58edo: Difference between revisions

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== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4""
{{Interval table|58edo}}
|-
! #
! Cents
! Approximate Ratios
! [[Ups and downs notation]]
|-
| 0
| 0.00
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 20.69
| [[56/55]], [[64/63]], [[81/80]], [[128/125]]
| {{UDnote|step=1}}
|-
| 2
| 41.38
| [[36/35]], [[49/48]], [[50/49]], [[55/54]]
| {{UDnote|step=2}}
|-
| 3
| 62.07
| [[26/25]], [[27/26]], [[28/27]], [[33/32]]
| {{UDnote|step=3}}
|-
| 4
| 82.76
| [[25/24]], [[21/20]], [[22/21]]
| {{UDnote|step=4}}
|-
| 5
| 103.45
| [[16/15]], [[17/16]], [[18/17]]
| {{UDnote|step=5}}
|-
| 6
| 124.14
| [[14/13]], [[15/14]], [[27/25]]
| {{UDnote|step=6}}
|-
| 7
| 144.83
| [[12/11]], [[13/12]]
| {{UDnote|step=7}}
|-
| 8
| 165.52
| [[11/10]]
| {{UDnote|step=8}}
|-
| 9
| 186.21
| [[10/9]]
| {{UDnote|step=9}}
|-
| 10
| 206.90
| [[9/8]], [[17/15]]
| {{UDnote|step=10}}
|-
| 11
| 227.59
| [[8/7]]
| {{UDnote|step=11}}
|-
| 12
| 248.28
| [[15/13]]
| {{UDnote|step=12}}
|-
| 13
| 268.97
| [[7/6]]
| {{UDnote|step=13}}
|-
| 14
| 289.66
| [[13/11]], [[20/17]]
| {{UDnote|step=14}}
|-
| 15
| 310.34
| [[6/5]]
| {{UDnote|step=15}}
|-
| 16
| 331.03
| [[17/14]]
| {{UDnote|step=16}}
|-
| 17
| 351.72
| [[11/9]], [[16/13]]
| {{UDnote|step=17}}
|-
| 18
| 372.41
| [[21/17]]
| {{UDnote|step=18}}
|-
| 19
| 393.10
| [[5/4]]
| {{UDnote|step=19}}
|-
| 20
| 413.79
| [[14/11]]
| {{UDnote|step=20}}
|-
| 21
| 434.48
| [[9/7]]
| {{UDnote|step=21}}
|-
| 22
| 455.17
| [[13/10]], [[17/13]], [[22/17]]
| {{UDnote|step=22}}
|-
| 23
| 475.86
| [[21/16]]
| {{UDnote|step=23}}
|-
| 24
| 496.55
| [[4/3]]
| {{UDnote|step=24}}
|-
| 25
| 517.24
| [[27/20]]
| {{UDnote|step=25}}
|-
| 26
| 537.93
| [[15/11]]
| {{UDnote|step=26}}
|-
| 27
| 558.62
| [[11/8]], [[18/13]]
| {{UDnote|step=27}}
|-
| 28
| 579.31
| [[7/5]]
| {{UDnote|step=28}}
|-
| 29
| 600.00
| [[17/12]], [[24/17]]
| {{UDnote|step=29}}
|-
| 30
| 620.69
| [[10/7]]
| {{UDnote|step=30}}
|-
| 31
| 641.38
| [[13/9]], [[16/11]]
| {{UDnote|step=31}}
|-
| 32
| 662.07
| [[22/15]]
| {{UDnote|step=32}}
|-
| 33
| 682.76
| [[40/27]]
| {{UDnote|step=33}}
|-
| 34
| 703.45
| [[3/2]]
| {{UDnote|step=34}}
|-
| 35
| 724.14
| [[32/21]]
| {{UDnote|step=35}}
|-
| 36
| 744.83
| [[20/13]], [[26/17]], [[17/11]]
| {{UDnote|step=36}}
|-
| 37
| 765.52
| [[14/9]]
| {{UDnote|step=37}}
|-
| 38
| 786.21
| [[11/7]]
| {{UDnote|step=38}}
|-
| 39
| 806.90
| [[8/5]]
| {{UDnote|step=39}}
|-
| 40
| 827.59
| [[34/21]]
| {{UDnote|step=40}}
|-
| 41
| 848.28
| [[13/8]], [[18/11]]
| {{UDnote|step=41}}
|-
| 42
| 868.97
| [[28/17]]
| {{UDnote|step=42}}
|-
| 43
| 889.66
| [[5/3]]
| {{UDnote|step=43}}
|-
| 44
| 910.34
| [[22/13]], [[17/10]]
| {{UDnote|step=44}}
|-
| 45
| 931.03
| [[12/7]]
| {{UDnote|step=45}}
|-
| 46
| 951.72
| [[26/15]]
| {{UDnote|step=46}}
|-
| 47
| 972.41
| [[7/4]]
| {{UDnote|step=47}}
|-
| 48
| 993.10
| [[16/9]], [[30/17]]
| {{UDnote|step=48}}
|-
| 49
| 1013.79
| [[9/5]]
| {{UDnote|step=49}}
|-
| 50
| 1034.48
| [[20/11]]
| {{UDnote|step=50}}
|-
| 51
| 1055.17
| [[11/6]], [[24/13]]
| {{UDnote|step=51}}
|-
| 52
| 1075.86
| [[13/7]], [[28/15]]
| {{UDnote|step=52}}
|-
| 53
| 1096.55
| [[15/8]], [[32/17]], [[17/9]]
| {{UDnote|step=53}}
|-
| 54
| 1117.24
| [[48/25]], [[40/21]], [[21/11]]
| {{UDnote|step=54}}
|-
| 55
| 1137.93
| [[25/13]], [[52/27]], [[27/14]], [[64/33]]
| {{UDnote|step=55}}
|-
| 56
| 1158.62
| [[35/18]], [[96/49]], [[49/25]], [[108/55]]
| {{UDnote|step=56}}
|-
| 57
| 1179.31
| [[55/28]], [[63/32]], [[160/81]], [[125/64]]
| {{UDnote|step=57}}
|-
| 58
| 1200.00
| [[2/1]]
| {{UDnote|step=58}}
|}


