Music (and Music Theory)
I graduated from
Washington University in St. Louis
in 1988 with a bachelor's degree in music. I play (in decreasing order
of skill) cello, violin, and keyboards, but primarily I am a singer. I
am currently involved in music in the following ways:
I also lurk in and occasionally post to various music-related newsgroups and mailing lists.
- I work at Wash. U.'s
Gaylord Music Library,
- For a long time I sang in the Saint Louis Symphony
Chorus, but in the
last decade or so I only returned in those seasons in which we were
singing music from Howard Shore's The Lord of the Rings
Until we had a falling-out in 1996, I conducted the Saint Louis
Taiwanese Youth Chamber Orchestra (STYCO),
I sang in Offbeat, a vocal jazz ensemble, for a while until
that group broke up, but currently am in a similar group, Java Jived.
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
- 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).
This chart shows consistency levels for all ETs from 1 to 1200, to all
relevant harmonic limits. A "." means a consistency level of 1.
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
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).
This is a continuation of the first (consist.txt), including ETs from
1200 up through 2400.
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