104edo

104edo divides the octave into 104 parts of size 11.54 cents each.

Theory

104edo has two different equally viable 5-limit vals, and both are useful. The flat major third val, ⟨104 165 241] (patent val), tempers out 3125/3072, and supports magic temperament. The sharp major third val, ⟨104 165 242] (104c val), tempers out 2048/2025 and supports diaschismic temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the optimal patent val for 11-limit magic and the 13-limit magic extension necromancy. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

17-limit Regular Temperaments

todo: clarify this table

Degree	Cents
2	23.08
3	34.615
4	46.15
5	57.69
7	80.77
8	92.31
9	103.85
10	115.385
11	126.92
12	138.46
13	150
14	161.54
15	173.08
16	184.615
17	196.15
18	207.69
20	230.77
21	242.31
22	253.85
23	265.385
25	288.46
26	300
27	311.54
28	323.08
29	334.615
30	346.15
31	357.69
32	369.23
33	380.77
34	392.31
35	403.85
36	415.385
38	438.46
39	450
40	461.54
41	473.08
43	496.15
45	519.23
46	530.77
47	542.31
48	553.85
50	576.92
51	588.45
52	600
53	611.54
54	623.08
56	646.15
57	657.69
58	669.23
59	680.77
61	703.85
63	726.92
64	738.46
65	750
66	761.54
67	773.08
68	784.615
69	796.15
70	807.69
71	819.23
72	830.77
73	842.31
74	853.85
75	865.385
76	876.92
77	888.46
78	900
79	911.54
81	934.615
82	946.15
83	957.69
84	969.23
86	992.31
87	1003.85
88	1015.385
89	1026.92
90	1038.46
91	1050
92	1061.54
93	1073.08
95	1096.15
96	1107.69
97	1119.23
99	1142.31
100	1153.85
101	1165.385
102	1176.92