104edo: Difference between revisions

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== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.
{| class="wikitable center-all"
|+ 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]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 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
| rowspan="17" | 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
| rowspan="31" | 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
|}
|}


Revision as of 21:12, 30 May 2023

← 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

Script error: No such module "primes_in_edo".

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
… … … … …
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