Enharmonic genus

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The enharmonic genus has historically been the most mysterious and controversial of the three Greek genera (genres) of tetrachords. 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.

Notation

Modern notation for enharmonic notes requires two special symbols for raised and lowered quarter-tones or half-halfsteps or quarter-steps. The symbol for a quartertone flat is ♭ or (half-flat). For a quartertone sharp, either ‡ or is used. Hence modern notation for an enharmonic tetrachord would be:

D E♭ F G ,

or

A B♭ C D .

The double-flat symbol () is used for the third tone in the tetrachord to keep scale notes in letter sequence, and to remind the reader that the third tone in an enharmonic tetrachord (say F, shown above) was not tuned quite the same as the second note in a diatonic or chromatic scale (the E♭ expected instead of F).

Like the diatonic scale, the ancient Greek enharmonic scale also had seven notes to the octave (assuming alternating conjunct and disjunct tetrachords), not 24 as one might imagine by analogy to the modern chromatic scale. A scale generated from two disjunct enharmonic tetrachords would be:

D E♭ F G - A B♭ C D ,

with the corresponding conjunct tetrachords forming

A B♭ C D E♭ F G .

Ancient Greek notation^[1], transferred to modern notation might be more analogous to:

D D D♯ G - A A A♯ D

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

References

1. ^ M. L. West (1992) Ancient Greek Music, pp. 254–273. Oxford, UK: Clarendon Press. ISBN 0-19-814975-1

See also

• Diatonic genus
• Chromatic genus
• Diatonic and chromatic