58edo
← 57edo | 58edo | 59edo → |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/63/Wikipedia-logo.png/44px-Wikipedia-logo.png)
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
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 | 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 quarter-tone accidentals:
Step Offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Sharp Symbol | ||||||||||||||
Flat Symbol |
Sagittal
The following table shows sagittal notation accidentals in one apotome for 58edo.
Step Offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|
Symbol | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Hemipyth notation
# | 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.
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
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).
Scales
Instruments
- Lumatone mapping for 58edo
- 15\58 × 2\58 isomorphic instrument layout
- 15\58 × 4\58 isomorphic instrument layout
- 17\58 × 2\58 isomorphic instrument layout