58edo

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58edo
Prime factorization 2 × 29
Step size 20.68966¢
Fifth 34\58 (703.45¢) (→ 17\29)
Major 2nd 10\58 (207¢)
Semitones (A1:m2) 6:4 (124¢ : 83¢)
Consistency limit 17
Monotonicity limit 23

The 58 equal divisions of the octave (58edo), or the 58(-tone) equal temperament (58tet, 58et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 58 equally-sized steps of about 20.7 cents each.

Theory

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, and 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 uniquely consistent in the 11-odd-limit (the first equal temperament to map the entire 11-limit tonality diamond to distinct scale steps), and hence the first et which can define a version of the famous 43-note Genesis scale of Harry Partch. It supports hemififths, myna, diaschismic, harry, mystery, buzzard and thuja temperaments, and supplies the optimal patent val for 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. 58 = 2 × 29, and 58 shares the same excellent fifth with 29edo.


Approximation of prime intervals in 58 EDO
Prime number 2 3 5 7 11 13 17 19
Error absolute (¢) +0.00 +1.49 +6.79 +3.59 +7.30 +7.75 -1.51 -7.86
relative (%) +0 +7 +33 +17 +35 +37 -7 -38
Steps (reduced) 58 (0) 92 (34) 135 (19) 163 (47) 201 (27) 215 (41) 237 (5) 246 (14)

Intervals

# Cents Approximate Ratios
0 0.00 1/1
1 20.69 56/55, 64/63, 81/80, 128/125
2 41.38 36/35, 49/48, 50/49, 55/54
3 62.07 26/25, 27/26, 28/27, 33/32
4 82.76 25/24, 21/20, 22/21
5 103.45 16/15, 17/16, 18/17
6 124.14 14/13, 15/14, 27/25
7 144.83 12/11, 13/12
8 165.52 11/10
9 186.21 10/9
10 206.90 9/8, 17/15
11 227.59 8/7
12 248.28 15/13
13 268.97 7/6
14 289.66 13/11, 20/17
15 310.34 6/5
16 331.03 17/14
17 351.72 11/9, 16/13
18 372.41 21/17
19 393.10 5/4
20 413.79 14/11
21 434.48 9/7
22 455.17 13/10, 17/13, 22/17
23 475.86 21/16
24 496.55 4/3
25 517.24 27/20
26 537.93 15/11
27 558.62 11/8, 18/13
28 579.31 7/5
29 600.00 17/12, 24/17
30 620.69 10/7
31 641.38 13/9, 16/11
32 662.07 22/15
33 682.76 40/27
34 703.45 3/2
35 724.14 32/21
36 744.83 20/13, 26/17, 17/11
37 765.52 14/9
38 786.21 11/7
39 806.90 8/5
40 827.59 34/21
41 848.28 13/8, 18/11
42 868.97 28/17
43 889.66 5/3
44 910.34 22/13, 17/10
45 931.03 12/7
46 951.72 26/15
47 972.41 7/4
48 993.10 16/9, 30/17
49 1013.79 9/5
50 1034.48 20/11
51 1055.17 11/6, 24/13
52 1075.86 13/7, 28/15
53 1096.55 15/8, 32/17, 17/9
54 1117.24 48/25, 40/21, 21/11
55 1137.93 25/13, 52/27, 27/14, 64/33
56 1158.62 35/18, 96/49, 49/25, 108/55
57 1179.31 55/28, 63/32, 160/81, 125/64
58 1200.00 2/1

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 is lower in relative error than any previous equal temperaments in the 13-limit, and the next ET that does better in this subgroup is 72.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Period
per Octave
Generator Temperament
1 1\58
1 3\58
1 5\58
1 7\58
1 9\58
1 11\58 Gorgik
1 13\58 Infraorwell
1 15\58 Myna
1 17\58 Hemififths
1 19\58
1 21\58
1 23\58 Vulture / buzzard
Subfourth
1 25\58
1 27\58 Thuja
2 1\58 Commatic
2 2\58
2 3\58
2 4\58 Harry
2 5\58 Srutal / diaschismic
2 6\58
2 7\58
2 8\58 Echidna
Supers
2 9\58 Secant
2 10\58
2 11\58
2 12\58 Sruti
Semihemi
2 13\58
2 14\58
29 1\58 Mystery

Scales