== Notation ==
== Notation ==

Revision as of 23:15, 19 January 2024

← 57edo 58edo 59edo →
Prime factorization 2 × 29
Step size 20.6897 ¢ 
Fifth 34\58 (703.448 ¢) (→ 17\29)
Semitones (A1:m2) 6:4 (124.1 ¢ : 82.76 ¢)
Consistency limit 17
Distinct consistency limit 11
English Wikipedia has an article on:

Template:EDO intro

Theory

58edo is a strong system in the 11-, 13- and 17-limit. It is the smallest edo which is consistent through the 17-odd-limit, and is also the smallest distinctly consistent in the 11-odd-limit (the first equal temperament to map the entire 11-odd-limit tonality diamond to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note Genesis scale of Harry Partch.

58et tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539. It supports hemififths, myna, diaschismic, harry, mystery, buzzard and thuja temperaments, and supplies the optimal patent val for the 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments thrush, bluebird, aplonis and jofur.

While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system.

Of all edos which map the syntonic comma (81/80) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1 ¢ narrower than the just interval.

Prime harmonics

Approximation of prime harmonics in 58edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.49 +6.79 +3.59 +7.30 +7.75 -1.51 -7.86 -7.58 +4.91 -7.10
Relative (%) +0.0 +7.2 +32.8 +17.3 +35.3 +37.4 -7.3 -38.0 -36.7 +23.7 -34.3
Steps
(reduced) 		58
(0) 		92
(34) 		135
(19) 		163
(47) 		201
(27) 		215
(41) 		237
(5) 		246
(14) 		262
(30) 		282
(50) 		287
(55)

Subsets and supersets

58 = 2 × 29, and 58edo shares the same excellent fifth with 29edo.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 20.7 ^D, v3E♭
2 41.4 ^^D, vvE♭
3 62.1 27/26, 28/27, 29/28, 30/29 ^3D, vE♭
4 82.8 21/20, 22/21 vvD♯, E♭
5 103.4 17/16, 18/17 vD♯, ^E♭
6 124.1 14/13, 29/27 D♯, ^^E♭
7 144.8 ^D♯, v3E
8 165.5 11/10 ^^D♯, vvE
9 186.2 10/9, 29/26 ^3D♯, vE
10 206.9 9/8 E
11 227.6 8/7, 33/29 ^E, v3F
12 248.3 15/13 ^^E, vvF
13 269 7/6 ^3E, vF
14 289.7 13/11, 32/27 F
15 310.3 ^F, v3G♭
16 331 23/19, 29/24 ^^F, vvG♭
17 351.7 11/9, 27/22 ^3F, vG♭
18 372.4 26/21 vvF♯, G♭
19 393.1 vF♯, ^G♭
20 413.8 14/11, 33/26 F♯, ^^G♭
21 434.5 9/7 ^F♯, v3G
22 455.2 13/10 ^^F♯, vvG
23 475.9 29/22 ^3F♯, vG
24 496.6 4/3 G
25 517.2 27/20, 31/23 ^G, v3A♭
26 537.9 15/11 ^^G, vvA♭
27 558.6 29/21 ^3G, vA♭
28 579.3 7/5 vvG♯, A♭
29 600 17/12, 24/17 vG♯, ^A♭
30 620.7 10/7 G♯, ^^A♭
31 641.4 29/20 ^G♯, v3A
32 662.1 22/15 ^^G♯, vvA
33 682.8 ^3G♯, vA
34 703.4 3/2 A
35 724.1 ^A, v3B♭
36 744.8 20/13 ^^A, vvB♭
37 765.5 14/9 ^3A, vB♭
38 786.2 11/7 vvA♯, B♭
39 806.9 vA♯, ^B♭
40 827.6 21/13, 29/18 A♯, ^^B♭
41 848.3 18/11, 31/19 ^A♯, v3B
42 869 33/20 ^^A♯, vvB
43 889.7 ^3A♯, vB
44 910.3 22/13, 27/16 B
45 931 12/7 ^B, v3C
46 951.7 26/15 ^^B, vvC
47 972.4 7/4 ^3B, vC
48 993.1 16/9 C
49 1013.8 9/5 ^C, v3D♭
50 1034.5 20/11 ^^C, vvD♭
51 1055.2 ^3C, vD♭
52 1075.9 13/7 vvC♯, D♭
53 1096.6 17/9, 32/17 vC♯, ^D♭
54 1117.2 21/11 C♯, ^^D♭
55 1137.9 27/14, 29/15 ^C♯, v3D
56 1158.6 ^^C♯, vvD
57 1179.3 ^3C♯, vD
58 1200 2/1 D

