104edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 165 -104 }}
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| {{mapping| 104 165 }}
| -0.597
| &minus;0.597
| 0.596
| 0.596
| 5.17
| 5.17
Line 38: Line 30:
| 2048/2025, {{monzo| 0 22 -15 }}
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| {{mapping| 104 165 242 }} (104c)
| -1.258
| &minus;1.258
| 1.054
| 1.054
| 9.14
| 9.14
Line 45: Line 37:
| 126/125, 2048/2025, 117649/116640
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| {{mapping| 104 165 242 292 }} (104c)
| -0.980
| &minus;0.980
| 1.032
| 1.032
| 8.95
| 8.95
Line 52: Line 44:
| 126/125, 176/175, 896/891, 14641/14580
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| {{mapping| 104 165 242 292 360 }} (104c)
| -0.930
| &minus;0.930
| 0.929
| 0.929
| 8.05
| 8.05
Line 59: Line 51:
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| {{mapping| 104 165 242 292 360 385 }} (104c)
| -0.855
| &minus;0.855
| 0.864
| 0.864
| 7.49
| 7.49
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
|+ style="font-size: 105%;" | Patent val
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperament
! Temperament
|-
|-
Line 92: Line 85:
|}
|}


==== In 104c val ====
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
|+ style="font-size: 105%;" | 104c val
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperament
! Temperament
|-
|-
Line 131: Line 125:
|-
|-
| 2
| 2
| 43\104<br>(9\104)
| 43\104<br />(9\104)
| 496.15<br>(103.85)
| 496.15<br />(103.85)
| 4/3<br>(17/16)
| 4/3<br />(17/16)
| [[Diaschismic]]
| [[Diaschismic]]
|-
|-
| 8
| 8
| 49\104<br>(2\104)
| 49\104<br />(2\104)
| 565.38<br>(34.62)
| 565.38<br />(34.62)
| 168/121<br>(55/54)
| 168/121<br />(55/54)
| [[Octowerck]] / octowerckis
| [[Octowerck]] / octowerckis
|-
|-
| 26
| 26
| 43\104<br>(1\104)
| 43\104<br />(1\104)
| 496.15<br>(11.54)
| 496.15<br />(11.54)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Bosonic]]
| [[Bosonic]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Intervals ==
== Intervals ==
Line 154: Line 148:
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2" | #
! rowspan="2" | &#35;
! 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
Line 487: Line 481:
|}
|}


[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Magic]]
[[Category:Necromancy]]
[[Category:Necromancy]]

Revision as of 06:13, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

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

Template:Orf

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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