth;" it simply wouldn't come up.

Likewise, if a person familiar with the new-style names heard a traditionalist describe an interval as a "diminished fifth," the new-stylist would say, "WTF? Why would you use a chromatic fifth there, instead of a diatonic fifth?" and after just one such interaction, the name-mapping would be clear to both parties in a very memorable way.

Hence, the ease-of-learning benefits of the proposed new names are very likely to outweigh the compatibility cost of the change.

Voronoi Expert Found!

2009-09-04T10:09:00.005-05:00

Alan Shaw noticed my blog post Help! Voronoi expert wanted and kindly volunteered to share his 20+ years of Voronoi-specific experience.

From my blog post's description of the problem (plus a few other details), he was able to develop a Flash applet to show the Voronoi relationships among regular lattice cells as the lattice is systematically deformed. This applet is a great example of data visualization.

By playing with this applet, Bill Sethares and Andy Milne (my partners in musical heresy) were able to deduce the relevant mathematical relationships. They're now working out the details & proofs with Alan.

Data visualization is remarkably powerful, isn't it? As the Overview to Wikipedia's article on data visualization states (or, at least, as it stated when I wrote this):

"The main goal of data visualization is to communicate information clearly and effectively through graphical means...To convey ideas effectively, both aesthetic form and functionality need to go hand in hand, providing insights into a rather sparse and complex data set by communicating its key-aspects in a more intuitive way. "

Fundamentally, JIMS is nothing more than a tool for data visualization (and control), whose aim is to expose, to its users' intuition, the invariant relationships in music just as Alan's applet exposed the invariant properties of a systematically-deformed lattice.

Thanks, Alan! :-)

What Killed Thumtronics?

2009-09-03T17:03:00.003-05:00

I killed Thumtronics, as its CEO, through my own inexperience.

Two major errors — both mine — killed Thumtronics, thus preventing the Thummer from reaching the market.

These errors were:
1) Starting Thumtronics is the wrong location.
2) Failing to observe the KISS Principle ("Keep It Simple, Stupid").

Location
I started Thumtronics in a tiny hick town (Busselton, Western Australia). Great place to semi-retire, but a lousy place to start a high-tech company. I believed that the world had become flat. However, if you're trying to get a start-up off the ground, geography still matters. Your first step must be to relocate to an appropriate start-up hub.

For Thumtronics, not relocating was fatal. Most of my other mistakes, large and small, could have been avoided simply by starting up in (for example) Austin, and taking advantage of its excellent start-up infrastructure.

KISS
I was initialy attracted to the Thummer, as an investment of my own time and money, because it was "old wine in new bottles," in which the bottle provided all of the added value. All I needed to do was wrap off-the-shelf parts in a new instrument-shape, and voila! — I'd have an inexpensive, expressive new instrument that was easy to learn, fully compatible with all existing (USB-)MIDI-based hardware and software, and patentable. Everything in the Thummer would be off-the-shelf except for its user interface, which was the only remaining source of value in the musical instrument industry's value chain (everything else having been commoditized).

Even better, every performance of the Thummer — whether live or in a video — would implicitly "endorse" the Thummer's unique abilities. Further, because the Thummer would look so unmistakably different from everything else, every performance would also be an "advertisement" for the Thummer. Our marketing expenses could be very low, because our customers would advertise the Thummer for us, simply by using it. (This approach doesn't work for guitar makers because all guitars look alike to non-guitarists. The Thummer, however, looks totally unique, even to non-musicians.)

Indeed, the Thummer was a "purple cow" — a product so different that it would attract attention effortlessly (which was later borne out by the Thummer's ability to attract press from such non-musical publications as the Wall Street Journal).

With this in mind, I should have focused exclusively on "getting version 1.0 to market ASAP, while spending as little on R&D as possible," in order to (a) keep its price low and (b) jump-start the use/endorse/advertise cycle ASAP.

Why did I not maintain this obviously-correct focus?

Because I was also aware that the Internet dramatically increased the effect of word-of-mouth communications (hence "word of mouse"). If Thummer v1 sucked, then its use/endorse/advertise cycle would never start — or, worse, an anti-use/endorse/advertise cycle would begin, "poisoning the well" for version 1 and all future versions, too.

I came to belive that, in order to ensure positive word of mouse, the Thummer v1 had to be "the most expressive instrument on the planet." It had to exceed its customers' expectations by such a wide margin that it would attract evangelically-enthusiastic word of mouse. This led me to elevate expressive potential over KISS, and therefore to invest time and money in two features that required R&D: (a) motion sensing and (b) key velocity/aftertouch.

Motion Sensing
Today, motion sensing (using accelerometers and gyroscopes) is as cheap as dirt, because it's implemented in off-the-shelf chips. Such chips are in every modern console game controller, such as Nintendo's Wii Remote and Sony's SixAxis/DualShock 3 controller.

But back then, in 2003-2005 when we were developing the Thummer, there were no cheap off-the-shelf motion-sensing solutions. Because of this, we should have written off motion-sensing as "a great feature for a later version, once motion-sensing chips became available off-the-shelf." Pretty obvious, right?

The problem was that people LOVED the motion-sensing prototype Thummers. Even skeptics became enthusiasts after seeing them demonstrated. Motion sensing made musician's expressive actions visible to the audience, which was something a tiny thumb-operated joystick could never do. Motion sensing was clearly the Thummer's killer feature.

If we could just implement motion sensing in Thummer v1 (we thought), then we'd have a hit, Hit, HIT!

However, with the crude and expensive motion-sensing chips available back then, there was no way we were going to make a motion-sensing Thummer. It took us months, and hundreds of thousands of dollars, to realize just how hard it was going to be to pull together a "complete solution" from those crude chips. Had Thumtronics been in Austin, I would have had access to people who knew that "complete solution" chips were just a couple of years away. Integrating the new chips into a Thummer would have required one-tenth the R&D effort by Thumtronics. Making the decision to wait would have been much, much easier, had I known that such chips were coming soon. (Chip advances are sporadic, so it's not easy to predict what the next year or two will bring, even if you know that "chips are getting better all the time.")

