Whoa: Major and minor 4th and 5th

(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

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.

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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!

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To which I responded (in part):

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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.

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Her response:

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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

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

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.