Commas

I guess my bias on the "pythagorean thirds" issue is that while just
intonation is beautiful, and open-ended tunings are bountiful, for music
that goes by at a reasonable pace there is a lot to be gained from simple
structures that exploit enharmonic "punning." For example, we can look at
some common commas and the (equal-tempered) tuning systems that do and do
not distinguish them:

_5-limit_

81/80 (four perfect fifths up, two octaves and a major third down) (syntotic
comma) --
distinguished in: 15, 22, 27, 32, 33, 34
not distinguished in: 12, 19, 26, 31 (meantone systems)

128/125 (one octave up, three major thirds down) (diesis) --
distinguished in: 19, 22, 26, 31, 32, 34
not distinguished in: 12, 15, 27, 33 (divisible by 3)


_7-limit_

36/35 (two perfect fifths up, a harmonic seventh and a major third down)
(septimal quarter tone) (syntotic comma plus septimal comma) --
distinguished in: 15, 19, 22, 26, 27, 31, 32, 34
not distinguished in: 12, 33

49/48 (two harmonic sevenths up, an octave and a perfect fifth down) (larger
septimal sixth tone) --
distinguished in: 12, 22, 26, 27, 31, 32, 33
not distinguished in: 15, 19, 34

50/49 (an octave and two major thirds up, two harmonic sevenths down)
(smaller septimal sixth tone) --
distinguished in: 15, 19, 27, 31, 33, 34
not distinguished in: 12, 22, 26, 32

64/63 (two octaves up, a harmonic seventh and two perfect fifths down)
(septimal comma) --
distinguished in: 19, 26, 31, 33, 34
not distinguished in: 12, 15, 22, 27, 32

126/125 (a harmonic seventh and two perfect fifths up, an octave and three
major thirds down) (diesis minus septimal comma) --
distinguished in: 26, 32, 33
not distinguished in: 12, 15, 19, 22, 27, 31, 34

225/224 (two perfect fifths and two major thirds up, a harmonic seventh and
an octave down) (septimal quarter tone minus diesis) --
distinguished in: 15, 26, 27, 34
not distinguished in: 12, 19, 22, 31, 32, 33

245/243 (an octave, two harmonic sevenths and a major third up, five perfect
fifths down) (larger septimal sixth tone minus syntotic comma) --
distinguished in: 12, 15, 26, 31, 32, 33, 34
not distinguished in: 19, 22, 27

The commas could have been defined in terms of different constituent
intervals (such as minor thirds), and some of the results would have come
out differently, but the main result is the same: all of the equal
temperaments considered here are capable of some enharmonic punning, while
representing other commas with a nonzero number of scale degrees. This
number is not always positive -- you should be able to see that from the
above. In each equal temperament, then, certain harmonic progressions will
center around a smaller number of notes (a scale), while others will create
an abundance of chromatic (or hyperchromatic) motion.


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