In any case, I should have stuck to my initial vision of "old wine in new bottles," and ignored motion sensing until it became "old wine," in the form of off-the-shelf chips. Deciding to spend R&D resources on motion-sensing was a mistake.

Key velocity/aftertouch
The harder you strike a piano key, the harder its strings are struck. This one extra expressive variable — "key velocity" — was enough to cause the piano to out-compete the harpsichord, pipe organ, and all other previous keyboard instruments.

"Aftertouch" goes a step further, by allowing an instrument to sense the pressure with which you continue to press a key after the initial strike.

Although there was ample off-the-shelf technology available to measure key-velocity in an electronic instrument that used a piano-like keys, there was none available for concertina-like button-field instruments (and there still is none today). The movement of a button is quite different than that of a piano-like key, so we couldn't use piano-based technology.

We figured that, if the Thummer v1 didn't implement key velocity, then it would suck, and ruin our word of mouse. Therefore, we decided to reverse-engineer the pressure-sensitive buttons of the Sony PlayStation video game controllers, which would give us both key-velocity and aftertouch. However, reverse-engineering this button-system turned out to be beyond the capabilities of our back-of-beyond, hick town company. It soaked up much more of our resources than we could afford. By the time we realized that the end of this R&D task was not in sight, the end of our capital was.

Attempting to implement key velocity/aftertouch was a mistake for three reasons. First, it required R&D, and the "old wine in new bottles" game plan was specifically designed to minimize R&D. Second, it simply wasn't necessary. An alternative feature, called "channel pressure," would have been (a) good enough, and (b) brain-dead simple/cheap/fast to implement. We were focused on key velocity/aftertouch because we listened too hard to our piano-playing beta-testers, who said it was a "must have." Third, even if the Thummer needed more expressive power to succeed, the "killer" way to get that expressive power was through motion sensing, not key velocity/aftertouch.

Bad Decisions => Lack of Cash => Death
These bad decisions cost Thumtronics time, and time is money. If I had not made these bad decisions, Thumtronics would have been able to bring v1 of the Thummer to market by Christmas 2005, at which time it still had enough capital to live cheap and market hard while sales ramped up.

Having made these errors, however, I had to attempt to raise more money. This fund-raising effort failed. Presumably, potential investors decided that if I hadn't brough the Thummer to market after a $1.5 million dollar investment — which should have been ample — then perhaps it was simply a bad idea, or I was simply a bad entrepreneur. They were probably right, on the latter point, at least (although I'd say "inexperienced" rather than "bad").

Bad Location => Bad Decsions
Had I started the company in the right location, and thus been able to assemble a board of directors (and suppliers, partners, employees, etc.) with the right experience, then they are very likely to have been able to help me (a) resist the temptation to elevate "excellence" over KISS, and (b) stick to my "old wine in new bottles" game plan, thereby getting Thummer v1 onto the market by Christmas 2005.

Purple cows don't need to be excellent, in their first version. They just need to be very, very purple...and commercially available. The Thummer would have been very bright purple indeed, even without motion sensing or key velocity/aftertouch. All it needed was to get to market, so that it could find its niche. Each subsequent version could have"sucked less," growing the niche, and climbing the Long Tail into the mainstream.

I never should have elevated "expressive potential" over KISS. Darn it.

Woulda, Coulda, Shoulda...
Let's imagine for a moment that I had not made either of these two major errors. How would the Thummer have worked out? No one can know for sure, however, but here's one possible scenario.

Thummer version 1 would have been available for sale in time for Christmas 2005, without motion sensing or key velocity/aftertouch. Although sales would not have been explosive by any means, v1 would have sold enough over the next 12 months to make Thumtronics cash-flow positive by the end of the year, with a clear growth curve. (This is especially true because we would never have employed the staff that we hired to implement motion-sensing and key velocity/aftertouch, thereby keeping our costs lower.) Sales of the open-source Monome, and of Yamaha's Tenori-On, show that demand existed for alternative instruments such as the Thummer. A history of real, proveable, black-and-white sales growth would have allowed us to attract growth capital. Growth capital is much easier to get than start-up capital, and there was no shortage of growth capital in the USA back then.

With that growth capital, we could have accelerated sales in 2006. Also, motion sensing chips became widely available in 2007, so we could have added motion-sensing to version 2 for Christmas 2007.

Thummer v2 would have been a truly excellent product, not only due to motion sensing but also due to lots of little refinements that users would have suggested after using version 1. As you can see from Ken Rushton and others, the very idea of the Thummer creates "evangelically enthusiastic" supporters. Think how much stronger this enthusiasm would be, and how much more broadly-based, if Thummers actually existed, so that one's enthusiasm could be based on experience (which Ken's now is, more or less, using his excellent DIY jammer) rather than expectation.

With motion-sensing driving the sales of Thummer v2 in 2008, we would have had the cash-flow to add channel pressure to Thummer v3, probably in time for Christmas 2008.

At least as importantly, with Thummers actually being available, the development of open-source software synths that exploited Thummer-only effects (such as dynamic tonality) would have proceeded much faster than they have in today's real time-line. By mid-2009, it is quite possible that dynamic tonality would have started showing up in pop music. (Consider this use of a Monome, or this use of a Reactable. Creative artists LOVE new gadgets.)

The more the Thummer was used by pop musicians, the more rapidly it would have ascended the Long Tail into the mainstream. That's when we would have started seeing Thummer-players being invited to join mainstream bands, or bursting into the commercial music business with Thummer-based bands. That's also when Yamaha, Fender, Roland, etc. would start considering offering their own Thummers, which Thumtronics would probably have encouraged through a patent & trademark license and reference design package.

Woulda, coulda, shoulda. Sigh.

It was all there, in the palm of my hand, but I screwed it up. By starting Thumtronics in the back of beyond, and by placing "expressive power" above KISS, I wasted my own and my investors' capital, and quite literally the opportunity of a lifetime.

