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== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
58edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.  
[[Stein–Zimmermann–Gould notation]] for 58edo uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, then arrows can be attached to quartertone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
58edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.  
{{Ups and downs sharpness}}
{{Ups and downs sharpness}}


Half-sharps and half-flats can be used to avoid triple arrows:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Ups and downs sharpness|58|true}}
{{Ups and downs sharpness|58|true}}
Alternatively, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.
{{Sharpness-sharp6}}
If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}


=== Ivan Wyschnegradsky's notation ===
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
{{Sharpness-sharp6-iw}}
{{Sharpness-sharp6-iw}}


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|+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)
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! Note names<br>on D
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Revision as of 14:23, 12 May 2026

← 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:

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. Since 58 = 2 × 29, 58edo shares the same excellent perfect fifth with 29edo. It is the last edo to have exactly one diatonic perfect fifth and no 5edo or 7edo fifths.

The 19th and 23rd harmonics are very flat, so it makes sense to use their second-best approximations in the 58hi val. This val is, in fact, one of the first to be diamond monotone in the 23-odd-limit, past the idiosyncratic 53e val. However, its accuracy is questionable, with primes 19 and 23 being about 13 cents sharp, so one may want to use a larger system like 62edo for the 23-limit instead, which has the added benefit of being meantone.

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)

As a tuning of other temperaments

As an equal temperament, 58et tempers out 2048/2025 in the 5-limit; 126/125, 1728/1715, and 5120/5103 in the 7-limit; 176/175, 243/242, 441/440, 540/539, and 896/891 in the 11-limit; 144/143, 351/350, 364/363 in the 13-limit. 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.

Subsets and supersets

58edo contains 2edo and 29edo as subsets.

Intervals

# Cents Approximate ratios* Ups and downs notation
0 0.0 1/1 D
1 20.7 56/55, 64/63, 81/80, 91/90, 105/104 ^D, v3E♭
2 41.4 36/35, 40/39, 45/44, 49/48, 50/49, 55/54 ^^D, vvE♭
3 62.1 26/25, 27/26, 28/27, 33/32 ^3D, vE♭
4 82.8 21/20, 22/21, 25/24 vvD♯, E♭
5 103.4 16/15, 17/16, 18/17 vD♯, ^E♭
6 124.1 14/13, 15/14 D♯, ^^E♭
7 144.8 12/11, 13/12 ^D♯, v3E
8 165.5 11/10 ^^D♯, vvE
9 186.2 10/9 ^3D♯, vE
10 206.9 9/8, 17/15 E
11 227.6 8/7 ^E, v3F
12 248.3 15/13 ^^E, vvF
13 269.0 7/6 ^3E, vF
14 289.7 13/11, 20/17 F
15 310.3 6/5 ^F, v3G♭
16 331.0 17/14, 40/33 ^^F, vvG♭
17 351.7 11/9, 16/13 ^3F, vG♭
18 372.4 21/17, 26/21 vvF♯, G♭
19 393.1 5/4 vF♯, ^G♭
20 413.8 14/11 F♯, ^^G♭
21 434.5 9/7 ^F♯, v3G
22 455.2 13/10, 17/13, 22/17 ^^F♯, vvG
23 475.9 21/16 ^3F♯, vG
24 496.6 4/3 G
25 517.2 27/20 ^G, v3A♭
26 537.9 15/11 ^^G, vvA♭
27 558.6 11/8, 18/13 ^3G, vA♭
28 579.3 7/5 vvG♯, A♭
29 600.0 17/12, 24/17 vG♯, ^A♭
30 620.7 10/7 G♯, ^^A♭
31 641.4 13/9, 16/11 ^G♯, v3A
32 662.1 22/15 ^^G♯, vvA
33 682.8 40/27 ^3G♯, vA
34 703.4 3/2 A
35 724.1 32/21 ^A, v3B♭
36 744.8 17/11, 20/13, 26/17 ^^A, vvB♭
37 765.5 14/9 ^3A, vB♭
38 786.2 11/7 vvA♯, B♭
39 806.9 8/5 vA♯, ^B♭
40 827.6 21/13, 34/21 A♯, ^^B♭
41 848.3 13/8, 18/11 ^A♯, v3B
42 869.0 28/17, 33/20 ^^A♯, vvB
43 889.7 5/3 ^3A♯, vB
44 910.3 17/10, 22/13 B
45 931.0 12/7 ^B, v3C
46 951.7 26/15 ^^B, vvC
47 972.4 7/4 ^3B, vC
48 993.1 16/9, 30/17 C
49 1013.8 9/5 ^C, v3D♭
50 1034.5 20/11 ^^C, vvD♭
51 1055.2 11/6, 24/13 ^3C, vD♭
52 1075.9 13/7, 28/15 vvC♯, D♭
53 1096.6 15/8, 17/9, 32/17 vC♯, ^D♭
54 1117.2 21/11, 40/21, 48/25 C♯, ^^D♭
55 1137.9 25/13, 27/14, 52/27, 64/33 ^C♯, v3D
56 1158.6 35/18, 39/20, 49/25, 88/45, 96/49, 108/55 ^^C♯, vvD
57 1179.3 55/28, 63/32, 160/81, 180/91, 208/105 ^3C♯, vD
58 1200.0 2/1 D

* As a 17-limit temperament, inconsistently mapped intervals in italic

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation for 58edo uses sharps and flats combined with quartertone accidentals and arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

If double arrows are not desirable, then arrows can be attached to quartertone accidentals:

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

Kite's ups and downs notation

58edo can also be notated with Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

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

Half-sharps and half-flats can be used to avoid triple arrows:

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

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

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

Sagittal notation

Evo flavor

58-EDO_Evo_Sagittal.svg

Evo-SZ flavor

58-EDO_Evo-SZ_Sagittal.svg

Revo flavor

58-EDO_Revo_Sagittal.svg

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
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 3\58 62.1 28/27 Unicorn / alicorn / qilin
1 11\58 227.6 8/7 Gorgik
1 13\58 269.0 7/6 Infraorwell
1 15\58 310.3 6/5 Myna
1 17\58 351.7 49/40 Hemififths
1 19\58 393.1 64/51 Emmthird
1 23\58 475.9 21/16 Buzzard / subfourth
1 27\58 558.6 11/8 Thuja
2 3\58 62.1 28/27 Monocerus
2 1\58 20.7 81/80 Bicommatic
2 9\58 186.2 10/9 Secant
2 17\58
(12\58) 		351.7
(248.3) 		11/9
(15/13) 		Sruti
2 21\58
(8\58) 		434.5
(165.5) 		9/7
(11/10) 		Echidna
2 24\58
(5\58) 		496.6
(103.4) 		4/3
(17/16) 		Diaschismic
2 25\58
(4\58) 		517.2
(82.8) 		27/20
(21/20) 		Harry
29 19\58
(1\58) 		393.1
(20.7) 		5/4
(91/90) 		Mystery

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

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

Octave stretch or compression

58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as 92edt, 150ed6 or 289zpi.

Scales

Instruments

Music

Jeff Brown
Bryan Deister
Francium
Cam Taylor
Xotla
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