104edo: Difference between revisions

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→Theory: +octave stretch
→Intervals: reduce cent values to one decimal place, per discussion on Discord; sort ratios; -some excessively complex ratios
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== Intervals ==
== Intervals ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2" | #
! rowspan="2" | #
! rowspan="2" | Cents
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
! colspan="3" | Approximate ratios
|-
|-
! Of 2.3.7.11.13.17.19.25<br />subgroup
! Of 2.3.7.11.13.17.19.25<br>subgroup
! Additional ratios of 5<br />tending sharp (104c val)
! Additional ratios of 5<br>tending sharp (104c val)
! Additional ratios of 5<br />tending flat (patent val)
! Additional ratios of 5<br>tending flat (patent val)
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| ''[[126/125]]''
|  
| ''[[225/224]]'', ''[[100/99]]''
|  
|-
|-
| 1
| 1
| 11.538
| 11.5
| [[225/224]], [[100/99]]
| [[100/99]], [[225/224]]
|
|
|
|
|-
|-
| 2
| 2
| 23.077
| 23.1
| [[64/63]]
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| [[81/80]], ''[[225/224]]''
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|-
|-
| 3
| 3
| 34.615
| 34.6
| [[49/48]], [[50/49]]
| [[49/48]], [[50/49]]
|
|
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|-
|-
| 4
| 4
| 46.154
| 46.2
|
|
| [[36/35]], ''[[50/49]]''
| [[36/35]], ''[[50/49]]''
Line 199: Line 198:
|-
|-
| 5
| 5
| 57.692
| 57.7
| ''[[28/27]]'', [[33/32]]
| ''[[28/27]]'', [[33/32]]
|
|
Line 205: Line 204:
|-
|-
| 6
| 6
| 69.231
| 69.2
| [[25/24]]
| [[25/24]]
|
|
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|-
|-
| 7
| 7
| 80.769
| 80.8
| [[22/21]]
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]''
| ''[[20/19]]''
|-
|-
| 8
| 8
| 92.308
| 92.3
| [[19/18]]
| [[19/18]]
| [[20/19]]
| [[20/19]]
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|-
|-
| 9
| 9
| 103.846
| 103.8
| [[17/16]], [[18/17]]
| [[17/16]], [[18/17]]
| ''[[16/15]]''
| ''[[16/15]]''
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|-
|-
| 10
| 10
| 115.385
| 115.4
|
|
|
|
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|-
|-
| 11
| 11
| 126.923
| 126.9
| [[14/13]]
| [[14/13]]
| ''[[15/14]]''
| ''[[15/14]]''
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|-
|-
| 12
| 12
| 138.462
| 138.5
| [[13/12]]
| [[13/12]]
|
|
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|-
|-
| 13
| 13
| 150.000
| 150.0
| [[12/11]]
| [[12/11]]
|
|
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|-
|-
| 14
| 14
| 161.538
| 161.5
|
|
| [[11/10]]
| [[11/10]]
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|-
|-
| 15
| 15
| 173.077
| 173.1
| [[21/19]]
| [[21/19]]
|
|
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|-
|-
| 16
| 16
| 184.615
| 184.6
|
|
| [[10/9]]
| [[10/9]]
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|-
|-
| 17
| 17
| 196.154
| 196.2
| [[28/25]], [[19/17]]
| [[19/17]], [[28/25]]
|
|
|
|
|-
|-
| 18
| 18
| 207.692
| 207.7
| [[9/8]]
| [[9/8]]
| ''[[17/15]]''
| ''[[17/15]]''
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|-
|-
| 19
| 19
| 219.231
| 219.2
| [[25/22]]
| [[25/22]]
|
|
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|-
|-
| 20
| 20
| 230.769
| 230.8
| [[8/7]]
| [[8/7]]
|
|
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|-
|-
| 21
| 21
| 242.308
| 242.3
|
|
|
|
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|-
|-
| 22
| 22
| 253.846
| 253.8
| [[22/19]]
| [[22/19]]
| ''[[15/13]]''
| ''[[15/13]]''
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|-
|-
| 23
| 23
| 265.385
| 265.4
| [[7/6]]
| [[7/6]]
|
|
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|-
|-
| 24
| 24
| 276.923
| 276.9
| [[75/64]]
| [[75/64]]
|
|
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|-
|-
| 25
| 25
| 288.462
| 288.5
| [[32/27]], [[13/11]]
| [[32/27]], [[13/11]]
| ''[[20/17]]''
| ''[[20/17]]''
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|-
|-
| 26
| 26
| 300.000
| 300.0
| [[25/21]], [[19/16]]
| [[25/21]], [[19/16]]
|
|
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|-
|-
| 27
| 27
| 311.538
| 311.5
|
|
| [[6/5]]
| [[6/5]]
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|-
|-
| 28
| 28
| 323.077
| 323.1
|
|
|
|
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|-
|-
| 29
| 29
| 334.615
| 334.6
| [[17/14]]
| [[17/14]]
| [[40/33]]
| [[40/33]]
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|-
|-
| 30
| 30
| 346.154
| 346.2
| [[11/9]], [[39/32]]
| [[11/9]], [[39/32]]
|
|
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|-
|-
| 31
| 31
| 357.692
| 357.7
| [[27/22]], [[16/13]]
| [[16/13]], [[27/22]]
|
|
|
|
|-
|-
| 32
| 32
| 369.231
| 369.2
| [[26/21]], [[21/17]]
| [[21/17]], [[26/21]]
|
|
|
|
|-
|-
| 33
| 33
| 380.769
| 380.8
|
|
|
|
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|-
|-
| 34
| 34
| 392.308
| 392.3
|
|
| ''[[5/4]]''
| ''[[5/4]]''
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|-
|-
| 35
| 35
| 403.846
| 403.8
| [[63/50]], [[24/19]]
| [[24/19]], [[63/50]]
| [[19/15]]
| [[19/15]]
|
|
|-
|-
| 36
| 36
| 415.385
| 415.4
| [[81/64]], [[14/11]]
| [[14/11]]
|
|
| ''[[19/15]]''
| ''[[19/15]]''
|-
|-
| 37
| 37
| 426.923
| 426.9
| [[32/25]]
| [[32/25]]
|
|
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|-
|-
| 38
| 38
| 438.462
| 438.5
| [[9/7]]
| [[9/7]]
|
|
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|-
|-
| 39
| 39
| 450.000
| 450.0
| [[22/17]]
| [[22/17]]
| [[13/10]]
| [[13/10]]
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|-
|-
| 40
| 40
| 461.538
| 461.5
| [[17/13]]
| [[17/13]]
|
|
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|-
|-
| 41
| 41
| 473.077
| 473.1
| [[21/16]]
| [[21/16]]
|
|
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|-
|-
| 42
| 42
| 484.615
| 484.6
|
|
|
|
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|-
|-
| 43
| 43
| 496.154
| 496.2
| [[4/3]]
| [[4/3]]
|
|
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|-
|-
| 44
| 44
| 507.692
| 507.7
|
|
|
|
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|-
|-
| 45
| 45
| 519.231
| 519.2
|
|
| [[27/20]]
| [[27/20]]
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|-
|-
| 46
| 46
| 530.769
| 530.8
| [[19/14]]
| [[19/14]]
|
|
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|-
|-
| 47
| 47
| 542.308
| 542.3
| [[26/19]]
| [[26/19]]
| [[15/11]]
| [[15/11]]
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|-
|-
| 48
| 48
| 553.846
| 553.8
| [[11/8]]
| [[11/8]]
|
|
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|-
|-
| 49
| 49
| 565.385
| 565.4
| [[18/13]]
| [[18/13]]
|
|
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|-
|-
| 50
| 50
| 576.923
| 576.9
|
|
| [[7/5]]
| [[7/5]]
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|-
|-
| 51
| 51
| 588.462
| 588.5
|
|
|
|
| [[45/32]], ''[[7/5]]''
| ''[[7/5]]'', [[45/32]]
|-
|-
| 52
| 52
| 600.000
| 600.0
| [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
| ''[[45/32]]'', ''[[64/45]]''