A skeptic might say that Thumtronics' experience proves that "new musical instruments always fail," no matter how much "better" the new instrument might be. This is absolutely the wrong interpretation of the facts, however. The Thummer has never had the chance to succeed or fail in the marketplace. Commercially speaking, it is completely untested.

What's really frustrating is that today, motion sensing could be incorporated into the Thummer for next to nothing. Version 1 of such a Thummer, with motion sensing and channel pressure (but not key velocity/aftertouch) could be a hit product from Day 1. I've still got the key patents. It's still doable. But now I'm broke, angel investors are broke, VCs are broke, and it's just not going to happen. Argh.

Ah, well. Thumtronics is dead. Long live iGetIt Music! :-)

What is a "perfect" interval, really?

2009-09-02T14:24:00.005-05:00

Why does tonal music include two different interval-naming classes, one ("perfect" intervals) with three variations (dim, perfect, aug) and another ("imperfect" intervals) with four (dim, min, maj, aug)? From what underlying cause does this artifact arise? How?

I’m trying to figure out how to explain the traditional interval-naming system, but it makes no sense to me, so I’m having trouble explaining it. There seems to be a pattern to the names, but I can’t quite grasp it. I have never seen any explanation of the interval-naming rules that made any sense whatsoever.

This is a typical example of an "explanation" which explains nothing. It defines perfect as follows: "The term 'perfect' explicitly indicates an interval which has not been modified, and is usually only applied to the fourth or fifth." This "explanation" raises more questions than it answers:
- Modified from what? Its width in the major scale? But...none of the intervals in the major scale are modifed from their widths in the major scale; why aren't they all called "perfect"?
- To what other intervals, besides the fourth and fifth, can the term "perfect" be applied in 'unusual' cases?

The above "explanation" doesn’t explain anything; it is incomplete; it doesn't even make sense.

Another "explanation" is that "perfect intervals are the same in major and minor." If that's the rule, then why isn't the major second called the "perfect" second? It's the same in major and minor, too.

Another "explanation" is that "those intervals that sound most consonant are called 'perfect'." But this begs the question: why do consonant intervals have only three variations (dim, perfect, aug) when imperfect intervals have four (dim, min, maj, aug)? How does consonance produce this important structural difference? Is there perhaps a deeper cause that links consonance and interval-naming classes?

Another approach to "explaining" the interval-naming system is to simply give up and say, "the following information must be memorized..." This is an egregious cop-out. It shows a failure to understand, let alone explain.

The bottom line is that tonal music seems to include two different classes of intervals:
- one with three variations (diminished, perfect, augmented) and
- one with four variations (diminished, minor, major, augmented).

Think of it this way. One could keep diminishing or augmenting either class of intervals ad infinitum, just be adding double-flats, triple-sharps, etc., so the absolute number of inter-variations isn't what matters. What matters is whether the series is centered ON ONE note (perfect), or centered BETWEEN TWO notes (minor and major).

These two interval classes do not appear to me to be an artifact of arbitrary naming rules. They seem to arise from music's deeper structure, but I can't see how.
- Does a given temperament have as many interval-classes as it has generators?
- Are the "perfect" intervals the ones that are defined by no more than one of each of the temperament' generators? That rule works for the syntonic temperament (generated by the octave and tempered perfect fifth, such that P8: [1, 0]; P5: [0, 1]; P4: [-1, 1]), but I don't know enough about other temperaments (Magic, Miracle, Hanson, etc.) to know whether it's generally true.
- Or is there some other cause?

I'm hoping to someday be able to answer more questions than I ask...

Help! Voronoi expert wanted

2009-08-27T15:29:00.005-05:00

If you know anything about the mathematics of Voronoi diagrams, I need your help.

The Isomorphic Conspiracy (Andy Milne, Bill Sethares, and myself, with various single-paper collaborators) has published three peer-reviewed journal papers so far, and a bunch of conference papers, on isomorphic keyboards and their unique musical capabilities. In that work, we have, so far, used an approach that tied the notions of "note-layout" and "button-arrangement" inextricably together.

For example, consider the Wicki-note-layout, mapped to a perfectly-regular hexagonal button-arrangement. If you squish the rows of the button-arrangement to be a smidgeon closer together (as the Thummer does), then the result wouldn't be the Wicki note-layout anymore, under our original definition, because the button-arrangement was no longer strictly hexagonal. It would be something else, albeit something closely related, but we had way to describe the relationship mathematically.

We're hot on the trail of a new approach that separates note-layouts from button-arrangements. The mathematics of Voronoi diagrams are central to the new approach, because they can help us define the conditions under which one button is adjacent to another, as a regular lattice of buttons is systematically deformed away from perfect regularity. We're making pretty good progress, but the work would procede much faster if we could bring someone into the Conspiracy, for this one paper at least, who knew Voronoi mathematics inside-out.

If you know a Voronoi-math expert, preferably one who also has an interest music, please let me know.

Thummer design docs

2009-08-27T15:23:00.002-05:00

Are here: http://www.thummer.com/OpenSource/DesignDocs.zip.

Modulation

2009-08-17T17:49:00.003-05:00

If JIMS'staff notates music using "Moveable Do with a La-based minor," how does it notate modulations? How does one execute such modulations on a JIMS-compatible keyboard?

Here's a link to a forum post on that topic. It refers to a hand-written document that I whipped up for the purpose, which I probably shoud have taken more time to draw up carefully.

Why La-based minor?

2009-08-17T15:13:00.004-05:00

[Note: This post is wrong. See this update.]

SHORT ANSWER

Movable Do with a Do-based minor provides an invariant mapping of solfa names to mode degrees.

Movable Do with a La-based minor, on the other hand, provides an invariant mapping of tonic solfa names to the positions of notes in a stack of tempered perfect fifths, which—combined with octaves—gives a unique name to each point in a two-dimensional tuning space [a space which includes standard 12-tone equal temperament tuning, and many other historically and culturally important tunings, too].

