114edo

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114edo

115edo →

Prime factorization	2 × 3 × 19
Step size	10.5263 ¢
Fifth	67\114 (705.263 ¢)
Semitones (A1:m2)	13:7 (136.8 ¢ : 73.68 ¢)
Consistency limit	7
Distinct consistency limit	7

114edo is the equal division of the octave into 114 parts, each of 10.52632 cents. In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120, 176/175 and 117440512/117406179, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for shrutar temperament; it is in fact the optimal patent val for shrutar in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

Prime harmonics

Approximation of odd harmonics in 114edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+3.31	+3.16	-0.40	-3.91	-3.95	+1.58	-4.06	+0.31	-2.78	+2.90	+3.30
	Relative (%)	+31.4	+30.0	-3.8	-37.1	-37.5	+15.0	-38.6	+2.9	-26.4	+27.6	+31.4
Steps (reduced)		181 (67)	265 (37)	320 (92)	361 (19)	394 (52)	422 (80)	445 (103)	466 (10)	484 (28)	501 (45)	516 (60)

Period of 19-limit Shrutar

Degree	Cents	Difference from 68edo
2	21.05263	3.40557¢
3	31.57895	-3.71517¢
5	52.63158	-0.3096¢
7	73.68421	3.096¢
8	84.21053	-4.02477¢
10	105.26316	-0.619195¢
12	126.31579	2.78638¢
13	136.842105	-4.334365¢
15	157.89474	-0.9288¢
17	178.94737	2.47678¢
18	189.47369	-4.644¢
20	210.52632	-1.23839¢
22	231.57895	2.16718¢
23	242.10526	-4.953560372
25	263.157895	-1.548¢
27	284.21053	1.857585¢
29	305.26316	5.26316¢
30	315.78947	-1.857585¢
32	336.842105	1.548¢
34	357.89474	4.95356¢
35	368.42105	-2.16718¢
37	389.47368	1.23839¢
39	410.52632	4.64396¢
40	421.05263	-2.47678¢
42	442.10526	0.92879¢
44	463.157895	4.334365¢
45	473.68421	-2.78638¢
47	494.73684	0.619195¢
49	515.78947	4.02477¢
50	526.31579	-3.095975¢
52	547.36842	0.3096¢
54	568.42105	3.71517¢
55	578.94737	-3.40557¢

Circulating temperaments

Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.

Circulating temperaments in 114edo

Tones	Pattern	L:s
5	4L 1s	23:22
6	6edo	equal
7	2L 5s	17:16
8	2L 6s	15:14
9	6L 3s	13:12
10	4L 6s	12:11
11	4L 7s	11:10
12	6L 6s	10:9
13	10L 3s	9:8
14	2L 12s
15	9L 6s	8:7
16	2L 14s
17	12L 5s	7:6
18	6L 12s
19	19edo	equal
20	14L 6s	6:5
21	9L 12s
22	4L 18s
23	22L 1s	5:4
24	18L 6s
25	14L 11s
26	10L 16s
27	6L 21s
28	2L 26s
29	27L 2s	4:3
30	24L 6s
31	21L 10s
32	18L 14s
33	15L 18s
34	12L 22s
35	9L 26s
36	6L 30s
37	3L 34s
38	38edo	equal
39	36L 3s	3:2
40	34L 6s
41	32L 9s
42	30L 12s
43	28L 15s
44	26L 18s
45	24L 21s
46	22L 24s
47	20L 27s
48	18L 30L
49	16L 33s
50	14L 36s
51	12L 39s
52	10L 42s
53	8L 45s
54	6L 48s
55	4L 52s
56	2L 54s
57	57edo	equal
58	56L 2s	2:1
59	55L 4s
60	54L 6s
61	53L 8s
62	52L 10s
63	51L 12s
64	50L 14s
65	49L 16s
66	48L 18s
67	47L 20s
68	46L 22s
69	45L 24s
70	44L 26s
71	43L 28s
72	42L 30s
73	41L 32s
74	40L 34s
75	39L 36s
76	38L 38s
77	37L 40s
78	36L 42s
79	35L 44s
80	34L 46s
81	33L 48s
82	32L 50s
83	31L 52s
84	30L 54s
85	29L 56s
86	28L 58s
87	27L 60s
88	26L 62s
89	25L 64s
90	24L 66s
91	23L 68s