103edo

Prime factorization

103 (prime)

Step size

11.6505¢

Fifth

60\103 (699.029¢)

Semitones (A1:m2)

8:9 (93.2¢ : 104.9¢)

Consistency limit

7

Distinct consistency limit

7

103 equal divisions of the octave (103edo), or 103-tone equal temperament (103tet), 103 equal temperament (103et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 103 equal parts of about 11.7 ¢ each.

Theory

103edo is a good miracle tuning, especially for the 7-, and for 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.

In 103edo, all intervals within the 17-odd-limit are consistent, with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

Prime harmonics

Approximation of prime harmonics in 103edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0.00	-2.93	-1.85	-1.84	-3.75	-1.69	-0.10	+5.40	+0.85	-4.33	-3.29
Error	relative (%)	+0	-25	-16	-16	-32	-15	-1	+46	+7	-37	-28
Steps (reduced)		103 (0)	163 (60)	239 (33)	289 (83)	356 (47)	381 (72)	421 (9)	438 (26)	466 (54)	500 (88)	510 (98)

Subsets and supersets

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
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	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 8ve	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	37\103	431.06	77/60	Lockerbie
1	38\103	442.708	162/125	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

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¢

Music

Forest Tribe Dance by Francium