Tetrachord - Pythagorean Tunings

Pythagorean Tunings

Here are the traditional Pythagorean tunings of the diatonic and chromatic tetrachords:

Diatonic Play hypate parhypate lichanos mese 4/3 81/64 9/8 1/1 | 256/243 | 9/8 | 9/8 | -498 -408 -204 0 cents Chromatic Play hypate parhypate lichanos mese 4/3 81/64 32/27 1/1 | 256/243 | 2187/2048 | 32/27 | -498 -408 -294 0 cents

Since there is no reasonable Pythagorean tuning of the enharmonic genus, here is a representative tuning due to Archytas:

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

The number of strings on the classical lyre varied at different epochs, and possibly in different localities – four, seven and ten having been favorite numbers. Originally, the lyre had only five to seven strings(see also the Kithara, a larger form), so only a single tetrachord was needed. Larger scales are constructed from conjunct or disjunct tetrachords. Conjunct tetrachords share a note, while disjunct tetrachords are separated by a disjunctive tone of 9/8 (a Pythagorean major second). Alternating conjunct and disjunct tetrachords form a scale that repeats in octaves (as in the familiar diatonic scale, created in such a manner from the diatonic genus), but this was not the only arrangement.

The Greeks analyzed genera using various terms, including diatonic, enharmonic, and chromatic, the latter being the color between the two other types of modes which were seen as being black and white. Scales are constructed from conjunct or disjunct tetrachords: the tetrachords of the chromatic genus contained a minor third on top and two semitones at the bottom, the diatonic contained a minor second at top with two major seconds at the bottom, and the enharmonic contained a major third on top with two quarter tones at the bottom, all filling in the perfect fourth of the fixed outer strings. However, the closest term used by the Greeks to our modern usage of chromatic is pyknon, the density ("condensation") of chromatic or enharmonic genera.

Didymos chromatic tetrachord	16:15, 25:24, 6:5
Eratosthenes chromatic tetrachord	20:19, 19:18, 6:5
Ptolemy soft chromatic	28:27, 15:14, 6:5
Ptolemy intense chromatic	22:21, 12:11, 7:6
Archytas enharmonic	28:27, 36:35, 5:4

This is a partial table of the superparticular divisions by Chalmers after Hofmann.