Notation

Sagittal

The following table shows sagittal notation accidentals in one apotome for 58edo.

Steps 0 1 2 3 4 5 6
Symbol

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 58edo. Prime harmonics are in bold. As 58edo is consistent in the 15-odd-limit, the results by direct approximation and patent val mapping are the same. The following table shows how 15-odd-limit intervals are represented in 58edo. Prime harmonics are in bold.

As 58edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 58edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/11, 22/13 0.445 2.2
11/10, 20/11 0.513 2.5
15/13, 26/15 0.535 2.6
9/7, 14/9 0.601 2.9
13/10, 20/13 0.958 4.6
15/11, 22/15 0.980 4.7
3/2, 4/3 1.493 7.2
7/6, 12/7 2.095 10.1
9/8, 16/9 2.987 14.4
7/5, 10/7 3.202 15.5
7/4, 8/7 3.588 17.3
11/7, 14/11 3.715 18.0
9/5, 10/9 3.803 18.4
13/7, 14/13 4.160 20.1
11/9, 18/11 4.316 20.9
15/14, 28/15 4.695 22.7
13/9, 18/13 4.762 23.0
5/3, 6/5 5.296 25.6
11/6, 12/11 5.809 28.1
13/12, 24/13 6.255 30.2
5/4, 8/5 6.790 32.8
11/8, 16/11 7.303 35.3
13/8, 16/13 7.748 37.4
15/8, 16/15 8.283 40.0

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5 2048/2025, 1594323/1562500 [⟨58 92 135]] -1.29 1.22 5.89
2.3.5.7 126/125, 1728/1715, 2048/2025 [⟨58 92 135 163]] -1.29 1.05 5.10
2.3.5.7.11 126/125, 176/175, 243/242, 896/891 [⟨58 92 135 163 201]] -1.45 1.00 4.83
2.3.5.7.11.13 126/125, 144/143, 176/175, 196/195, 364/363 [⟨58 92 135 163 201 215]] -1.56 0.94 4.56
2.3.5.7.11.13.17 126/125, 136/135, 144/143, 176/175, 196/195, 364/363 [⟨58 92 135 163 201 215 237]] -1.28 1.10 5.33

58et has a lower relative error than any previous equal temperaments in the 13-limit, and the next equal temperament that does better in this subgroup is 72.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Period
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated Ratio
(Reduced) 		Temperament
1 3\58 62.07 28/27 Unicorn / alicorn / qilin
1 11\58 227.59 8/7 Gorgik
1 13\58 268.97 7/6 Infraorwell
1 15\58 310.34 6/5 Myna
1 17\58 351.72 49/40 Hemififths
1 19\58 393.10 64/51 Emmthird
1 23\58 475.86 21/16 Buzzard / subfourth
1 27\58 558.62 11/8 Thuja
2 3\58 62.07 28/27 Monocerus
2 1\58 20.69 81/80 Commatic
2 9\58 186.21 10/9 Secant
2 17\58
(12\58) 		351.72
(248.28) 		11/9
(15/13) 		Sruti
2 21\58
(8\58) 		434.48
(165.52) 		9/7
(11/10) 		Echidna
2 24\58
(5\58) 		496.55
(103.45) 		4/3
(17/16) 		Diaschismic
2 25\58
(4\58) 		517.24
(82.76) 		27/20
(21/20) 		Harry
29 19\58
(1\58) 		393.10
(20.69) 		5/4
(91/90) 		Mystery

58et can also be detempered to semihemi (58 & 140), supers (58 & 152), condor (58 & 159), and eagle (58 & 212).

Scales

Instruments

Music

Jeff Brown
Cam Taylor
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