Music (and Music Theory)

• I work at Wash. U.'s Gaylord Music Library,
• I sing in the Saint Louis Symphony Chorus,
• Until we had a falling-out in 1996, I conducted the Saint Louis Taiwanese Youth Chamber Orchestra (STYCO),
• I sing in Offbeat, a vocal jazz ensemble, which unfortunately is currently on hiatus. (What can I say--we weren't getting gigs.)

I believe I have discovered an interesting result in the theory of well-temperaments.

The following will only make sense/be of interest if you are conversant with microtonal terminology for equal-tempered scales and harmonic limits.

N-TET is level-P consistent at the M-limit IFF:

For any triad whose three intervals can each be expressed as a product of P or fewer primary (or "consonant") M-limit intervals, the sum of the N-TET approximations of the two smaller intervals equals the (N-TET) approximation of their sum (the larger interval).

http://library.wustl.edu/~manynote/consist.txt

This chart shows consistency levels for all ETs from 1 to 1200, to all relevant harmonic limits. A "." means a consistency level of 1.

http://library.wustl.edu/~manynote/consist2.txt

This chart only shows those ETs which have a higher consistency level at some harmonic limit than all lower-numbered ETs, and goes up to 10000TET.

http://library.wustl.edu/~manynote/consist3.txt

This chart shows the stepsize and the greatest error among the primary or "consonant" intervals within the harmonic limit, both in cents, for all ETs up to 200TET, for all harmonic limits within which they are consistent (meaning, level 1 or better).

http://library.wustl.edu/~manynote/consist4.txt

This is a continuation of the first (consist.txt), including ETs from 1200 up through 2400.

http://library.wustl.edu/~manynote/consist5.txt

This is a modification of the second (consist2.txt), covering the same range but including more ETs. Specifically, each ET included betters all previous ETs for consistency level at some limit, but compared individually, not collectively. E.g. 87TET is included here but not in consist2.txt because it is level 4 consistent at the 5-limit, and level 1 consistent to the 15-limit, and while 19TET matches the first and 27TET the second, 87TET is the first to do both.

The following chart contains data on another concept which I call completeness. An ET is complete at a given harmonic limit if the basic intervals within that limit form a basis which spans that ET, if you think of the ET as a space or group. Example: 24TET is incomplete at the 5-limit because 5/4 is approximated by 8 steps and 3/2 by 14, which means no matter what combination of 5/4s and 3/2s (or 6/5s) you use, you can never generate those intervals which contain an odd number of steps.

So what do the numbers mean? Only those combinations of ET and (odd) limit have entries which are both consistent and complete. (I stopped, somwhat arbitrarily, at 300TET.) x/y means that the ET is x-level consistent at that limit (as in the above charts), and y is the diameter at which the ET is completed.

So what, in turn, does diameter mean? If n-ET has diameter y at the m-limit, that means that there is at least one interval which requires combining y m-limit (primary) intervals to derive it, but no intervals which require more. Example: the primary intervals (consonances) within the 5-limit (the senario) are represented in 12TET by 3, 4, 5, 7, 8, and 9 steps. 1 can be expressed as 4-3 (or 5-4, etc.), 2 by 5-3 etc, and 6 by 3+3. (The derivations of 10 and 11 are analogous to those of their complements 2 and 1.) That completes the gamut of 12TET, therefore the diameter of 12TET at the 5-limit is 2.

A contrasting example: in 19TET, combinations of exactly 2 of the primary 5-limit consonances (5, 6, 8, 11, 13, 14) give you all the rest except for 4 (and its complement 15). 4 does, however, have a ternary derivation (4=5+5-6), so the diameter of 19TET at the 5-limit is 3.

This chart is like the previous in that it gives consistency, completeness, and diameter information, but only those combinations of ET and limit are included for which the consistency level is greater than or equal to the diameter. It's rather small. Although I let the program run to 300TET again, note that the last ET in the chart is only 171TET.