Revision as of 13:36, 9 April 2025

← 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

104 equal divisions of the octave (abbreviated 104edo or 104ed2), also called 104-tone equal temperament (104tet) or 104 equal temperament (104et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 104 equal parts of about 11.5 ¢ each. Each step represents a frequency ratio of 21/104, or the 104th root of 2.

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)

Octave stretch

104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as 269ed6, which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as 165edt.

Subsets and supersets

Since 104 factors into primes as 23 × 13, 104edo 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

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
104c val
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		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
26 43\104
(1\104) 		496.15
(11.54) 		4/3
(225/224) 		Bosonic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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 0.0 1/1
1 11.5 100/99, 225/224
2 23.1 64/63 81/80, 225/224 50/49
3 34.6 49/48, 50/49 81/80, 126/125
4 46.2 36/35, 50/49
5 57.7 28/27, 33/32 25/24, 36/35
6 69.2 25/24
7 80.8 22/21 21/20, 25/24 20/19
8 92.3 19/18 20/19 21/20
9 103.8 17/16, 18/17 16/15
10 115.4 16/15, 15/14
11 126.9 14/13 15/14
12 138.5 13/12
13 150.0 12/11
14 161.5 11/10
15 173.1 21/19 10/9, 11/10
16 184.6 10/9
17 196.2 19/17, 28/25
18 207.7 9/8 17/15
19 219.2 25/22 17/15
20 230.8 8/7
21 242.3 15/13
22 253.8 22/19 15/13
23 265.4 7/6
24 276.9 75/64 20/17
25 288.5 32/27, 13/11 20/17
26 300.0 25/21, 19/16
27 311.5 6/5
28 323.1 6/5, 40/33
29 334.6 17/14 40/33
30 346.2 11/9, 39/32
31 357.7 16/13, 27/22
32 369.2 21/17, 26/21
33 380.8 5/4
34 392.3 5/4
35 403.8 24/19, 63/50 19/15
36 415.4 14/11 19/15
37 426.9 32/25
38 438.5 9/7
39 450.0 22/17 13/10
40 461.5 17/13 13/10
41 473.1 21/16
42 484.6
43 496.2 4/3
44 507.7
45 519.2 27/20
46 530.8 19/14 27/20, 15/11
47 542.3 26/19 15/11
48 553.8 11/8
49 565.4 18/13
50 576.9 7/5
51 588.5 7/5, 45/32
52 600.0 17/12, 24/17 45/32, 64/45
… … … … …
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