Enharmonic genus
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The enharmonic genus has historically been the most mysterious and controversial of the three Greek genera. Its characteristic interval is a major third, leaving the remainder of the tetrachord (the pyknon) to be divided by two intervals smaller than a semitone (approximately quarter tones). Because it is not easily represented by Pythagorean tuning or meantone temperament, there was much fascination with it in the Renaissance.
[edit] Tunings of the enharmonic
There is no reasonable Pythagorean tuning of the enharmonic (the simplest recognizable enharmonic has two notes separated by a Pythagorean comma). It is thought that the pyknon was originally undivided, resulting in a pentatonic scale identical to the Japanese iwato. Only later was the semitone split into two microtones.
Archytas 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
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
