58edo

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58edo

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Prime factorization	2 × 29
Step size	20.6897¢
Fifth	34\58 (703.448¢) (→17\29)
Semitones (A1:m2)	6:4 (124.1¢ : 82.76¢)
Consistency limit	17
Distinct consistency limit	11

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58 equal temperament

58 equal divisions of the octave (abbreviated 58edo or 58ed2), also called 58-tone equal temperament (58tet) or 58 equal temperament (58et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 58 equal parts of about 20.7 ¢ each. Each step represents a frequency ratio of 2^1/58, or the 58th root of 2.

Theory

58edo 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 distinctly consistent in the 11-odd-limit (the first equal temperament to map the entire 11-odd-limit tonality diamond to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note Genesis scale of Harry Partch.

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 58edo shares the same excellent fifth with 29edo.

As an equal temperament, 58et 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. It supports hemififths, myna, diaschismic, harry, mystery, buzzard, thuja temperaments plus a number of gravity family extensions, and supplies the optimal patent val for the 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.

Of all edos which map the syntonic comma (81/80) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1 ¢ narrower than the just interval.

Prime harmonics

Approximation of prime harmonics in 58edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+1.49	+6.79	+3.59	+7.30	+7.75	-1.51	-7.86	-7.58	+4.91	-7.10
	Relative (%)	+0.0	+7.2	+32.8	+17.3	+35.3	+37.4	-7.3	-38.0	-36.7	+23.7	-34.3
Steps (reduced)		58 (0)	92 (34)	135 (19)	163 (47)	201 (27)	215 (41)	237 (5)	246 (14)	262 (30)	282 (50)	287 (55)

Subsets and supersets

58edo contains 2edo and 29edo as subsets.

Intervals

#	Cents	Approximate ratios	Ups and downs notation
0	0.00	1/1	D
1	20.69	56/55, 64/63, 81/80, 128/125	^D, v3E♭
2	41.38	36/35, 49/48, 50/49, 55/54	^^D, vvE♭
3	62.07	26/25, 27/26, 28/27, 33/32	^3D, vE♭
4	82.76	25/24, 21/20, 22/21	^4D, E♭
5	103.45	16/15, 17/16, 18/17	^5D, v5E
6	124.14	14/13, 15/14, 27/25	D♯, v4E
7	144.83	12/11, 13/12	^D♯, v3E
8	165.52	11/10	^^D♯, vvE
9	186.21	10/9	^3D♯, vE
10	206.90	9/8, 17/15	E
11	227.59	8/7	^E, v3F
12	248.28	15/13	^^E, vvF
13	268.97	7/6	^3E, vF
14	289.66	13/11, 20/17	F
15	310.34	6/5	^F, v3G♭
16	331.03	17/14	^^F, vvG♭
17	351.72	11/9, 16/13	^3F, vG♭
18	372.41	21/17	^4F, G♭
19	393.10	5/4	^5F, v5G
20	413.79	14/11	F♯, v4G
21	434.48	9/7	^F♯, v3G
22	455.17	13/10, 17/13, 22/17	^^F♯, vvG
23	475.86	21/16	^3F♯, vG
24	496.55	4/3	G
25	517.24	27/20	^G, v3A♭
26	537.93	15/11	^^G, vvA♭
27	558.62	11/8, 18/13	^3G, vA♭
28	579.31	7/5	^4G, A♭
29	600.00	17/12, 24/17	^5G, v5A
30	620.69	10/7	G♯, v4A
31	641.38	13/9, 16/11	^G♯, v3A
32	662.07	22/15	^^G♯, vvA
33	682.76	40/27	^3G♯, vA
34	703.45	3/2	A
35	724.14	32/21	^A, v3B♭
36	744.83	20/13, 26/17, 17/11	^^A, vvB♭
37	765.52	14/9	^3A, vB♭
38	786.21	11/7	^4A, B♭
39	806.90	8/5	^5A, v5B
40	827.59	34/21	A♯, v4B
41	848.28	13/8, 18/11	^A♯, v3B
42	868.97	28/17	^^A♯, vvB
43	889.66	5/3	^3A♯, vB
44	910.34	22/13, 17/10	B
45	931.03	12/7	^B, v3C
46	951.72	26/15	^^B, vvC
47	972.41	7/4	^3B, vC
48	993.10	16/9, 30/17	C
49	1013.79	9/5	^C, v3D♭
50	1034.48	20/11	^^C, vvD♭
51	1055.17	11/6, 24/13	^3C, vD♭
52	1075.86	13/7, 28/15	^4C, D♭
53	1096.55	15/8, 32/17, 17/9	^5C, v5D
54	1117.24	48/25, 40/21, 21/11	C♯, v4D
55	1137.93	25/13, 52/27, 27/14, 64/33	^C♯, v3D
56	1158.62	35/18, 96/49, 49/25, 108/55	^^C♯, vvD
57	1179.31	55/28, 63/32, 160/81, 125/64	^3C♯, vD
58	1200.00	2/1	D

Notation

Ups and downs notation

In 58edo, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from Helmholtz–Ellis notation can be used to fill in the gaps.

