From: Anton Sherwood (dasher@netcom.com)
Date: Sat Sep 20 1997 - 01:58:22 MDT
Anders writes
: Otherwise, I think the twelfth root of 2 is involved in music as
: the pythagorean comma, but I know too little about it.
Good heavens no. The Pythagoreans flipped when the learned that
2//2 is irrational; you think they'd have anything to do with 2//12?
The most important musical interval, after the octave (1:2), is the
fifth (2:3). The Pythagorean scale is built entirely of 2s and 3s
tiered up and down -- 8:9, 64:81, 3:4, 2:3, 16:27, 128:243, 1:2 ...
The only consonant intervals within the octave, in the Pythagorean
tradition, are what we now call the fourth (3:4) and the fifth (2:3).
Sometime in the late Middle Ages, English church composers began using
a consonant version of the major third, 4:5 (which by combination and
inversion with other intervals implies the major sixth 3:5, the minor
third 5:6 and the minor sixth 5:8). The true major third differs from
the Pythagorean major third by 80:81, which is the interval most often
called a comma.
Musical notation, as now standard, is based on what's called the
meantone scale, which replaces every factor of 3 with 2*5//4 to
make the approximation 80=81 come out right.
Scales are formed of tones (2:5//2) and semitones (5**(5/4):8).
Octaves, major thirds and major sixths are true; fifths, fourths,
minor thirds and minor sixths are a bit off but tolerable.
On this scale C# (16:5**(7/4), approximating 24:25 or 128:135 or
243:250) is distinct from Db (5**(5/4):8, approximating 15:16 or
25:27 or 243:256), and Fb (25:32) is distinct from E (4:5).
Unfortunately, the keyboard instruments that became important in the
17th century didn't have room for all the notes distinguished in
meantone notation, so tuners had to choose the twelve most important
notes for each octave. (I suspect it was usually C Db D Eb E F F# G
Ab A Bb B, but have no evidence.) If you played on your organ a piece
containing notes not tuned for, you'd get occasional "wolf chords"
so called because of the howling beats of their dissonance.
So to limit the worst-case dissonance, tuners finally spread the error
around all keys, resulting in the now-usual equal temperament, 2//12.
This makes very good fifths and fourths (each .02 semitone away from
true, better than meantone), but poor thirds (.14 semitone sharp) etc.
Now, with synthesizers, we could make keyboard instruments that
adjust their tuning on the fly so that all chords are true.
I've invented a theoretical improvement to the meantone scale, in
which the error in any interval is bounded by an amount proportional
to the Euler complexity of its factors - if you really want to know
what that means, I'll tell you later. 1:2 is sharpened by 1/16 comma;
1:3 flattened by 1/8 comma; 1:5 sharpened by 1/4 comma; my tone is
32//4:(15//4)(125//16), my semitone is 5(3//2)(5//8):128//2.
I haven't tested this scale, and might not be able to distinguish it
from classical meantone by ear alone.
Anton Sherwood *\\* +1 415 267 0685 *\\* DASher@netcom.com
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