Using the "La-based minor" note-naming system allows any tonal structure (that is, any sequence or combination of tonal intervals) to retain its "shape" on each of JIMS geometric elements—that is, JIMS' staff, keyboard, and tonnetz—in every octave, key, and tuning . Using the "Do-based minor" naming system would change the shapes of these tonal structures with every mode. Hence, JIMS uses the "movable Do with a La-based minor" note-naming system, to keep its shapes invariant.

LONG ANSWER

All musical notational systems must provide naming systems for three separate concepts, one way or the other:

Absolute pitch
Mode degree
Locus: the unique position of a given note within the constellation of tonal intervals (called proprietas by Guido d'Arezzo, inventor of the musical staff, c. 1020).

Locus

I can't find an English word that means what Guido meant by proprietas—that is, "the unique position of a given note within the constellation of tonal intervals"—so I'm declaring locus to have that meaning. (Neither "function" nor "role" is quite right; worse, each of these alternatives already has far too many musical meanings.) In both Latin and English, locus means "place," which is evocative of "position in tonal space," which is ultimately what locus is all about. I'm going to italicize locus throughout this document, but only because this new musical meaning of locus is novel to this document. In subsequent postings & documents, I won’t italicize it, nor should you.

To hold one of these concepts to be invariant in a notational system , the other two must be allowed to vary. For example, the traditional staff holds the notation of pitch invariant. An A4 is notated in just one location on the treble clef, for example. One can determine the locus of that A4 from the key signature, but the tonic of that key, and hence A's mode degree, cannot be determined directly from the traditional staff or its key signature. That is, traditional notation barely notates locus at all.

Each of the three above-listed concepts is held invariant by one of the three versions of solfege:

Absolute pitch is held invariant by Fixed Do (Do == C)
Mode degree is held invariant by Movable Do with a Do-based minor (Do == tonic)
Locus is held invariant by Movable Do with a La-based minor (Do == the note that is a major second below Re)

"Re" is the only diatonic note around which the diatonic scale is symmetrical, so it is the reasonable point of reference for defining the locus of diatonic notes, and by chromatic extension, of all other notes.

The idea that "Re is the only note in the diatonic scale around which the diatonic scale is symmetrical" captures the essence of locus: that one can describe notes in terms of their relationships to the other notes of the diatonic scale, and by extension, to those notes' chromatic alterations, independently of their octave, key, mode, scale, or tuning.

WHAT IS A "NOTE"?

In JIMS, a note is a unique point in two-dimensional tuning space. Any given note N can be described by a two-dimensional point [j, k]. The origin note is the point [0, 0]. A reference frequency is associated with the origin note. The frequency of the note [j, k] is j*P8 + k*P5 cents higher than the pitch of the reference frequency, where P8 = the width of the tempered perfect octave, and P5 = the width of the tempered perfect fifth.

In the discussion below, I'll calculate intervals widths under the assumption that P8 = 1200 cents and P5 = 700 cents, which is equivalent to 12-tone equal temperament.

With these definitions,

- One executes a "key change" by changing the reference frequency.

- One executes a "tuning bend" by changing the width of P5 (and/or P8).

DO-BASED MINOR

Ask yourself this question: "What is the interval between Re and Mi in Do-based minor? Specifically, is it a minor second, or a major second? "

(I pause, while you actually ask & answer this question.....Are you done answering it? OK, moving on...)

The answer is, "It depends on the mode.

In major/Ionian, the Re-Mi interval is [-1, 2] (that is, (-1)*P8 + 2*P5 = (-1)*1200 + 2*700 = -1200 + 1400 = 200 = a major second), but
in minor/Aeolian, the Re-Mi interval is [3, -5] (that is, 3*P8 - 5*P5 = 3*1200 - 5*700 = 3600 – 3500 = 100 = a minor second)."

Now, look at Figure 8 in the Spectral Tools paper. It shows "the coordinates [j, k] of the Thummer's button lattice, when using its default Wicki note layout."

Looking at that figure, ask yourself, "which buttons should be labeled 'Mi'?" More specifically, using the numbering system shown in the figure,

should the buttons [j, 2] (for any j) be labeled "Mi", indicating that Mi is a major second higher than Re? Or
should the buttons [j, -5] be labeled "Mi," indicating that Mi is a minor second higher than Re?

The answer is, again, that "it depends on the mode." In Do-based minor, the naming of mode degrees is invariant, but the naming of [j, k] notes varies. In Do-based minor, Do means 1st degree, Re means 2nd degree, Mi means 3rd degree, and so on. The "Mi" label would have to move from one button to another every time the mode changed, in order to keep Mi's scale degree constant.

In short, for the Do-based minor note-naming system to hold mode degree invariant, it must allow the locus of a mode degree to vary.

LA-BASED MINOR

Looking again at Figure 8 from the Spectral Tools paper, and comparing it to Figure 3 in the Perception paper, we can see that

all of the notes labeled [j, 2] (for any j) in SpecTools' Figure 8 are labeled "Mi" in Perception's Figure 3, and
all of the notes labeled [j, -5] in SpecTools are labeled "Me" in Perception.

What, then, is La-based minor holding invariant?

Not pitch, which differs among keys.
Not degree, which differs among modes.
Not interval width, which differs among tunings.

Instead, the one and only thing that La-based minor holds invariant – despite changes to octave, key, mode, scale, and tuning – is the mapping between note-names and [j, k] points in tuning space.

Alternatively put, what "movable Do with a La-based minor" holds invariant is locus.

For example, in La-based minor, Mi is always a major second higher (that is, [-1, 2]) than the same octave's Re. Mi is never [3, -5] higher than Re. The note that's a minor second higher (that is, [3, -5]) than Re is Me (with an 'e'), not Mi (with an 'i'). The relationship between Re and Mi is invariant in La-based minor. The same is true for any other pair of note-names when using La-based minor: the relationship between them is invariant, no matter what changes might occur in key, mode, tuning, etc.

In short, for the La-based minor note-naming system to hold locus invariant, it must allow the mode degree of a given locus to vary.

