Unison and octave?

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).