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enharmonic genus

[Joe Monzo]

In ancient Greek theory, one of the three basic type of genus, with a characteristic interval of approximately a "major 3rd" at the top of the tetrachord, then 2 successive intervals of approximately a quarter-tone at the bottom, making up a 4/3 "perfect 4th".

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

enharmonic genus - comparative graph of pitch height for ancient authors

Below is a detailed examination of the enharmonic genera proposed by each writer.

. . . . . . . . .
Archytas
string-length proportions:    1512 : 1890 : 1944 : 2016  (reduced 84:105:108:112)


	note       ratio     ~ cents

	mese        1/1       0
	                                    >   4:5   ~ 386.3137139 cents
	lichanos    4/5   - 386.3137139
	                                    >  35:36  ~  48.7703814 cents
	parhypate   7/9   - 435.0840953
	                                    >  27:28  ~  62.96090387 cents
	hypate      3/4   - 498.0449991
			

Archytas gives the characteristic interval of his enharmonic genus as the 5-limit 5:4 "major-3rd", as described disparagingly by Aristoxenos, who preferred the Pythagorean ditone. His lichanos is thus also a 16:15 "diatonic semitone" above the bottom note hypate.

Archytas divided his enharmonic pyknon, with the string-length proportion 15:16, by multiplying these numbers by 7 (resulting in 105:112) and inserting the parhypate in between by locating it 3 units from the upper boundary of the pyknon (lichanos) and 4 units from the lower boundary (hypate), giving 105:108:112.

. . . . . . . . .
Eratosthenes
string-length proportions:    30 : 38 : 39 : 40

	note       ratio     ~ cents

	mese        1/1       0
	                                    >  15:19   ~ 409.2443014 cents
	lichanos   15/19  - 409.2443014
	                                    >  38:39   ~  44.9696465 cents
	parhypate  10/13  - 454.2139479
	                                    >  39:40   ~  43.83105123 cents
	hypate      3/4   - 498.0449991
			

Eratosthenes ingeniously replaced the typical Pythagorean "ditone" for the enharmonic "characteristic interval", using instead the 19:15 "enneadecimal major-3rd" which greatly resembles the ditone; this was in order to have superparticular ratios in his pyknon, since it places lichanos a 20:19 "enneadecimal semitone" above the lowest note hypate. Then he multiplied these terms by 2 and divided his pyknon with the arithmetic mean, to give 38:39:40.

. . . . . . . . .
Didymus
string-length proportions:    24 : 30 : 31 : 32


	note       ratio     ~ cents

	mese        1/1       0
	                                    >   4:5   ~ 386.3137139 cents
	lichanos    4/5   - 386.3137139
	                                    >  30:31  ~  56.76685773 cents
	parhypate  24/31  - 443.0805716
	                                    >  31:32  ~  54.96442754 cents
	hypate      3/4   - 498.0449991
			

Didymus followed Archytas in giving 5:4 as his characteristic interval, placing lichanos a 16:15 5-limit "diatonic semitone" above the lowest note hypate. Then he used Eratosthenes's method of multiplying these terms by 2 and dividing the pyknon with the arithmetic mean, to give 30:31:32.

. . . . . . . . .
Ptolemy
string-length proportions:    276 : 345 : 360 : 368

	note       ratio     ~ cents

	mese        1/1       0
	                                    >   4:5    ~ 386.3137139 cents
	lichanos    4/5   - 386.3137139
	                                    >  23:24   ~  73.6806536 cents
	parhypate  23/30  - 459.9943675
	                                    >  45:46   ~  38.05063167 cents
	hypate      3/4   - 498.0449991
			

Ptolemy also retained the 5:4 5-limit "major-3rd" "characteristic interval" of Archytas and Didymus. However, he did not use the simple expedient of finding the arithmetic mean of his pyknon to locate parhypate. The fact that the proportions for the pyknon cannot be reduced further than 345:360:368 leads me to believe that Ptolemy was trying hard to give a measurement which reflected actual practice in his time.

. . . . . . . . .
Boethius
string-length proportions:    384 : 486 : 499 : 512

	note       ratio     ~ cents

	mese         1/1         0
	                                      >   64:81    ~ 407.8200035 cents
	lichanos    64/81    - 407.8200035
	                                      >  486:499   ~  45.70020208 cents
	parhypate  384/499   - 453.5202055
	                                      >  499:512   ~  44.5247936 cents
	hypate       3/4     - 498.0449991
			

Boethius reverted to the Pythagorean 81:64 "ditone" for his "characteristic interval". Then he located his parhypate simply by finding the arithmetical mean between his other ratios, using the formula parhypate = lichanos + ((hypate-lichanos)/2).

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