JIMS

In designing JIMS, my guiding principle has been – as Guido d'Arezzo's was – that locus was the most important property of tonal music, so it should hold locus to be invariant.

If JIMS holds locus invariant, then everything else must be allowed to vary. Therefore,

When you change mode, the tonic "moves" to another keyboard button, staff location, and tonnetz location.
When you change key, the pitch of each note "moves" to another button, staff location, and tonnetz location.
When you change tuning (i.e., the width of the tempered P5),
the pitches of all notes (except the tonic) change, and
the widths of the intervals between all non-octave note-pairs changes.

To paraphrase the guiding principle of Western jurisprudence, "Let locus be invariant, though the heavens fall."

In JIMS' case, the "fall of the heavens" takes the form of incompatibility with traditional instruments. JIMS is great for isomorphic keyboards and the voice, but incompatible with essentially all other musical instruments. However, given the widespread availability of both vocal chords and the standard computer keyboard, both of which are entirely compatible with JIMS, there is considerable opportunity to benefit from JIMS' use.

Going back to the issue of notating the three separate concepts in music, JIMS uses three separate naming systems:

Absolute pitch: Traditional pitch names (A4 = 440Hz)
Mode degree: Arabic numerals (1 = 1st degree, 2 = 2^nd degree, etc.)
Locus: Movable Do with a La-based minor (Re == the diatonic scale's note of symmetry)

Whereas traditional staff notation notates pitches directly, JIMS notates locus directly. However, JIMS provides much more context, including a stack of scale dots indicating the current scale, and an indication of the tonic's position within that scale (i.e., the current mode).

LOCUS AND THE ORIGINAL MUSICAL STAFF

When Guido d'Arezzo invented the musical staff c. 1020 C.E., he did not use it to notate absolute pitches (A4=440Hz) as we do today. Rather, he used it to notate locus (which he called proprietas, which is Latin for "properties"). Guido intended his staff to be used only by singers; it was the use of his staff by instrumentalists, later, that forced its switch from notating locus to notating pitch.

JIMS can be seen as a return to Guido's locus-based notation, with the addition of isomorphic keyboards and the tonnetz, both of which were invented centuries after Guido's death.

A Neurological basis for tonal space

2009-08-02T17:23:00.003-05:00

I recently thought of a rough hypothesis for the neurological basis of tonal music: the application of place cells and grid cells to the navigation of tonal space. A lack of relevant hits on the hypothesis' key phrases using Google suggests that it has not previously been proposed.

To simplify egregiously, place cells, found in the hippocampus, remember a "place" in the spatial environment, while grid cells, found in the entorhinal cortex, form a hexagonal grid, and remember the relationships among objects in the spatial environment.

From an evolutionary perspective, having good spatial memory confers a considerable survival advantage on an individual, so it's no surprise that humans have neurological hardware that is optimized for this purpose. Such optimized hardware is often borrowed for related task. I hypothesize that the mind borrows this spatial-memory hardware to process musical information, thereby enabling music to be processed as "movement through tonal space."

Tonal space can also be represented as hexagonal grid, for example as a isomorphic keyboard, of which a subset of notes form a hexagonal tonnetz. If such a tonnetz is tied to specific pitches, then it maps to a pitch space.

However, only one fixed pitch is necessary to map a interval-based tonal space to a pitch space. Consider, for example, the hexagonal isomorphic note-layouts (keyboards) described in the spreadsheet JIMS_Note.xls. These all describe intervals, not pitches. To describe pitches, the origin note [0, 0] needs to be associated with a specific reference frequency (e.g., 440Hz).

With a grid cell describing an isomorphic interval-layout, any given interval, sequence of intervals (melody), or stack of intervals (chord) could be described/recognized by a specific pattern of points on that grid. If any one such point were associated with a specific frequency via a place cell, then the same interval-pattern could be described/recognized by the same grid cells, despite its transposition by octave, key, or tuning.

I have no evidence whatsoever for this hypothesis, nor any counter-evidence. I only thought of it earlier this week.

Many studies have shown that musical training stimulates the development of those parts of the brain dedicated to spatial-temporal processing. The grid & place cell system may be the mechanism by which this stimulation is effected.

But, what do I know? ;-)

Note-layout spreadsheet

2009-08-02T15:56:00.003-05:00

I just posted, to the iGetItMusic.com website, a spreadsheet that's useful for exploring isomorphic note-layouts: JIMS_Note.xls. There has been some discussion of isomorphic note-layouts in the Music Notation Project's forums, which prompted this blog posting.

The spreadsheet workbook starts with a "Variables" sheet, which contains six highlighted cells (variables):
Alpha (Period)
Beta (Generator)
H_Alpha
V_Alpha
H_Beta
V_Beta

Some years ago, I posted a document that described isomorphic note-layouts in terms of the adjacency intervals H & V. That document defined H and V in semitones. That's fine for 12-tone equal temperament (12-tet), but the "semitone" has no musical meaning outside of that one tuning. ("Augmented unison" has meaning; "minor second" has meaning; but the "semitone" is, in 12-tet, a conflation of those two meaningful terms into one meaningless artifact.)

To make the description of adjacency intervals more general, the JIMS_Note spreadsheet defines adjacency intervals as vectors [a, b], where a is the number of alphas, and b is the number of betas. The interval [a, b] is ((a*alpha) + (b*beta)) cents wide. Alpha and beta are the two intervals that define a rank-2 tuning of p-limit just intonation. In the syntonic temperament, alpha is the tempered octave (c. 1200 cents), and beta is the tempered perfect fifth (c. 702 cents). 12-tet is just one specific tuning of the syntonic temperament, in which alpha is exactly 1200 cents and beta is exactly 700 cents.

Every note in such a rank-2 temperament can be defined as a note[a, b] where both a and b are integers. This two-dimensional definition of notes is a good fit with two-dimensional hexagonal-grid keyboards, exactly as the one-dimensional (i.e., stack of semitones) definition of notes is a good fit with the one-dimensional piano-style keyboard.

