18edf

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18edf

19edf →

Prime factorization	2 × 32
Step size	38.9975¢
Octave	31\18edf (1208.92¢)
Twelfth	49\18edf (1910.88¢)
Consistency limit	4
Distinct consistency limit	4

18EDF is the equal division of the just perfect fifth into 18 parts of 38.9975 cents each, corresponding to 30.7712 edo. It is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.

Lookalikes: 31edo, 49edt

Intervals

degree	cents value	corresponding JI intervals	comments
0		exact 1/1
1	38.9975	45/44
2	77.995
3	116.9925	16/15
4	155.99	128/117
5	194.9875	28/25
6	233.985	8/7
7	272.9825	7/6
8	311.98	6/5
9	350.9775	60/49, 49/40
10	389.975	5/4
11	428.9725	9/7
12	467.97
13	506.9675	75/56
14	545.965
15	584.9625
16	623.96
17	662.9575	22/15
18	701.955	exact 3/2	just perfect fifth
19	740.9525	135/88
20	779.95
21	818.9475	8/5
22	857.945	64/39
23	896.9425	42/25
24	935.94	12/7
25	974.9375	7/4
26	1013.935	9/5
27	1052.9325	90/49, 147/80
28	1091.93	15/8
29	1130.9275	27/14
30	1169.925
31	1208.9225	225/112
32	1247.92
33	1286.9175
34	1325.915
35	1364.9125
36	1403.91	exact 9/4

Related regular temperaments

The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.

7-limit 31&369

Commas: 2401/2400, 8589934592/8544921875

POTE generator: ~5/4 = 386.997

Mapping: [<1 19 2 7|, <0 -54 1 -13|]

EDOs: 31, 369, 400, 431, 462

11-limit 31&369

Commas: 2401/2400, 5632/5625, 46656/46585

POTE generator: ~5/4 = 386.999

Mapping: [<1 19 2 7 37|, <0 -54 1 -13 -104|]

EDOs: 31, 369, 400, 431, 462

13-limit 31&369

Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585

POTE generator: ~5/4 = 387.003

Mapping: [<1 19 2 7 37 -35|, <0 -54 1 -13 -104 120|]

EDOs: 31, 369, 400, 431, 462

Scale tree

If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger EDOs pop up in our continuum.

Generator range: 28.09524 cents (4\7/18 = 2\63) to 40 cents (3\5/18 = 1\30)

Fifth							Cents	Comments
4\7							38.0952
						27\47	38.2979
					23\40		38.3
						42\73	38.3562
				19\33			38.38
						53\92	38.4058
					34\59		38.4181
						49\85	38.4314
			15\26				38.4615
						56\97	38.4880
					41\71		38.49765
						67\116	38.50575
				26\45			38.518	Flattone is in this region
						63\109	38.5321
					37\64		38.5417
						48\83	38.5542
		11\19					38.5965
						51\88	38.63
					40\69		38.6473
						69\119	38.6555
				29\50			38.6
						76\131	38.6768	Golden meantone (696.2145¢)
					47\81		38.6831
						65\112	38.6905
			18\31				38.7097	Meantone is in this region
						61\105	38.7302
					43\74		38.738
						68\117	38.7464
				25\43			38.7597
						57\98	38.7755
					32\55		38.78
						39\67	38.8060
	7\12						38.8
						38\65	38.9743
					31\53		38.9937	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	39.0071	Garibaldi / Cassandra
				24\41			39.0244
						65\111	39.039
					41\70		39.0476
						58\99	39.0572
			17\29				39.0805
						61\104	39.1026
					44\75		39.1
						71\121	39.1185	Golden neogothic (704.0956¢)
				27\46			39.1304	Neogothic is in this region
						64\109	39.1437
					37\63		39.1534
						47\80	39.16
		10\17					39.2157
						43\73	39.2694
					33\56		39.2857
						56\95	39.29825
				23\39			39.3162
						59\100	39.3
					36\61		39.3443
						49\83	39.3574
			13\22				39.39	Archy is in this region
						42\71	39.4366
					29\49		39.4558
						45\76	39.4737
				16\27			39.5062
						35\59	39.5480
					19\32		39.583
						22\37	39.639
3\5							40.0000

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.