113edo

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113edo

114edo →

Prime factorization	113 (prime)
Step size	10.6195 ¢
Fifth	66\113 (700.885 ¢)
Semitones (A1:m2)	10:9 (106.2 ¢ : 95.58 ¢)
Consistency limit	13
Distinct consistency limit	13

The 113 equal divisions of the octave (113edo), or the 113(-tone) equal temperament (113tet, 113et) when viewed from a regular temperament perspective, is the equal division of the octave into 113 parts of about 10.6 cents each.

Theory

113edo is distinctly consistent in the 13-odd-limit with a flat tendency. As a temperament, it tempers out the amity comma and the ampersand in the 5-limit; 225/224, 1029/1024 and 1071875/1062882 in the 7-limit; 243/242, 385/384, 441/440 and 540/539 in the 11-limit; 325/324, 364/363, 729/728, and 1625/1617 in the 13-limit. It notably supports the 5-limit amity temperament, 7-limit amicable temperament, 7- and 11-limit miracle temperament, and 13-limit manna temperament.

113edo is the 30th prime EDO.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-179 113⟩	[⟨113 179]]	+0.338	0.338	3.18
2.3.5	1600000/1594323, 34171875/33554432	[⟨113 179 262]]	+0.801	0.712	6.70
2.3.5.7	225/224, 1029/1024, 1071875/1062882	[⟨113 179 262 317]]	+0.820	0.617	5.81
2.3.5.7.11	225/224, 243/242, 385/384, 980000/970299	[⟨113 179 262 317 391]]	+0.604	0.700	6.59
2.3.5.7.11.13	225/224, 243/242, 325/324, 385/384, 1875/1859	[⟨113 179 262 317 391 418]]	+0.575	0.643	6.05

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	4\113	42.48	40/39	Humorous
1	6\113	63.72	28/27	Sycamore / betic
1	8\113	84.96	21/20	Amicable / pseudoamical / pseudoamorous
1	11\113	116.81	15/14~16/15	Miracle / manna
1	13\113	138.05	27/25	Quartemka
1	22\113	233.63	8/7	Slendric
1	27\113	286.73	13/11	Gamity
1	29\113	307.96	3200/2673	Familia
1	32\113	339.82	243/200	Amity / houborizic
1	34\113	360.06	16/13	Phicordial
1	37\113	392.92	2744/2187	Emmthird
1	47\113	499.12	4/3	Gracecordial

Scales

Since 113edo has a step of 10.6195 cents, it also allows one to use its MOS scales as circulating temperaments^{[clarification needed]}. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament.

Circulating temperaments in 113edo

Tones	Pattern	L:s
5	3L 2s	23:22
6	5L 1s	19:18
7	1L 6s	17:16
8	1L 7s	15:14
9	5L 4s	13:12
10	3L 7s	12:11
11	3L 8s	11:10
12	5L 7s	10:9
13	9L 4s	9:8
14	1L 13s
15	7L 8s	8:7
16	1L 15s
17	11L 6s	7:6
18	5L 13s
19	18L 1s	6:5
20	13L 7s
21	8L 13s
22	3L 19s
23	21L 2s	5:4
24	17L 7s
25	13L 12s
26	9L 17s
27	5L 22s
28	1L 27s
29	26L 3s	4:3
30	23L 7s
31	20L 11s
32	17L 15s
33	14L 19s
34	11L 23s
35	8L 27s
36	5L 31s
37	2L 35s
38	37L 1s	3:2
39	35L 4s
40	33L 7s
41	31L 10s
42	29L 13s
43	27L 16s
44	25L 19s
45	23L 22s
46	21L 25s
47	19L 28s
48	17L 31s
49	15L 34s
50	13L 37s
51	11L 40s
52	9L 43s
53	7L 46s
54	5L 49s
55	3L 52s
56	1L 55s
57	56L 1s	2:1
58	55L 3s
59	54L 5s
60	53L 7s
61	52L 9s
62	51L 11s
63	50L 13s
64	49L 15s
65	48L 17s
66	47L 19s
67	46L 21s
68	45L 23s
69	44L 25s
70	43L 27s
71	42L 29s
72	41L 31s
73	40L 33s
74	39L 35s
75	38L 37s
76	37L 39s
77	36L 41s
78	35L 43s
79	34L 45s
80	33L 47s
81	32L 49s
82	31L 51s
83	30L 53s
84	29L 55s
85	28L 57s
86	27L 59s
87	26L 61s
88	25L 63s
89	24L 65s
90	23L 67s