In the JIMS_Note spreadsheet, one can
- specify values for alpha and beta (thus specifying the tuning), and also
- specify the adjacency intervals H & V using the generalized [a, b] intervals rather than semitone counts.

Using this spreadsheet, you can plug any adjacency intervals you like into the "Variable" sheet's definitions of H & V, and see the resulting isomorphic note-layouts.

The "Variables" sheet also contains a table of adjacency intervals for common isomorphic keyboards, including the Wicki, Janko, Chromatic Button Accordion (both type B and type C), Wesley, and Triad (aka Harmonic Table).

When plugging new values into the spreadsheet, it helps to know how to express common tonal intervals in [a, b] form.
A1: [-4, 7]
m2: [ 3, -5]
M2: [-1, 2]
m3: [ 2, -3]
M3: [-2, 4]
d4: [ 5, -8]
P4: [ 1, -1]
A4: [-3, 6]
d5: [ 4, -6]
P5: [ 0, 1]
A5: [-4, 8]
...and P8: [1, 0]

Many of these intervals are enharmonic in 12-tet, such as A1/m2, d4/M3, and A4/d5. However, that's an artifact of 12-tet; in all other tunings, these pairs are NOT enharmonic. Defining these intervals using the [a, b] approach makes Dynamic tonality possible.

Taubes & the Semmelweis Reflex

2009-07-28T09:40:00.004-05:00

Here's a current example of the Semmelweis Reflex: today's ongoing diet wars.

Gary Taubes is an award-winning science journalist whose New York Times Magazine article, What if it's all been a big fat lie?, sparked a firestorm of controversy. In his article, and in his subsequent book Good Calories, Bad Calories, Taubes argues that
(a) the USA's dietary science community (henceforth, the "dietary Establishment") has accepted a single hypothesis as its guiding paradigm despite a lack of credible supporting evidence; and that
(b) an alternative hypothesis fits the observed facts much better, and is supported by evidence that is more scientifically credible.

This two-hour video of Taubes presenting his ideas (at UC Berkeley) is a good overview of his 600-page book.

One might reasonably expect that a community of scientists, when presented with evidence supporting an alternative hypothesis, would be all too eager to design and perform experiments aimed at testing the validity of this alternative, in order to determine the objective truth. After all, a "scientist" is, by definition, someone who practices the scientific method, which is all about the formulation and testing of hypotheses.

But that would be naïve. Instead, the response of the dietary Establishment was, and continues to be, a classic example of the Semmelweis Reflex in action. When Taubes' article and book were published, the dietary Establishment's immune system went into overdrive, engaging in every possible defense against the invading hypothesis and anyone who might sympathize with it. These immune responses weren't coldy rational, but were instead hotly personal.

For example, the dietary Establishment excoriated Taubes for getting a $700,000 advance for his book Good Calories, Bad Calories. They chose to view this as "selling out" rather than being, in effect, a research grant. The dietary Establishment did the same thing to Dr. Robert Atkins, who similarly challenged the validity of its prevailing hypothesis for 30 years starting in the 1970's. The dietary Establishment doled out generous servings of federal research funds to supporters of its chosen hypothesis, but not to anyone who might seriously challenge that hypothesis, such as Dr. Atkins. Instead, they said that he could fund his own experiments from the sales of his diet book. Of course, had he done so, then the Establishment would have discounted those studies' results as "advocacy," due to their not having been funded by an "unbiased" source. It was a classic Catch-22.

Maybe the Establishment's prevailing hypothesis is right; maybe Taubes' alternative hypothesis is right. We don't know, because so far, the dietary Establishment has refused to fund the kinds of experiments that might reveal the objective truth. Why should it? It already knows the objective truth, after all, as the defenders of any discipline's dominant paradigm inevitably claim to do.

When I was getting a BS in Geology back in the late 1970's, many of my professors were adamantly opposed to the newfangled plate tectonics hypothesis. They were invicibly ignorant, clinging to the previously-dominant geosyncline hypothesis despite mounting evidence against it.

On the one hand, this was an exciting introduction to paradigm shifts; on the other, I ended up with a weird education, half-geosyncline and half-tectonics.

I suspect that this experience has led me to be unusually open to criticisms of any domain's dominant paradigm, whether in geology, diet, music, or anything else.

Go, Taubes! As the Romans would have said, Fiat scientia, ruat coelum! That is, "Let science be done, though the heavens fall." :-)

[Interestingly, Google returns zero hits for the above-quoted English phrase. It is hard to imagine that I am the first to think of applying the well-known judicial phrase "Let justice be done, though the heavens fall" to a scientific context, as this result implies.]

[Also -- lest my Latin teacher spin in her grave -- let me point out that the Latin word scientia translates more to "knowledge" than to the modern concept of "science." Gimme just a little poetic license, OK?]

Proposal: The Learning Faster XPrize

2009-07-15T09:53:00.003-05:00

A three-way partnership between the College Board, the XPrize Foundation, and the Gates Foundation could provide an effective infrastructure for disruptive innovation in educational efficiency.

The basic idea is this: a multi-million-dollar prize would be announced in the field of educational efficiency, using the College Board’s Advanced Placement exams as benchmarks. The College Board would define the Prize; the XPrize Foundation would determine the winner; and the Gates Foundation would provide the prize money.

The first team to achieve a specified metric of improvement to student success on a current AP exam would be awarded the prize.

What should the prize-winning metric be? It seems to me that three variables are involved in educational efficiency:
1. The educational outcome (i.e., exam scores).
2. The percentage of the total student population that attains a given outcome.
3. The cost of attaining a given outcome (including study materials, teacher time, and students’ study & practice time).

Consider the College Board’s Calculus Advanced Placement examination as an example. One could increase educational efficiency by
A. Increasing the average exam score for the current percentage of students at the current cost.
B. Increasing the percentage of students that attained the current average score at the current cost.
C. Reducing the cost of attaining the current average score for the current percentage of students.

One way to define the prize-winning metric would be to simply pick variable A, B, or C above, and to define the amount of improvement needed over the status quo that is necessary to win the prize. For example, one might define the prize-winning metric to be “reducing the cost of attaining a passing grade by 50%.”

