Theory
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. It 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