104edo: Difference between revisions

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


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=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|104}}
{{Harmonics in equal|104}}
 
=== Subsets and supersets ===
Since 104 factors into 2<sup>3</sup> × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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==== In patent val ====
==== In patent val ====
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods <br> per octave
! Periods <br> per 8ve
! Generator
! Generator
! Cents
! Cents
! Associated ratio
! Associated Ratio
! Temperament
! Temperament
|-
|-
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==== In 104c val ====
==== In 104c val ====
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods <br> per octave
! Periods<br>per 8ve
! Generator <br> (reduced)
! Generator<br>(Reduced)
! Cents <br> (reduced)
! Cents<br>(Reduced)
! Associated ratio <br> (reduced)
! Associated Ratio<br>(Reduced)
! Temperament
! Temperament
|-
|-

Revision as of 06:07, 31 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

Approximation of prime harmonics in 104edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.89 -5.54 +0.40 +2.53 +1.78 -1.11 +2.49 -5.20 -2.65 -2.73
Relative (%) +0.0 +16.4 -48.1 +3.5 +21.9 +15.4 -9.6 +21.6 -45.0 -23.0 -23.6
Steps
(reduced) 		104
(0) 		165
(61) 		241
(33) 		292
(84) 		360
(48) 		385
(73) 		425
(9) 		442
(26) 		470
(54) 		505
(89) 		515
(99)

Subsets and supersets

Since 104 factors into 23 × 13, it has subset edos 2, 4, 8, 13, 26, and 52.

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 8ve 		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 8ve 		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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