58edo

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 cent 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.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 58edo (ordered by absolute error).

Interval, complement	Error (abs., in cents)
13/11, 22/13	0.445
11/10, 20/11	0.513
15/13, 26/15	0.535
9/7, 14/9	0.601
13/10, 20/13	0.958
15/11, 22/15	0.98
4/3, 3/2	1.493
7/6, 12/7	2.095
9/8, 16/9	2.987
7/5, 10/7	3.202
8/7, 7/4	3.588
14/11, 11/7	3.715
10/9, 9/5	3.803
14/13, 13/7	4.16
11/9, 18/11	4.316
15/14, 28/15	4.695
18/13, 13/9	4.762
6/5, 5/3	5.296
12/11, 11/6	5.809
13/12, 24/13	6.255
5/4, 8/5	6.79
11/8, 16/11	7.303
16/13, 13/8	7.748
16/15, 15/8	8.283

Scales

hemif7

hemif10

hemif17

Intervals

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

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