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.

Rank two temperaments

In patent val

Periods per octave	Generator	Cents	Associated ratio	Temperament
1	33\104	380.769	5/4	Magic / necromancy / divination
1	51\104	588.462	7/5	Untriton
4	9\104	103.846		Undim

In 104c val

Periods per octave	Generator	Cents	Associated ratio	Temperament
1	21\104	242.308	147/128	Septiquarter
1	27\104	311.538	6/5	Oolong
1	47\104	542.308	15/11	Casablanca / marrakesh
2	43\104	496.154	4/3	Diaschismic
8	50\104	576.923	121/84	Octowerck (7- or 11-limit)

Intervals

#	Cents	Approximate Ratios
		of 2.3.7.11.13.17.19.25 Subgroup	Additional Ratios of 5 Tending Sharp (104c Val)	Additional Ratios of 5 Tending Flat (Patent Val)
0	0.000	1/1	126/125	225/224, 100/99
1	11.538	225/224, 100/99
2	23.077	64/63	81/80, 225/224	50/49
3	34.615	49/48, 50/49		81/80, 126/125
4	46.154		36/35, 50/49
5	57.692	28/27, 33/32		25/24, 36/35
6	69.231	25/24
7	80.769	22/21	25/24, 21/20	20/19
8	92.308	19/18	20/19	21/20
9	103.846	17/16, 18/17	16/15
10	115.385			16/15, 15/14
11	126.923	14/13	15/14
12	138.462	13/12
13	150.000	12/11
14	161.538		11/10
15	173.077	21/19		10/9, 11/10
16	184.615		10/9
17	196.154	28/25, 19/17
18	207.692	9/8	17/15
19	219.231	25/22		17/15
20	230.769	8/7
21	242.308			15/13
22	253.846	22/19	15/13
23	265.385	7/6
24	276.923	75/64		20/17
25	288.462	32/27, 13/11	20/17
26	300.000	25/21, 19/16
27	311.538		6/5
28	323.077			6/5
29	334.615	17/14
30	346.154	11/9, 39/32
31	357.692	27/22, 16/13
32	369.231	26/21, 21/17
33	380.769			5/4
34	392.308		5/4
35	403.846	63/50, 24/19	19/15
36	415.385	81/64, 14/11		19/15
37	426.923	32/25
38	438.462	9/7
39	450.000	22/17	13/10
40	461.538	17/13		13/10
41	473.077	21/16
42	484.615
43	496.154	4/3
44	507.692
45	519.231		27/20
46	530.769	19/14		27/20, 15/11
47	542.308	26/19	15/11
48	553.846	11/8
49	565.385	18/13
50	576.923		7/5
51	588.462			45/32, 7/5
52	600.000	17/12, 24/17	45/32, 64/45
…	…	…	…	…