58edo
| ← 57edo | 58edo | 59edo → |
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 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
| 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 | +7 | +33 | +17 | +35 | +37 | -7 | -38 | -37 | +24 | -34 | |
| 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, v3Eb |
| 2 | 41.38 | 36/35, 49/48, 50/49, 55/54 | ^^D, vvEb |
| 3 | 62.07 | 26/25, 27/26, 28/27, 33/32 | ^3D, vEb |
| 4 | 82.76 | 25/24, 21/20, 22/21 | ^4D, Eb |
| 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, v3Gb |
| 16 | 331.03 | 17/14 | ^^F, vvGb |
| 17 | 351.72 | 11/9, 16/13 | ^3F, vGb |
| 18 | 372.41 | 21/17 | ^4F, Gb |
| 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, v3Ab |
| 26 | 537.93 | 15/11 | ^^G, vvAb |
| 27 | 558.62 | 11/8, 18/13 | ^3G, vAb |
| 28 | 579.31 | 7/5 | ^4G, Ab |
| 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, v3Bb |
| 36 | 744.83 | 20/13, 26/17, 17/11 | ^^A, vvBb |
| 37 | 765.52 | 14/9 | ^3A, vBb |
| 38 | 786.21 | 11/7 | ^4A, Bb |
| 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, v3Db |
| 50 | 1034.48 | 20/11 | ^^C, vvDb |
| 51 | 1055.17 | 11/6, 24/13 | ^3C, vDb |
| 52 | 1075.86 | 13/7, 28/15 | ^4C, Db |
| 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 | 
|
|
|
|
|
|
|
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.
| Interval, 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 (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
- 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