This approach could exclude potentially prize-winning solutions that focused on improving the other variables. That’s not good.

However, this downside is ameliorated by the inter-dependency of the variables. Reducing the cost of passing a given exam, for example, will tend to enable a higher percentage of students to pass it. Likewise, those students whose study-investment remained constant, despite the lower cost of passing, would be likely to achieve higher scores, thus raising the average score.

Hence, I suggest that the prize-winning metric should focus on cost-reduction.

If the metric uses the “current average score,” however, then one way to win would be to identify the easiest-to-learn subset of material tested in the examination, teach that material only, and ignore the rest – in short, to “dumb down” the material. That’s not the Prize’s goal at all!

Therefore, rather than using the “current average score,” a much more-demanding metric such as “the current 80th percentile score” should be used.
The metric I propose, then, would be “reducing, by 50%, the cost of attaining a score equal to the 80th percentile score on the previous year’s exam.”

In the previously-mentioned case of AP Calculus, awarding the prize would indicate that students could now learn calculus twice as quickly as before, and that despite this lower investment of time, would learn more than they did before. That’s a very significant outcome, worthy of a multi-million dollar prize.

The Gates Foundation is not, by any means, the only source of funding for such a prize. The US Government has recently expressed interest in offering such prizes, and of course there are many other education-focused philanthropies -- but the Gates Foundation has the size and the focus to help make such a prize successful.

Notational Load

2009-07-11T17:50:00.003-05:00

I define notational load as the cognitive load imposed on a learner by the notation in which a concept is expressed, rather than by the concept itself.

As examples of notational load, consider:

Roman numerals vs. Arabic numerals: Arithmetic operations such as long division are much easier to teach, learn, and apply using Arabic numerals than Roman numerals. Indeed, the Romans’ numeric notation is often given as the main reason why the Romans contributed almost nothing to mathematics per se, whereas the Arabs made great advances in mathematics.
Chinese writing vs. Korean writing: The use of Chinese ideograms for Korean writing restricted literacy to Korean elites until the invention of the Korean-specific Hangul writing system, which made it possible for “a bright Korean-speaking student to become literate in one day, and a slow student in ten.” Smilarly, the Cherokee-specific writing system produced a similar jump in literacy rates within the Cherokee-speaking population of the early 1800's.
The musical staff vs. neumes: Guido d’Arrezo’s invention of sight-singing, including the musical staff and solmization, is credited with reducing the training time of Church singers from ten years to “one, or at most two” – thus reducing the cost of music education by between 80% and 90% without sacrificing quality.

The development of other notational systems, such as calculus, the Periodic Table, and Feynman diagrams, have similarly contributed to significant increases in the efficiency of education.

Perhaps of even greater importance, the development of new notations has often led to new discoveries that were literally “inconceivable” using the previous notations, because notations invariably constrain, in addition to reflecting, patterns of thought. Examples include the Arabs’ use of their numerals in developing algebra and algorithms (the names of which reflect their Arabic roots), Mendeleev’s prediction of new elements based on “holes” in his Periodic Table, and Feynman’s use of his own diagrams in making major contributions to quantum mechanics.

Cognitive load theory recognizes three kinds of cognitive loads (from an educational perspective): intrinsic load (inherent to the subject being studied), extraneous load (irrelevant to the subject being studied), and germane load (which is extraneous to the current lesson in isolation, but which reduces of the overall load of the lesson-sequence as a whole).

Notational load seems to me to be an entirely extraneous load. In opposition to this position, one could argue that the mastery of a given domain’s traditional notation is required for communication with other professionals within that domain, and that this “communication conformity requirement” makes mastery of a domain’s traditional notation intrinsic. For example, without the ability to read traditional music notation, musicians cannot read the works of other composers.

Or…can they? Using modern music notation software programs, musicians can convert any given piece of written music to alternative, non-traditional notations such as guitar tab. Many of these programs support a “plug-in” architecture that enables the developers of alternative notations to retroactively upgrade the software to support new notations The potential availability of such notation-translation software, in any given domain, and the ease of distributing it over the Internet, significantly reduces the communication conformity requirement, supporting the claim that notational load is extraneous.

Changes in notation are not easy to effect in any domain, especially among tightly inter-connected professionals. However, a dramatic increase in learning efficiency may make it possible for non-professionals to rapidly gain knowledge previously restricted to a domain’s professionals. For example, more people in the USA now read Guitar Hero’s scrolling tablature than read traditional music notation, and are, as a result, learning more about music than they otherwise would.

JIMS paper rejected (MTO)

2009-07-08T14:26:00.003-05:00

On Monday, I received an email notifying me that my paper on JIMS Isomorphic Music System (JIMS) was rejected by the peer-reviewed journal Music Theory Online.

The rejection letter read as follows:

------------------

From: Matthew Shaftel
Sent: Monday, July 06, 2009 1:01 PM
To: Jim Plamondon
Subject: Re: MTO Submissions

Dear Jim,

I have just spoken informally with both reviewers and we are in agreement that your submission is not appropriate for MTO. It's underlying scenario is not appropriate to our audience (who mostly teach students with musical backgrounds), and the technology you describe seems far too cumbersome for entry-level theory at either the High-School or College level. In addition, we agreed that the articles advocacy of the Thummer (and subsequent open-source iterations thereof), is simply not appropriate for an academic journal like MTO.

I am sorry that we cannot give you any better news, but I wish you luck in your continued endeavors.

