Microtonal, just intonation, electronic music software Microtonal, just intonation, electronic music software

Encyclopedia of Microtonal Music Theory

@ 00 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Login   |  Encyclopedia Index

diatonic genus

[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.

diatonic genus - pitch height graphic showing proposals of ancient Greek and Roman music theorists
. . . . . . . . .

Philolaus (and Eratosthenes, Ptolemy "ditonic diatonic", and Boethius)
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.

. . . . . . . . .
Archytas (and Ptolemy "tonic diatonic")
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.

. . . . . . . . .
Didymus
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".

. . . . . . . . .
Ptolemy "even diatonic" (homalon, also translated "equable" or "smooth")
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.

. . . . . . . . .
Ptolemy "tense diatonic" (syntonon, frequently translated poorly as "intense")
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.

. . . . . . . . .
Ptolemy "relaxed diatonic" (malakon, frequently translated poorly as "soft")
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.

. . . . . . . . .

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.

support level