58edo

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58edo
Prime factorization 2 × 29
Step size 20.6897¢
Fifth 34\58 (703.4¢) (→ 17\29)
Major 2nd 10\58 (206.9¢)
Semitones (A1:m2) 6:4 (124.1¢ : 82.8¢)
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.

Prime harmonics

Approximation of prime harmonics in 58edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0.00 +1.49 +6.79 +3.59 +7.30 +7.75 -1.51 -7.86 -7.58 +4.91
relative (%) +0 +7 +33 +17 +35 +37 -7 -38 -37 +24
Steps
(reduced)
58
(0)
92
(34)
135
(19)
163
(47)
201
(27)
215
(41)
237
(5)
246
(14)
262
(30)
282
(50)

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

Notation

Sagittal

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

Steps 0 1 2 3 4 5 6
Symbol Sagittal natural.png Sagittal pai.png Sagittal kai.png Sagittal pakai.png Sagittal sharp kao.png Sagittal sharp pao.png Sagittal sharp.png

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.

15-odd-limit intervals by patent val mapping
Interval, complement Error (abs, ¢) Error (rel, %)
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 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 Llywelyn / 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