103edo

The 103 equal divisions of the octave (103edo), or the 103(-tone) equal temperament (103tet, 103et) when viewed from a regular temperament perspective, is the equal division of the octave into 103 steps of size 11.6505 cents each.

Theory

103edo is a good miracle tuning, especially for the 7- and 13-limit and benediction and hemisecordite, two of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the 11-limit; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for marvel temperament as well as benediction and hemisecordite.

103edo is the 27th prime edo.

Intervals

Degree	Cents	Approximate Ratios
1	11.650	81/80, 126/125
2	23.301	65/64, 66/65, 78/77
3	34.951	49/48, 50/49, 64/63
4	46.602	33/32, 35/34, 36/35
5	58.252	27/26, 34/33
6	69.903	25/24, 26/25, 28/27
7	81.553	21/20, 22/21
8	93.204	18/17
9	104.854	17/16
10	116.505	15/14, 16/15
11	128.155	14/13
12	139.806	13/12
13	151.456	12/11
14	163.107	11/10
15	174.757	72/65
16	186.408	10/9
17	198.058	9/8
18	209.708
19	221.359	17/15, 25/22
20	233.010	8/7
21	244.660	15/13
22	256.311
23	267.961	7/6
24	279.712	20/17
25	291.262	13/11
26	303.013	25/21
27	314.563	6/5
28	326.214	63/52, 65/54
29	337.864	17/14, 39/32
30	349.615	11/9, 27/22
31	361.165	16/13, 21/17
32	372.816	26/21, 81/65
33	384.466	5/4
34	396.117	44/35
35	407.767	33/26
36	419.417	14/11
37	431.068	9/7
38	442.708	22/17
39	454.369	13/10
40	466.019	17/13, 21/16
41	477.670
42	489.320	65/49
43	500.971	4/3
44	512.621	27/20
45	524.272	65/48
46	535.922	15/11
47	547.573	11/8
48	559.223	18/13
49	570.874	25/18
50	582.524	7/5
51	594.175	24/17
…	…	…

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-163 103⟩	[⟨103 166]]	+0.923	0.924	7.92
2.3.5	78732/78125, 34171875/33554432	[⟨103 166 239]]	+0.881	0.757	6.49
2.3.5.7	225/224, 1029/1024, 78732/78125	[⟨103 166 239 289]]	+0.824	0.663	5.68
2.3.5.7.11	225/224, 243/242, 385/384, 43923/43750	[⟨103 166 239 289 356]]	+0.876	0.602	5.16
2.3.5.7.11.13	225/224, 243/242, 351/350, 385/384, 847/845	[⟨103 166 239 289 356 381]]	+0.806	0.571	4.90
2.3.5.7.11.13.17	225/224, 243/242, 273/272, 351/350, 375/374, 847/845	[⟨103 166 239 289 356 381 421]]	+0.694	0.595	5.10

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	3\103	34.951	1990656/1953125	Gammic (5-limit)
1	5\103	58.252	27/26	Hemisecordite
1	9\103	104.854	17/16	Septendesemi
1	10\103	116.505	15/14~16/15	Miracle / benediction
1	16\103	186.408	10/9	Mintone
1	20\103	233.010	8/7	Slendric
1	21\103	244.660	15/13	Subsemifourth
1	26\103	303.013	25/21	Quinmite
1	31\103	361.165	16/13	Phicordial
1	38\103	442.708	162/125	Sensipent / sensei
1	39\103	454.369	13/10	Fibo
1	40\103	466.019	55/42	Hemiseptisix
1	42\103	489.320	65/49	Catafourth
1	45\103	524.272	65/48	Widefourth
1	47\103	547.573	11/8	Heinz
1	48\103	559.223	242/175	Tritriple
1	50\103	582.524	7/5	Neptune

Scales

Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments^{[clarification needed]}.

