2L 5s: Difference between revisions

Revision as of 23:24, 24 March 2021

2L 5s, pelogic, or antidiatonic refers to the structure of octave-equivalent MOS scales with generators ranging from 3\7 (3 degrees of 7edo = 514.29¢) to 1\2 (one degree of 2edo = 600¢). In the case of 7edo, L and s are the same size; in the case of 2edo, s becomes so small it disappears (and all that remains are the two equal L's).

While antidiatonic is closely associated with mavila temperament, not every 2L 5s scale an instance of "mavila", since some of them extend to 2L 7s scales (like the 2L 5s generated by 11edo's 6\11 = 656.5657¢), not 7L 2s mavila superdiatonic scales.

In terms of harmonic entropy, the most significant minimum is at Liese/Triton, in which the generator is about 7/5 and three of them make a 3/1.

generator in degrees of an edo							generator in cents	tetrachord	L in cents	s in cents	L to s ratio	comments
3\7							514.3	1 1 1	171.4	171.4	1.00
						19\44	518.2	6 6 7	190.9	163.6	1.17
					16\37		518.9	5 5 6	194.6	162.2	1.20
				13\30			520.0	4 4 5	200.0	160.0	1.25	Mavila extends from here...
			10\23				521.7	3 3 4	208.7	156.5	1.33
				17\39			523.1	5 5 7	215.4	153.8	1.40
		7\16					525.0	2 2 3	225.0	150.0	1.50	Mavila in Armodue Optimum rank range (L/s=3/2)
							526.3	2 2 pi	231.5	147.4	pi/2
				18\41			526.8	5 5 8	234.1	146.3	1.60
							1200*5/(13-phi)	1 1 phi	235.7	145.7	phi	Golden mavila
					29\66		527.3	8 8 13	236.4	145.5	1.625
			11\25				528.0	3 3 5	240.0	144.0	1.67
							529.1	1 1 √3	245.6	141.8	√3
				15\34			529.4	4 4 7	247.1	141.2	1.75	...to somewhere around here
	4\9						533.3	1 1 2	266.7	133.3	2.00	Boundary of propriety (generators smaller than this are proper)
				13\29			537.9	3 3 7	289.7	124.1	2.33
			9\20				540.0	2 2 5	300.0	120.0	2.50
							541.4	1 1 phi+1	306.9	117.2	1 1 phi+1
				14\31			541.9	3 3 8	309.7	116.1	2.66
							542.5	1 1 e	321.55	115.0	e	L/s = e
		5\11					545.5	1 1 3	327.3	109.1	3.00	L/s = 3
							546.8	1 1 pi	334.1	106.35	pi	L/s = pi
				11\24			550.0	2 2 7	350.0	100.0	3.50
			6\13				553.8	1 1 4	369.2	92.3	4.00	Thuja is optimal around here L/s = 4
				7\15			560.0	1 1 5	400.0	80.0	5.00	ie. (11/8)^5 = 5/1
					8\17		564.7	1 1 6	423.5	70.6	6.00
						9\19	568.4	1 1 7	442.1	63.2	7.00	Liese/Triton is around here
1\2							600.0	0 0 1	600.0	0	—

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 '''2L 5s''', '''pelogic''', or '''antidiatonic''' refers to the structure of octave-equivalent [[MOS]] scales with generators ranging from 3\7 (3 degrees of [[7edo|7edo]] = 514.29¢) to 1\2 (one degree of [[2edo]] = 600¢). In the case of 7edo, L and s are the same size; in the case of 2edo, s becomes so small it disappears (and all that remains are the two equal L's).
-The word "mavila" is used in different ways by different folks. Not every user of the word would consider every 2L 5s scale an instance of "mavila." In particular, between 13\29 and 14\31, and centered on 9\20, is the albitonic scale for the 2.7.11.13 subgroup temperament [[Chromatic_pairs#Score|score]], which is not intended to be treated as having any kind of fifth, flat or otherwise.
+While antidiatonic is closely associated with [[mavila temperament]], not every 2L 5s scale an instance of "mavila", since some of them extend to [[2L 7s]] scales (like the 2L 5s generated by 11edo's 6\11 = 656.5657¢), not [[7L 2s]] mavila superdiatonic scales.
 In terms of harmonic entropy, the most significant minimum is at [[Meantone_family|Liese]]/Triton, in which the generator is about 7/5 and three of them make a 3/1.

2L 5s: Difference between revisions

Revision as of 23:24, 24 March 2021

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