Sincerely,

Matthew

------------------

Today I responded as follows:

------------------

From: Jim Plamondon
Sent: Wednesday, July 08, 2009 12:24 PM
To: 'Matthew Shaftel'
Subject: RE: MTO Submissions

Matthew –

To be more specific, it appears to me that your rejection is based on three claims:

JIMS is not compatible with traditional instruments and notation, and therefore not appropriate for students and teachers who have already mastered both. One could argue identically that Guido’s sight-reading technology was of no value to those who had already memorized the Church’s canon, as was the norm before his technology took root. Backward compatibility with established technologies is relevant only up to a point. Historically, if a new technology offers an efficiency-gain of 100% to 200% (i.e., twice or three times the previous efficiency), then compatibility with previous technologies becomes irrelevant.
JIMS is cumbersome. “Cumbersome” is defined as “unwieldy because of heaviness and bulk,” or “troublesome or onerous.” I can’t see how any aspect of JIMS can be fairly described as cumbersome, compared to traditional instruments & notation. I suspect that what you really mean is that JIMS “is not what I already know,” which is just a re-statement of Claim #1 above.
The paper’s advocacy [of the Thummer] is unacceptable. If an alternative technology is worthy of contending with the status quo, then it must provide sufficient benefits to overcome the inertia of the status quo. However, when an alternative technology is new, there has not yet been sufficient opportunity (by definition) to develop rigorous evidence to support its claimed benefits (else it would no longer be “new”). The factually-based description of *potential* advantages is, for a truly new technology, the most scientific discourse possible. Consider, for example, the metric system; before its adoption, there was considerable scientific discourse on its potential benefits – that is, “advocacy,” by your definition – which could not be rigorously proven until it had been adopted by at least one country. If JIMS is incompatible with traditional instruments, then it must be shown to be compatible with at least one novel instrument that delivers at least the expressive power of traditional instruments. This is not advocacy; it is counter-argument to the argument of incompatibility…which brings us back to Claim #1.

Hence, your only substantial criticism of the JIMS paper is that JIMS is incompatible with traditional instruments & notations. Yet this incompatibility will be shared by *any* paradigm-shifting improvement to the status quo in music theory and/or music theory education. To use this criticism as the basis of rejecting the JMS paper is to enshrine, as an editorial goal, the MTO’s reactionary defense of the status quo.

I sincerely doubt that this is your intent. You strike me as a good, reasonable, and serious person. I respect your good will, and hope that you respect mine, too. :-)

Consider your criticism of the JIMS paper’s “advocacy.” As you have noted, JIMS is incompatible with traditional notation and instruments. This incompatibility is likely to be noted by the paper’s readers, too, if only because I state it explicitly. A reader might then wonder if learning music using JIMS were a dead end, leaving no opportunity for expressive performance. Therefore, it is necessary, in any paper that introduces JIMS, to counter this inevitable argument with a counter-argument, to wit, that (a) Thummer-like instruments can be made; that (b) they can offer up to 10 degrees of freedom; that (c) this is more expressive potential than that offered by any other polyphonic instrument; and that (d) they offer the unique ability to control the novel effects of dynamic tonality. If, as your rejection suggests, the cost of incompatibility is high, then the benefits offered by any proposed alternative must be high, too. To argue that the description of such potential benefits is unacceptable “advocacy” is to require only the negative consequences of incompatibility to be discussed. This is clearly a reactionary, pro-status-quo bias.

Likewise, if the mere description of the potential advantages of an as-yet unimplemented technology is deemed to constitute unacceptable advocacy thereof, then no scientific cooperation in the implementation of such a technology is possible, because such cooperation requires exactly the shared awareness that’s blocked by this criterion. It’s a Catch-22.

Consider, for example, Mendeleev’s initial paper describing his Periodic Table, in which he described (a) a number of proposed corrections to previously-accepted atomic weights and (b) predictions that previously-unknown elements existed with the weights and properties described by “holes” in his Table. He had no scientific evidence whatsoever to support these claims; indeed, his claimed “corrections” to established atomic weights flew in the face of all previous evidence. Instead, his paper invited its readers to collaborate in conducting the experiments necessary to prove or disprove his radical claims. Those experiments subsequently bore out his claims, and enshrined the Periodic Table as one of the greatest discoveries of science.

Yet according to the standards of compatibility and advocacy you describe above, Mendeleev’s paper should not have been published, because it was incompatible with the status quo and “advocated” – that is, described – an alternative to it. Had his paper not been published, then no one would have known of its predictions, and hence none of the research to prove or disprove those predictions would have taken place (for decades, at least).

Likewise, when Wegner proposed his theory of continental drift – which has since become the basis of modern geology’s plate tectonics – his suggestions were derided as being incompatible with the prevailing theory of geosynclines, and his papers were often rejected on grounds similar to your claim of “advocacy,” because they described ways in which his theory resolved previously unresolved issues in geology, paleontology, and paleoclimatology – exactly as my JIMS paper described ways in which JIMS enables greater ease of learning, expressive potential, and freedom of tuning.

These are not isolated incidents. The history of science is rife with such examples; it is a well-recognized problem at the intersection of the peer-review system and paradigm-shifting ideas (see here, here, and here).

Your rejection of the JIMS paper on the grounds of incompatibility and advocacy is, I submit, an example of exactly this kind of implicit, reactionary, pro-status-quo bias. This is not because you’re a bad person, but rather because you are merely human, and have humanity’s inherent weaknesses…as do I. ;-)

I do not purport to have a solution to this systematic problem. I would, however, encourage you to look critically at the MTO’s use of “incompatibility and advocacy” as publication criteria; to consider these criteria’s roles in suppressing potentially paradigm-shifting innovation; and to reconsider the use of these criteria in the interest of truly advancing the state of the art.

Which is what such journals are all about, right?

Thanks! :-)

--- Jim

From: Jim Plamondon [mailto:jim@thumtronics.com]
Sent: Wednesday, July 08, 2009 9:18 AM
To: 'Matthew Shaftel'
Subject: RE: MTO Submissions

Bummer. Semmelweis reflex in action.

For Neda

2009-06-23T15:18:00.003-05:00

When the Nazis came for the communists, I did not protest;
I was not a communist.

Then they locked up the social democrats, I did not protest;
I was not a social democrat.

Then they came for the trade unionists, I did not protest;
I was not a trade unionist.

Then they came for the Jews, I did not protest;
I was not a Jew.

When they came for me,
there was no one left to protest.

--- Pastor Martin Niemöller

I am not an Iranian, either. But we are all Neda.