58edo

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← 57edo58edo59edo →
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:

58 equal divisions of the octave (abbreviated 58edo or 58ed2), also called 58-tone equal temperament (58tet) or 58 equal temperament (58et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 58 equal parts of about 20.7 ¢ each. Each step represents a frequency ratio of 21/58, or the 58th root of 2.

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, thuja temperaments plus a number of gravity family extensions, 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

# Cents Approximate Ratios Ups and downs notation
0 0.00 1/1 D
1 20.69 56/55, 64/63, 81/80, 128/125 ^D, v3E♭
2 41.38 36/35, 49/48, 50/49, 55/54 ^^D, vvE♭
3 62.07 26/25, 27/26, 28/27, 33/32 ^3D, vE♭
4 82.76 25/24, 21/20, 22/21 ^4D, E♭
5 103.45 16/15, 17/16, 18/17 ^5D, v5E
6 124.14 14/13, 15/14, 27/25 D♯, v4E
7 144.83 12/11, 13/12 ^D♯, v3E
8 165.52 11/10 ^^D♯, vvE
9 186.21 10/9 ^3D♯, vE
10 206.90 9/8, 17/15 E
11 227.59 8/7 ^E, v3F
12 248.28 15/13 ^^E, vvF
13 268.97 7/6 ^3E, vF
14 289.66 13/11, 20/17 F
15 310.34 6/5 ^F, v3G♭
16 331.03 17/14 ^^F, vvG♭
17 351.72 11/9, 16/13 ^3F, vG♭
18 372.41 21/17 ^4F, G♭
19 393.10 5/4 ^5F, v5G
20 413.79 14/11 F♯, v4G
21 434.48 9/7 ^F♯, v3G
22 455.17 13/10, 17/13, 22/17 ^^F♯, vvG
23 475.86 21/16 ^3F♯, vG
24 496.55 4/3 G
25 517.24 27/20 ^G, v3A♭
26 537.93 15/11 ^^G, vvA♭
27 558.62 11/8, 18/13 ^3G, vA♭
28 579.31 7/5 ^4G, A♭
29 600.00 17/12, 24/17 ^5G, v5A
30 620.69 10/7 G♯, v4A
31 641.38 13/9, 16/11 ^G♯, v3A
32 662.07 22/15 ^^G♯, vvA
33 682.76 40/27 ^3G♯, vA
34 703.45 3/2 A
35 724.14 32/21 ^A, v3B♭
36 744.83 20/13, 26/17, 17/11 ^^A, vvB♭
37 765.52 14/9 ^3A, vB♭
38 786.21 11/7 ^4A, B♭
39 806.90 8/5 ^5A, v5B
40 827.59 34/21 A♯, v4B
41 848.28 13/8, 18/11 ^A♯, v3B
42 868.97 28/17 ^^A♯, vvB
43 889.66 5/3 ^3A♯, vB
44 910.34 22/13, 17/10 B
45 931.03 12/7 ^B, v3C
46 951.72 26/15 ^^B, vvC
47 972.41 7/4 ^3B, vC
48 993.10 16/9, 30/17 C
49 1013.79 9/5 ^C, v3D♭
50 1034.48 20/11 ^^C, vvD♭
51 1055.17 11/6, 24/13 ^3C, vD♭
52 1075.86 13/7, 28/15 ^4C, D♭
53 1096.55 15/8, 32/17, 17/9 ^5C, v5D
54 1117.24 48/25, 40/21, 21/11 C♯, v4D
55 1137.93 25/13, 52/27, 27/14, 64/33 ^C♯, v3D
56 1158.62 35/18, 96/49, 49/25, 108/55 ^^C♯, vvD
57 1179.31 55/28, 63/32, 160/81, 125/64 ^3C♯, vD
58 1200.00 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 Sagittal natural.png Sagittal pai.png Sagittal kai.png Sagittal pakai.png Sagittal sharp kao.png Sagittal sharp pao.png Sagittal sharp.png

Ups and downs notation

58edo can also be notated using ups and downs notation. In this case, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from Helmholtz–Ellis notation can be used to fill in the gaps.

Step Offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp Symbol
Heji18.svg
Heji19.svg
Heji20.svg
HeQu1.svg
Heji23.svg
Heji24.svg
Heji25.svg
Heji26.svg
Heji27.svg
HeQu3.svg
Heji30.svg
Heji31.svg
Heji32.svg
Heji33.svg
Heji34.svg
Flat Symbol
Heji17.svg
Heji16.svg
HeQd1.svg
Heji13.svg
Heji12.svg
Heji11.svg
Heji10.svg
Heji9.svg
HeQd3.svg
Heji6.svg
Heji5.svg
Heji4.svg
Heji3.svg
Heji2.svg

If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:

Step Offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp Symbol
Heji18.svg
Heji19.svg
HeQu1-sd1.svg
HeQu1.svg
HeQu1-su1.svg
Heji24.svg
Heji25.svg
Heji26.svg
HeQu3-sd1.svg
HeQu3.svg
HeQu3-su1.svg
Heji31.svg
Heji32.svg
Heji33.svg
Flat Symbol
Heji17.svg
HeQd1-su1.svg
HeQd1.svg
HeQd1-sd1.svg
Heji12.svg
Heji11.svg
Heji10.svg
HeQd3-su1.svg
HeQd3.svg
HeQd3-sd1.svg
Heji5.svg
Heji4.svg
Heji3.svg

JI approximation

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 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