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. It has nothing to do with modern uses of the term enharmonic.

[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