Dividing the whole tone into twelfths

Since we're discussing Marchetto and his division the whole tone...

As I said before, I'm writing something in 72-tone again. I long considered the tuning to only be good for approximating 11-limit, which it does very well. But this time, it's 17-limit, which is doable in 72-equal. (Yes, I'm preaching to the choir, I know.)

The 200-cent whole tone, of course, is divided into twelve equal parts. Dividing a ninth of a string length approximates this fair enough. This gives you a (utonal) chain of harmonics, the 96th through the 108th.

108/108 (0 commas): reduces to 1/1, the unison.
108/107 (1)
108/106 (2): reduces to 54/53, but 53 is a high prime I don't use.
108/105 (3): reduces to 36/35, one of the two 11-limit quarter tones (the other is 33/32).
108/104 (4): reduces to 27/26, a type of third tone.
108/103 (5)
108/102 (6): reduces to al-Farabi's semitone, 18/17.
108/101 (7)
108/100 (8): reduces to 27/25, a two-thirds tone (13/12 is a simpler one).
108/99 (9): reduces to 12/11, al-Farabi's mujannab, the common three-quarter tone.
108/98 (10): reduces to 54/49, what al-Farabi called "Zalzal's mujannab". Similar to 11/10.
108/97 (11)
108/96 (12): reduces to 9/8, the whole tone.

Leaving out the excessively high primes (53, 97, 101, 103 and 107), all prime numbers from 2 to 17 are represented in factorizations, approximated in roughly decreasing precision in 72-tone. Third and quarter tones are expressed in simple ratios. It also covers many of al-Farabi's ratios in their divisions of the tetrachord, and touches on the 13-based ratios proposed by Avicenna.

Those four ratios I threw out because of the primes well beyond 17 can be replaced with others, typically 81/80, 64/63, 21/20, 16/15 and 10/9, so it is very possible to use all 72 pitch classes in a 72-edo composition, even within these constraints. But again, I'm NOT touting 72-tone as the greatest of all; it's just my personal favorite right now.

~D.