Enharmonic Genus - Tunings of The Enharmonic

Tunings of The Enharmonic

There is no known Pythagorean tuning of the enharmonic (the simplest recognizable enharmonic has two notes separated by a Pythagorean comma). Aristoxenus believed that the pyknon evolved from an originally pentatonic trichord in which a perfect fourth was divided by a single "infix" into a semitone plus major third (e.g., E, F, A). This became an enharmonic tetrachord by the division of the semitone into two quarter tones (E, E↑, F, A).

Archytas, according to Ptolemy, Harmonics, ii.14 (for no original writings by him survive), as usual gives a tuning with small-number ratios:

hypate parhypate lichanos mese 4/3 9/7 5/4 1/1 | 28/27 |36/35| 5/4 | -498 -435 -386 0 cents Play

Also according to Ptolemy, Didymus uses the same major third (5/4) but divides the pyknon with the arithmetic mean of the string lengths (so therefore the harmonic mean of the frequencies):

hypate parhypate lichanos mese 4/3 31/24 5/4 1/1 |32/31 |31/30 | 5/4 | -498 -443 -386 0 cents