104edo

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← 103edo 104edo 105edo →
Prime factorization 23 × 13
Step size 11.5385 ¢ 
Fifth 61\104 (703.846 ¢)
Semitones (A1:m2) 11:7 (126.9 ¢ : 80.77 ¢)
Consistency limit 3
Distinct consistency limit 3

The 104 equal divisions of the octave (104edo), or the 104(-tone) equal temperament (104tet, 104et) when viewed from a regular temperament perspective, divides the octave into 104 parts of size about 11.5 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-3 temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank-4 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, 352/351 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, 10648/10647, 16807/16731, 20449/20412, 21632/21609, and 26411/26364. 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.

Prime harmonics

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Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [165 -104⟩ [⟨104 165]] -0.597 0.596 5.17
2.3.5 2048/2025, [0 22 -15⟩ [⟨104 165 242]] (104c) -1.258 1.054 9.14
2.3.5.7 126/125, 2048/2025, 117649/116640 [⟨104 165 242 292]] (104c) -0.980 1.032 8.95
2.3.5.7.11 126/125, 176/175, 896/891, 14641/14580 [⟨104 165 242 292 360]] (104c) -0.930 0.929 8.05
2.3.5.7.11.13 126/125, 176/175, 196/195, 364/363, 2197/2187 [⟨104 165 242 292 360 385]] (104c) -0.855 0.864 7.49

Rank-2 temperaments

In patent val

Periods
per octave 		Generator Cents Associated ratio Temperament
1 33\104 380.77 5/4 Magic / necromancy / divination
1 51\104 588.46 7/5 Untriton
4 9\104 103.85 18/17 Undim

In 104c val

Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated ratio
(reduced) 		Temperament
1 11\104 126.92 27/25 Mowgli
1 21\104 242.31 147/128 Septiquarter
1 27\104 311.54 6/5 Oolong
1 47\104 542.31 15/11 Casablanca / marrakesh
2 21\104 242.31 121/105 Semiseptiquarter
2 43\104
(9\104) 		496.15
(103.85) 		4/3
(17/16) 		Diaschismic
8 49\104
(2\104) 		565.38
(34.62) 		168/121
(55/54) 		Octowerck / octowerckis

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, 40/33
29 334.615 17/14 40/33
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
… … … … …

Scales

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

Circulating temperaments in 104 EDO
Tones Pattern L:s
5 4L 1s 21:20
6 2L 4s 18:17
7 6L 1s 15:14
8 8 EDO equal
9 5L 4s 12:11
10 4L 6s 11:10
11 5L 6s 10:9
12 8L 4s 9:8
13 13 EDO 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 26 EDO 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 52 EDO 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
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