Circulating temperaments in 103edo

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

13-limit temperaments

Marvel and Benediction			Hemisecordite
Degree	cents	Difference from 72edo	Degree	cents	Difference from 62edo
1	11.6505	-5.016¢	2	23.301	3.946¢
3	34.9515	1.618¢	3	34.9515	-3.758¢
4	46.602	-3.398¢	5	58.252	0.188¢
6	69.903	3.236¢	7	81.553	4.134¢
7	81.553	-1.78¢	8	93.204	-3.57¢
9	104.854	4.854¢	10	116.505	0.376¢
10	116.5045	-0.162¢	12	139.806	4.322¢
11	128.155	-5.178¢	13	151.456	-3.382¢
13	151.456¢	1.456¢	15	174.757	0.563¢
14	163.107¢	-3.56¢	17	198.058	4.51¢
16	186.408	3.074¢	18	209.709	-3.1945¢
17	198.058	-1.942¢	20	233.01	0.751¢
19	221.359	4.693¢	22	256.311	4.698¢
20	233.01	-0.324¢	23	267.961	-3.007¢
21	244.66	-5.34¢	25	291.262	0.94¢
23	267.961	1.2945¢	27	314.563	4.886¢
24	279.612	-3.722¢	28	326.214	-2.819¢
26	302.913	2.913¢	30	349.515	1.1275¢
27	314.563	-2.104¢	32	372.8155	5.074¢
29	337.864	4.531¢	33	384.466	-2.631¢
30	349.515	-0.485¢	35	407.767	1.315¢
31	361.165	-5.502¢	37	431.068	5.2615¢
33	384.466	1.133¢	38	442.718	-2.443¢
34	396.1165	-3.8835¢	40	466.0190	1.503¢
36	419.4175	2.751¢	42	489.32	5.449¢
37	431.068	-2.265¢	43	500.971	-2.255¢
39	454.369	4.369¢	45	524.272	1.691¢
40	466.019	-0.647¢	47	547.573	5.637¢
41	477.67	-5.663¢	48	559.223	-2.067¢
43	500.971	0.971¢	50	582.524	1.879¢
44	512.621¢	-4.045¢	52	605.825	5.825¢
46	535.922¢	2.589¢	53	617.476	-1.879¢
47	547.573¢	-2.427¢	55	640.777¢	2.067
49	570.874¢	4.207¢	56	652.427	-5.637¢
50	582.524	-0.809¢	58	675.728	-1.691
52	605.825	5.825¢	60	699.029	2.255¢
53	617.475	0.809¢	61	710.68	-5.449¢
54	629.126¢	-4.207¢	63	733.981	-1.503¢
56	652.427¢	2.427¢	65	757.282	2.443¢
57	664.078	-2.589¢	66	768.932	-5.2615¢
59	687.379	4.045¢	68	792.233	-1.315¢
60	699.029	-0.971¢	70	815.534	2.631¢
62	722.33	5.663¢	71	827.1845	-5.074¢
63	733.981	0.647¢	73	850.485	-1.1275¢
64	745.631	-4.369¢	75	873.786	2.819¢
66	768.932	2.265¢	76	885.437	-4.886¢
67	780.5825	-2.751¢	78	908.738	-0.94¢
69	803.8835	3.8835¢	80	932.039	3.007¢
70	815.534	-1.133¢	81	943.689	-4.698¢
72	838.835	5.501¢	83	966.99	-0.752¢
73	850.485	0.485¢	85	990.291	3.1945¢
74	862.136	-4.531¢	86	1001.942	-4.51¢
76	885.439	2.104¢	88	1025.243	-0.564¢
77	897.087	-2.913¢	90	1048.544	3.382¢
79	920.388	3.722¢	91	1060.194	-4.322¢
80	932.039	-1.2945¢	93	1083.495	-0.376¢
82	955.34	5.34¢	95	1106.796	3.57¢
83	966.99	0.324¢	96	1118.447¢	-4.134¢
84	978.641	-4.693¢	98	1141.748	-0.188¢
86	1001.942	1.942¢	100	1165.0485	3.758¢
87	1013.592	-3.074¢	101	1176.699	-3.946¢
89	1036.893	3.56¢
90	1048.544	-1.456¢
92	1071.845	5.178¢
93	1083.495	0.162¢
94	1095.146	-4.854¢
96	1118.447	1.78¢
97	1130.097	-3.236¢
99	1153.398	3.398¢
100	1165.0485	-1.618¢
102	1188.3495	5.016¢