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chromatic genus

[Joe Monzo]

In ancient Greek theory, one of the three basic types of genus. It had a characteristic interval of approximately a minor 3rd at the top of the tetrachord, then 2 successive intervals of approximately a semitone at the bottom, making up the 4/3 perfect 4th.

Below is a graph showing the comparative structures of tetrachords for the chromatic genus as explained by various ancient theorists:

chromatic genera tetrachord structure - comparison of ancient authors
. . . . . . . . .
Thrasyllus
string-length proportions:   1728 : 2048 : 2187 : 2304

	note       ratio     ~ cents

	mese         1/1       0
	                                   >    27:32    ~ 294.1349974 cents
	lichanos    27/32  - 294.1349974
	                                   >  2048:2187  ~ 113.6850061 cents
	parhypate   64/81  - 407.8200035
	                                   >   243:256   ~  90.22499567 cents
	hypate       3/4   - 498.0449991
			

As with the classic Pythagorean diatonic described by Philolaus, Thrasyllus's chromatic, if extended beyond the single tetrachord, could be tuned using the method of "tuning by concords", using 3:2 perfect-5ths and 4:3 perfect-4ths up and down from the starting-note mese.

Thrasyllus's top interval is the Pythagorean 32:27 trihemitone minor-3rd, the middle interval is the Pythagorean 2187:2048 apotome "greater semitone", and the bottom interval is the Pythagorean 256:243 limma "lesser semitone". Parhypate lies a Pythagorean 81:64 ditone major-2nd below the top note mese, and lichanos lies a Pythagorean 9:8 "greater tone" above the bottom note hypate.

. . . . . . . . .
Archytas
string-length proportions:   1512 : 1792 : 1944 : 2016

	note       ratio     ~ cents
		mese        1/1        0
	                                    >   27:32    ~ 294.1349974 cents
	lichanos   27/32   - 294.1349974
	                                    >  224:243   ~ 140.9490979 cents
	parhypate   7/9    - 435.0840953
	                                    >   27:28    ~  62.96090387 cents
	hypate      3/4    - 498.0449991
			

Archytas measured the upper boundary of his pyknon as a limma below that of the diatonic, making his lichanos the same Pythagorean 32:27 trihemitone below the top note (mese) of the tetrachord as that described by Thrasyllus. This lichanos is also a 9:8 "tone" above the bottom note hypate; Archytas's chromatic pyknon thus has the string-length proportion 8:9; multiplied by 28 and with the middle note inserted the proportions are 224:243:252, with the respective arithmetic differences of 19 and 9.

Archytas's parhypate is the same as in his diatonic genus and enharmonic genus. (For a discussion of Archytas's idiosyncratic use of 28:27 as the lowest interval in all three of his genera, see the explanation of his diatonic genus.)

. . . . . . . . .
Eratosthenes
string-length proportions:    15 : 18 : 19 : 20

	note       ratio     ~ cents

	mese        1/1        0
	                                    >   5:6    ~ 315.641287 cents
	lichanos    5/6    - 315.641287
	                                    >  18:19   ~  93.6030144 cents
	parhypate   15/19  - 409.2443014
	                                    >  19:20   ~  88.80069773 cents
	hypate      3/4    - 498.0449991
			

For his characteristic interval at the top of the tetrachord, Eratosthenes replaced the Pythagorean "trihemitone" with the 5-limit 6:5 JI minor-3rd. The lichanos is therefore also a 5-limit 10:9 "lesser tone" above the bottom note hypate. Eratosthenes then multiplied these terms by 2 and divided this pyknon by its arithmetic mean to get the superparticular proportions 18:19:20. Note that his parhypate is the "enneadecimal major-3rd" which is very similar to the Pythagorean 81:64 "ditone", making the bottom part of the tetrachord resemble that of Thrasyllus.

. . . . . . . . .
Didymus
string-length proportions:    60 : 72 : 75 : 80

	note       ratio     ~ cents

	mese        1/1       0
	                                  >   5:6    ~ 315.641287 cents
	lichanos    5/6   - 315.641287
	                                  >  24:25   ~  70.67242686 cents
	parhypate   4/5   - 386.3137139
	                                  >  15:16   ~ 111.7312853 cents
	hypate      3/4   - 498.0449991
			

Didymus retained the 5-limit 6:5 minor-3rd "characteristic interval" introduced by Eratosthenes, then altered the other two ratios so that his entire chromatic genus is 5-limit, introducing the 25:24 "chromatic semitone" in the middle and (as he had done in his diatonic genus) the 16:15 "diatonic semitone" at the bottom of the tetrachord.

. . . . . . . . .
Ptolemy - tense (syntonon, frequently translated poorly as "intense")
string-length proportions:    66 : 77 : 84 : 88

	note       ratio     ~ cents

	mese        1/1        0
	                                    >   6:7    ~ 266.8709056 cents
	lichanos    6/7    - 266.8709056
	                                    >  11:12   ~ 150.6370585 cents
	parhypate  11/14   - 417.5079641
	                                    >  21:22   ~  80.53703503 cents
	hypate      3/4    - 498.0449991
			

In his "tense" version of the chromatic genus, Ptolemy used the narrow 7:6 "septimal minor-3rd" as his "characteristic interval" at the top of the tetrachord, thus simultaneously placing his lichanos a wide 8:7 "septimal tone" above the bottom note hypate. The middle interval is the 12:11 "undecimal neutral 2nd", functioning here as a very wide semitone.

. . . . . . . . .
Ptolemy - relaxed (malakon, frequently translated poorly as "soft")
string-length proportions:    105 : 126 : 135 : 140

	note       ratio     ~ cents

	mese        1/1       0
	                                    >   5:6   ~ 315.641287 cents
	lichanos    5/6   - 315.641287
	                                    >  14:15  ~ 119.4428083 cents
	parhypate   7/9   - 435.0840953
	                                    >  27:28  ~  62.96090387 cents
	hypate      3/4   - 498.0449991
			

For his "relaxed" version of the chromatic genus, Ptolemy chose for his "characteristic interval" the same 5-limit 6:5 "minor-3rd" as Eratosthenes and Didymus. Note Ptolemy's placement of parhypate a wide 9:7 "septimal major-3rd" below the top note mese, as in Archytas's version of the chromatic genus, and thus giving the same 28:27 as Archytas for his lowest interval. The middle interval is therefore the 15:14 "septimal semitone".

. . . . . . . . .
Boethius
string-length proportions:    192 : 228 : 243 : 256

	note       ratio     ~ cents

	mese         1/1        0
	                                     >   16:19    ~ 297.5130161 cents
	lichanos    16/19   - 297.5130161
	                                     >   76:81    ~ 110.3069873 cents
	parhypate   64/81   - 407.8200035
	                                     >  243:256   ~  90.22499567 cents
	hypate       3/4    - 498.0449991
			

Boethius's parhypate is an 81:64 Pythagorean "ditone" below the top note mese, and therefore a Pythagorean 256:243 limma "lesser semitone" above the lowest note hypate.

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