# 58edo

 ← 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
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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.

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

As an equal temperament, 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.

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

58edo contains 2edo and 29edo as subsets.

## 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

### Ups and downs notation

In 58edo, 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 Sharp Symbol Flat Symbol 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

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

 Step Offset Sharp Symbol Flat Symbol 0 1 2 3 4 5 6 7 8 9 10 11 12 13

### Sagittal

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

 Step Offset Symbol 0 1 2 3 4 5 6

### Hemipyth notation

Hemipyth notation for 58edo (SW3-style)
# Cents Note Names
on D
0 0.0 D
2 41.4 α𝄳
5 103.4 α
7 144.8 E𝄳
10 206.9 E
12 248.3 β𝄳
14 289.7 F
15 310.3 β
17 351.7 F‡
19 393.1 γ
22 455.2 γ‡
24 496.6 G
27 558.6 G‡
29 600.0 δ
31 641.4 A𝄳
34 703.4 A
36 744.8 ε𝄳
39 806.9 ε
41 848.3 B𝄳
43 889.7 ζ
44 910.3 B
46 951.7 ζ‡
48 993.1 C
51 1055.2 C‡
53 1096.6 η
56 1158.6 η‡
58 1200.0 D

## Approximation to JI

### 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* Cents* Associated
Ratio*
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

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

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

Jeff Brown
Cam Taylor