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-limits. It is the smallest equal temperament which is consistent through the 17-odd limit, and is also the smallest uniquely consistent in the 11-odd limit (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