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.

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-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.

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

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