93edo

← 92edo

93edo

94edo →

Prime factorization	3 × 31
Step size	12.9032 ¢
Fifth	54\93 (696.774 ¢) (→ 18\31)
Semitones (A1:m2)	6:9 (77.42 ¢ : 116.1 ¢)
Dual sharp fifth	55\93 (709.677 ¢)
Dual flat fifth	54\93 (696.774 ¢) (→ 18\31)
Dual major 2nd	16\93 (206.452 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

Approximation of odd harmonics in 93edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-5.18	+0.78	-1.08	+2.54	+3.52	-1.82	-4.40	-1.73	-0.74	-6.26	+3.98
	Relative (%)	-40.2	+6.1	-8.4	+19.7	+27.3	-14.1	-34.1	-13.4	-5.7	-48.6	+30.9
Steps (reduced)		147 (54)	216 (30)	261 (75)	295 (16)	322 (43)	344 (65)	363 (84)	380 (8)	395 (23)	408 (36)	421 (49)

93 = 3 * 31, and 93 is a contorted 31 through the 7 limit. In the 11-limit the patent val tempers out 4000/3993 and in the 13-limit 144/143, 1188/1183 and 364/363. It provides the optimal patent val for the 11-limit prajapati and 13-limit kumhar temperaments, and the 11 and 13 limit 43&50 temperament. It is the 13th no-3s zeta peak edo.

Since 93edo has good approximations of 13th, 17th and 19th harmonics unlike 31edo (as 838.710 ¢, 103.226 ¢, and 296.774 ¢ respectively, octave-reduced), it also allows one to give a clearer harmonic identity to 31edo's excellent approximation of 13:17:19.

Temperament properties

Since 93edo has a step of 12.903 ¢, it also allows one to use its MOS scales as circulating temperaments, which it is the first edo to do. It is also the first edo to allow one to use a syntonic or Mavila MOS scale or a 17 tone MOS scale similar to a median between Pelog and the theories of Sundanese composer-musicologist-teacher Raden Machjar Angga Koesoemadinata as a circulating temperament.

Circulating temperaments in 93edo

Tones	Pattern	L:s
5	3L 2s	19:18
6	3L 3s	16:15
7	2L 5s	14:13
8	5L 3s	12:11
9	3L 6s	11:10
10	3L 7s	10:9
11	5L 6s	9:8
12	9L 3s	8:7
13	2L 11s
14	9L 5s	7:6
15	3L 12s
16	13L 3s	6:5
17	8L 9s
18	3L 15s
19	17L 2s	5:4
20	13L 7s
21	9L 12s
22	5L 17s
23	1L 22s
24	21L 3s	4:3
25	18L 7s
26	15L 11s
27	12L 15s
28	9L 19s
29	6L 23s
30	3L 27s
31	31edo	equal
32	29L 3s	3:2
33	27L 6s
34	25L 9s
35	23L 12s
36	21L 15s
37	19L 18s
38	17L 21s
39	15L 24s
40	13L 27s
41	12L 29s
42	9L 33s
43	7L 36s
44	5L 39s
45	3L 42s
46	1L 45s
47	46L 1s	2:1
48	45L 3s
49	44L 5s
50	43L 7s
51	42L 9s
52	41L 11s
53	40L 13s
54	39L 15s
55	38L 17s
56	37L 19s
57	36L 21s
58	35L 23s
59	34L 25s
60	33L 27s
61	32L 29s
62	31L 31s
63	30L 33s
64	29L 35s
65	28L 37s
66	27L 39s
67	26L 41s
68	25L 43s
69	24L 45s
70	23L 47s
71	22L 49s
72	21L 51s
73	20L 53s
74	19L 55s

Scales

Meantone Chromatic

• 116.129
• 193.548
• 309.677
• 387.097
• 503.226
• 580.645
• 696.774
• 812.903
• 890.323
• 1006.452
• 1083.871
• 1200.000

Shailaja

• 270.968
• 709.677
• 761.290
• 980.645
• 1200.000

Subminor Hexatonic

• 219.355
• 270.968
• 490.323
• 709.677
• 980.645
• 1200.000

Subminor Pentatonic

• 270.968
• 490.323
• 709.677
• 980.645
• 1200.000

Superpyth Chromatic

• 51.613
• 219.355
• 270.968
• 438.710
• 490.323
• 658.065
• 709.677
• 761.290
• 929.032
• 980.645
• 1148.387
• 1200.000

See Also

• 93edo and stretched hemififths