58edo

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The 58 equal temperament, often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the octave into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents.

Theory

58edo 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-limits. It is the smallest equal temperament which is consistent through the 17-limit, and is also the first et 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 three 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.

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

Just approximation

Selected just intervals

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.00 +1.59 +6.79 +3.59 +7.30 +7.75 -1.51 -7.86 -7.58
relative (%) 0.0 +7.2 +32.8 +17.3 +35.3 +37.4 -7.3 -38.0 -36.7

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 58et.

3-limit 5-limit 7-limit 11-limit 13-limit 17-limit
Octave stretch (¢) -0.47 -1.29 -1.29 -1.45 -1.56 -1.28
Error absolute (¢) 0.47 1.22 1.05 1.00 0.94 1.10
relative (%) 2.28 5.89 5.10 4.83 4.56 5.33
  • 58et has a lower relative error than any previous ETs in the 13-limit. The next ET that does better in this subgroup is 72.

Rank two temperaments

Period Generator Name
1\1 1\58
3\58
5\58
7\58
9\58
11\58 Gorgik
13\58
15\58 Myna
17\58 Hemififths
19\58
21\58
23\58 Buzzard
25\58
27\58 Thuja
1\2 1\58
2\58
3\58
4\58 Harry
5\58 Srutal/Diaschismic
6\58
7\58
8\58 Echidna, Supers
9\58 Secant
10\58
11\58
12\58 Sruti
13\58
14\58
1\29 1\58 Mystery

Scales

hemif7

hemif10

hemif17