104edo

104edo divides the octave into 104 parts of size 11.5385 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
	51\104	588.462	7/5	Untriton
4	9\104	103.846	18/17	Undim

In 104c val

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio (reduced)	Temperament
1	21\104	242.308	147/128	Septiquarter
	27\104	311.538	6/5	Oolong
	47\104	542.308	15/11	Casablanca / marrakesh
2	43\104	496.154	4/3	Diaschismic
8	50\104 (2\104)	576.923 (23.077)	121/84 (78/77)	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
…	…	…	…	…

Since 104edo has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*13edo, it is the first edo where two smaller edos it allows one to use as circulating temperaments are Fibonacci edos.

Circulating temperaments in 104edo

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