Step Offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Sharp Symbol
Flat Symbol

If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:

Step Offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp Symbol
Flat Symbol

Sagittal

The following table shows sagittal notation accidentals in one apotome for 58edo.

Step Offset	0	1	2	3	4	5	6
Symbol

Hemipyth notation

Hemipyth notation for 58edo (SW3-style)

#	Cents	Note names on D
0	0.0	D
2	41.4	α𝄳
5	103.4	α
7	144.8	E𝄳
10	206.9	E
12	248.3	β𝄳
14	289.7	F
15	310.3	β
17	351.7	F‡
19	393.1	γ
22	455.2	γ‡
24	496.6	G
27	558.6	G‡
29	600.0	δ
31	641.4	A𝄳
34	703.4	A
36	744.8	ε𝄳
39	806.9	ε
41	848.3	B𝄳
43	889.7	ζ
44	910.3	B
46	951.7	ζ‡
48	993.1	C
51	1055.2	C‡
53	1096.6	η
56	1158.6	η‡
58	1200.0	D

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 58edo. Prime harmonics are in bold.

As 58edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 58edo

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/11, 22/13	0.445	2.2
11/10, 20/11	0.513	2.5
15/13, 26/15	0.535	2.6
9/7, 14/9	0.601	2.9
13/10, 20/13	0.958	4.6
15/11, 22/15	0.980	4.7
3/2, 4/3	1.493	7.2
7/6, 12/7	2.095	10.1
9/8, 16/9	2.987	14.4
7/5, 10/7	3.202	15.5
7/4, 8/7	3.588	17.3
11/7, 14/11	3.715	18.0
9/5, 10/9	3.803	18.4
13/7, 14/13	4.160	20.1
11/9, 18/11	4.316	20.9
15/14, 28/15	4.695	22.7
13/9, 18/13	4.762	23.0
5/3, 6/5	5.296	25.6
11/6, 12/11	5.809	28.1
13/12, 24/13	6.255	30.2
5/4, 8/5	6.790	32.8
11/8, 16/11	7.303	35.3
13/8, 16/13	7.748	37.4
15/8, 16/15	8.283	40.0

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 has a lower relative error than any previous equal temperaments in the 13-limit, and the next equal temperament that does better in this subgroup is 72.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	3\58	62.07	28/27	Unicorn / alicorn / qilin
1	11\58	227.59	8/7	Gorgik
1	13\58	268.97	7/6	Infraorwell
1	15\58	310.34	6/5	Myna
1	17\58	351.72	49/40	Hemififths
1	19\58	393.10	64/51	Emmthird
1	23\58	475.86	21/16	Buzzard / subfourth
1	27\58	558.62	11/8	Thuja
2	3\58	62.07	28/27	Monocerus
2	1\58	20.69	81/80	Commatic
2	9\58	186.21	10/9	Secant
2	17\58 (12\58)	351.72 (248.28)	11/9 (15/13)	Sruti
2	21\58 (8\58)	434.48 (165.52)	9/7 (11/10)	Echidna
2	24\58 (5\58)	496.55 (103.45)	4/3 (17/16)	Diaschismic
2	25\58 (4\58)	517.24 (82.76)	27/20 (21/20)	Harry
29	19\58 (1\58)	393.10 (20.69)	5/4 (91/90)	Mystery

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

58et can also be detempered to semihemi (58 & 140), supers (58 & 152), condor (58 & 159), and eagle (58 & 212).

Scales

• Hemif7
• Hemif10
• Hemif17

Instruments

• Lumatone mapping for 58edo
• 15\58 × 2\58 isomorphic instrument layout
• 15\58 × 4\58 isomorphic instrument layout
• 17\58 × 2\58 isomorphic instrument layout

Music

Jeff Brown

• Fruitbats in Formation

Cam Taylor

• 58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone