What is a "perfect" interval, really?
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...
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...
Labels: music theory
posted by JimPlamondon | 2:24 PM
5 Comments:
I don't know how much my experience as a musician qualifies me to comment but the little I've looked into this topic of the origin and relevance of interval names, be it asking private and school teachers, Google-ing, or just sitting there and pondering the possibilities, I have found an equally baffling amount of nothing. The system at least strikes one at first to be non arbitrary yet at the same time its lack of consistent and sensible explanation from any source leads me to believe that our current system is either a corrupt version of a previously sensible system, a combination of incompatible systems, or a by-product of a complete misunderstanding of music and tuning theory.
I'm still unsure why in music we decided to view all tones created by a tuning not in its applicable diatonic scales as deviations therefrom by means of augmentation of diminution, and not as just, well, other tones. If as a culture we had based our music on scales like the double harmonic, which to my understanding inherently uses tones outside of the diatonic scale of a tuning, would we have viewed every note outside of this scale as an accidental, sharp or flat, or what have you?
This is why I like solfege so much; it gives a different name to each tonally relevant note of a tuning including enharmonic equivalents, but I don't know of a solfege interval naming system.
I'm afraid I only bring to you more questions and no answers.
Andy Milne (www.tonalcentre.org), lead author of most of the scientific papers on which JIMS is based, sent me this cogent explanation.
----------------------------------
In the diatonic scale (indeed in any MOS scale [http://en.wikipedia.org/wiki/Generated_collection]), all interval classes come in one of two sizes: a "major" size, and a "minor" size. So it makes complete sense to talk of major and minor seconds, thirds, etc.. If this naming scheme were to be followed to its natural conclusion, there would be a minor and major fourth (the minor fourth would be what we call the perfect fourth, the major fourth would be what we call the augmented fourth; the minor fifth would be what we call the diminished fifth, the major fifth would be what we call the perfect fifth. Augmented and diminished apply to non-diatonic intervals. This is probably a more logical system of naming.
So why hasn't this naming system been adopted? I guess (I may be wrong) that in Medieval times, "perfect" was reserved for consonances (in those days only 1/1, 4/3, 3/2, 2/1, and their octave expansions were deemed consonant), the major third was called a ditone, for example. So the fourths and fifths already had an established name, which was not dropped when the major/minor naming scheme came into play.
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This suggests that the intervals could be named as follows:
Old Name New Name
----------------- -----------------
Diminished Fourth Diminished Fourth
Perfect Fourth Minor Fourth
Augmented Fourth Major Fourth
Double Augmented Fourth Augmented Fourth
Double Diminished Fifth Diminished Fifth
Diminished Fifth Minor Fifth
Perfect Fifth Major Fifth
Augmented Fifth Augmented Fifth
In the above alternative interval-naming system,
- all of the intervals of Fa-mode (Lydian) are called "major," while
- all of the intervals of Ti-mode (Locrian) are called "minor."
Well, that makes sense. If you list the modes in circle-of-fifths order, starting on Fa, you get
Fa-mode (Lydian) "most major"
Do-mode (Ionian)
So-mode (Mixolydian)
Re-mode (Dorian) "neutral"
La-mode (Aeolian)
Mi-mode (Phrygian)
Ti-mode (Locrian) "most minor"
As you proceed down the list from Fa-mode (Lydian) to Ti-mode (Locrian), a minor interval replaces a major interval at each step.
In this systems, as Andy stated above, all diatonic intervals are either major or minor, and both versions of each interval appear at least once.
BTW, JIMS' solfege naming system is independent of scale and tuning...sort of.
The syntonic temperament on which JIMS is based is generated by the octave and perfect fifth (which would be called the "major fifth" using the alternative naming system proposed by this thread, but I'll stick with "perfect fifth" for now). Tuning space is a square matrix of notes, each identified by its position [a, b] in the matrix, where a is the number of P8's it is away from [0, 0], and b is the number of P5's it is away from [0, 0].
So, the first octave of [0, 0] is at [1, 0]; the second is at [2, 0], the third is at [3, 0], and so on.
The circle of fifths from [0, 0] would pass through [0, 1], [0, 2], [0, 3], and so on.
Every value of b in the range -9 < b < 9 has a unique solfa name.
-9 De
-8 Se
-7 Ra
-6 Le
-5 Me
-4 Te
-3 Fa
-2 Do
-1 So
0 Re
1 La
2 Mi
3 Ti
4 Fi
5 Di
6 Si
7 Ri
8 Li
9 Mi
Every octave of [x, b] has the same name as [0, b]. If the above naming system is to be extended, 'u' can be used for doubly-diminished intervals, and 'y' for doubly-augmented intervals (although these rarely occur in tonal music, if you keep the tonic centered on the [Do, Re, Mi, Fa, So, La, Ti] diatonic set using electronic transposition, i.e., movable Do with a La-based minor).
The width of the perfect fifth can be changed across the syntonic temperament's wide tuning continuum, thereby changing the width of all intervals [a, b] (other than octaves), without changing the names of the notes, their layout on an isomorphic keyboard, their positions on the tonnetz, or their locations on the chromatic staff.
The second explanation works if you define a "minor Nth" as the octave inversion of a "major (9-N)th". Or define "the minor scale" as Mi-mode (Phrygian) instead of the usual La-mode (Aeolian).
The real confusing part of interval names is the inclusive counting and the resulting unorthodox arithmetic: two "fourths" make a "seventh"!
I agree -- the "inclusive counting" aspect of interval naming is confusing, as it is not the way European cultures count, anymore.
"Inclusive counting" in musical intervals is a hold-over from Roman times; that's how the Romans counted. Inclusive counting is part of a suite of inter-related Roman technologies and conventions which included the Roman abacus and Roman numerals. They all made perfectly good sense when used together. For example, Roman numerals are a near-perfect notation for inputting data into a Roman abacus, and copying down the output of that abacus (http://en.wikipedia.org/wiki/Roman_abacus), but without such an abacus, Roman numerals are a nightmare.
I thought about changing the interval names to be zero based, so that the first degree of the diatonic scale is a 0th, the next degree the 1st, the next degree the 2nd, and so on, which would make an octave the 7th degree of the diatonic scale. It seemed to introduce too much incompatibility for too little gain...but I don't have any firm data either way, at this point.
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