[Joe Monzo]

In ancient Greek theory, one of the three basic types of genus. It had a characteristic interval of approximately a "tone" at the top of the tetrachord, then two successive intervals of approximately a "tone" and then a semitone at the bottom, making up a 4/3 "perfect 4th".

Below is a graph showing the comparative structures of tetrachords for the diatonic genus as explained by various ancient theorists. A detailed examination of each proposal follows.

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string-length proportions: 192 : 216 : 243 : 256 note ratio ~ cents mese 1/1 0 > 8:9 ~ 203.9100017 cents lichanos 8/9 - 203.9100017 > 8:9 ~ 203.9100017 cents parhypate 64/81 - 407.8200035 > 243:256 ~ 90.22499567 cents hypate 3/4 - 498.0449991

The earliest measurement of the diatonic genus is by Philolaus. Descending from the top note, it divides the 4:3 "perfect-4th" bounding the tetrachord into two successive Pythagorean 9:8 tones and a Pythagorean 256:243 "limma" semitone.

Extended to a full "octave", this version of the diatonic genus could be tuned using the method of "tuning by concords", that is, successive 3:2 "perfect-5ths" and 4:3 "perfect-4ths" up and down from the starting note mese.

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string-length proportions: 1512 : 1701 : 1944 : 2016 (reduced 168:189:216:224) note ratio ~ cents mese 1/1 0 > 8:9 ~ 203.9100017 cents lichanos 8/9 - 203.9100017 > 7:8 ~ 231.1740935 cents parhypate 7/9 - 435.0840953 > 27:28 ~ 62.96090387 cents hypate 3/4 - 498.0449991

Archytas used the same tuning for the note above the bottom (parhypate) in all three of his genera, making it a 28:27 above the bottom note (hypate) and a 9:7 below the top note (mese); all other ancient Greek theorists allowed parhypate to be moveable along with lichanos, its placement depending on the genus. Winnington-Ingram speculated that perhaps the reason for this strange interval was that it was important for the note above the 28:27 to be in a 7:6 "septimal minor-3rd" ratio to the note which lies a 9:8 "tone" below its lower note.

Thus, whereas the highest interval in his diatonic tetrachord is the same Pythagorean 9:8 tone indicated by Philolaus, the middle interval is the larger 8:7 "septimal tone".

The next description to come down to us, by Eratosthenes, specifies the same ratios as Philolaus.

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string-length proportions: 24 : 27 : 30 : 32 note ratio ~ cents mese 1/1 0 > 8:9 ~ 203.9100017 lichanos 8/9 - 203.9100017 > 9:10 ~ 182.4037121 parhypate 4/5 - 386.3137139 > 15:16 ~ 111.7312853 hypate 3/4 - 498.0449991

Didymus kept the Pythagorean 9:8 tone as his highest interval, but replaced the middle interval with the 5-limit 10:9 "lesser tone", thus making the interval between parhypate and the top note (mese) the 5-limit 5:4 "major-3rd". The small interval at the bottom is thus the 5-limit 16:15 "diatonic semitone".

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string-length proportions: 9 : 10 : 11 : 12 note ratio ~ cents mese 1/1 0 > 9:10 ~ 182.4037121 cents lichanos 9/10 - 182.4037121 > 10:11 ~ 165.0042285 cents parhypate 9/11 - 347.4079406 > 11:12 ~ 150.6370585 cents hypate 3/4 - 498.0449991

The string-length proportions of Ptolemy's "even diatonic" have the smallest-number consecutive ratios which can describe a four-fold division of the 4:3 "perfect-4th". The top interval is thus the 5-limit 10:9 "lesser tone", the middle interval is the 11:10 "undecimal tone", and the bottom interval is the 12:11 "neutral 2nd" functioning as a very wide semitone.

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string-length proportions: 36 : 40 : 45 : 48 note ratio ~ cents mese 1/1 0 > 9:10 ~ 182.4037121 cents lichanos 9/10 - 182.4037121 > 8:9 ~ 203.9100017 cents parhypate 4/5 - 386.3137139 > 15:16 ~ 111.7312853 cents hypate 3/4 - 498.0449991

Ptolemy's "tense diatonic" is similar to Didymus in that it makes use of both size of "whole tone", the Pythagorean 9:8 and the 5-limit 10:9 -- however, Ptolemy uses them in reverse order, putting the smaller 10:9 at the top and the larger 9:8 in the middle. This leaves the 16:15 "diatonic semitone" at the bottom, as with Didymus.

Ptolemy's "ditonic diatonic" is the same as Philolaus's classic Pythagorean description, and his "tonic diatonic" is the same as Archytas's.

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note ratio ~ cents mese 1/1 0 > 7:8 ~ 231.1740935 cents lichanos 7/8 - 231.1740935 > 9:10 ~ 182.4037121 cents parhypate 63/80 - 413.5778057 > 20:21 ~ 84.46719347 cents hypate 3/4 - 498.0449991

Ptolemy's "relaxed diatonic" has the 8:7 "septimal tone" as its top interval, the 5-limit 10:9 "lesser tone" in the middle, and the small 21:20 semitone at the bottom.

Boethius gave the same classic Pythagorean description of the diatonic as that used by Philolaus.

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The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

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