58edo: Difference between revisions

← 57edo 58edo 59edo →
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

← Older edit

VisualWikitext

Latest revision as of 07:42, 26 May 2026

English Wikipedia has an article on:

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. Since 58 = 2 × 29, 58edo shares the same excellent perfect fifth with 29edo. It is the last edo to have exactly one diatonic perfect fifth and no 5edo or 7edo fifths.

The 19th and 23rd harmonics are very flat, so it makes sense to use their second-best approximations in the 58hi val. This val is, in fact, one of the first to be diamond monotone in the 23-odd-limit, past the idiosyncratic 53e val. However, its accuracy is questionable, with primes 19 and 23 being about 13 cents sharp, so one may want to use a larger system like 62edo for the 23-limit instead, which has the added benefit of being meantone.

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
Error	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)

As a tuning of other temperaments

As an equal temperament, 58et tempers out 2048/2025 in the 5-limit; 126/125, 1728/1715, and 5120/5103 in the 7-limit; 176/175, 243/242, 441/440, 540/539, and 896/891 in the 11-limit; 144/143, 351/350, 364/363 in the 13-limit. 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.

Subsets and supersets

58edo contains 2edo and 29edo as subsets.

Intervals

#	Cents	Approximate ratios*	Ups and downs notation
0	0.0	1/1	D
1	20.7	56/55, 64/63, 81/80, 91/90, 105/104	^D, v³E♭
2	41.4	36/35, 40/39, 45/44, 49/48, 50/49, 55/54	^^D, vvE♭
3	62.1	26/25, 27/26, 28/27, 33/32	^³D, vE♭
4	82.8	21/20, 22/21, 25/24	vvD♯, E♭
5	103.4	16/15, 17/16, 18/17	vD♯, ^E♭
6	124.1	14/13, 15/14	D♯, ^^E♭
7	144.8	12/11, 13/12	^D♯, v³E
8	165.5	11/10	^^D♯, vvE
9	186.2	10/9	^³D♯, vE
10	206.9	9/8, 17/15	E
11	227.6	8/7	^E, v³F
12	248.3	15/13	^^E, vvF
13	269.0	7/6	^³E, vF
14	289.7	13/11, 20/17	F
15	310.3	6/5	^F, v³G♭
16	331.0	17/14, 40/33	^^F, vvG♭
17	351.7	11/9, 16/13	^³F, vG♭
18	372.4	21/17, 26/21	vvF♯, G♭
19	393.1	5/4	vF♯, ^G♭
20	413.8	14/11	F♯, ^^G♭
21	434.5	9/7	^F♯, v³G
22	455.2	13/10, 17/13, 22/17	^^F♯, vvG
23	475.9	21/16	^³F♯, vG
24	496.6	4/3	G
25	517.2	27/20	^G, v³A♭
26	537.9	15/11	^^G, vvA♭
27	558.6	11/8, 18/13	^³G, vA♭
28	579.3	7/5	vvG♯, A♭
29	600.0	17/12, 24/17	vG♯, ^A♭
30	620.7	10/7	G♯, ^^A♭
31	641.4	13/9, 16/11	^G♯, v³A
32	662.1	22/15	^^G♯, vvA
33	682.8	40/27	^³G♯, vA
34	703.4	3/2	A
35	724.1	32/21	^A, v³B♭
36	744.8	17/11, 20/13, 26/17	^^A, vvB♭
37	765.5	14/9	^³A, vB♭
38	786.2	11/7	vvA♯, B♭
39	806.9	8/5	vA♯, ^B♭
40	827.6	21/13, 34/21	A♯, ^^B♭
41	848.3	13/8, 18/11	^A♯, v³B
42	869.0	28/17, 33/20	^^A♯, vvB
43	889.7	5/3	^³A♯, vB
44	910.3	17/10, 22/13	B
45	931.0	12/7	^B, v³C
46	951.7	26/15	^^B, vvC
47	972.4	7/4	^³B, vC
48	993.1	16/9, 30/17	C
49	1013.8	9/5	^C, v³D♭
50	1034.5	20/11	^^C, vvD♭
51	1055.2	11/6, 24/13	^³C, vD♭
52	1075.9	13/7, 28/15	vvC♯, D♭
53	1096.6	15/8, 17/9, 32/17	vC♯, ^D♭
54	1117.2	21/11, 40/21, 48/25	C♯, ^^D♭
55	1137.9	25/13, 27/14, 52/27, 64/33	^C♯, v³D
56	1158.6	35/18, 39/20, 49/25, 88/45, 96/49, 108/55	^^C♯, vvD
57	1179.3	55/28, 63/32, 160/81, 180/91, 208/105	^³C♯, vD
58	1200.0	2/1	D

* As a 17-limit temperament, inconsistently mapped intervals in italic

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation for 58edo uses sharps and flats combined with quartertone accidentals and arrows:

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 quartertone accidentals:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol														
Flat symbol														

Kite's ups and downs notation

58edo can also be notated with Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

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

Half-sharps and half-flats can be used to avoid triple arrows:

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

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

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

Sagittal notation

Evo flavor

Evo-SZ flavor

Revo flavor

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
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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	62.1	28/27	Unicorn / alicorn / qilin
1	11\58	227.6	8/7	Gorgik
1	13\58	269.0	7/6	Infraorwell
1	15\58	310.3	6/5	Myna
1	17\58	351.7	49/40	Hemififths
1	19\58	393.1	64/51	Emmthird
1	23\58	475.9	21/16	Buzzard / subfourth
1	25\58	517.2	27/20	Gravity / abergravity / gravid
1	27\58	558.6	11/8	Thuja
2	3\58	62.1	28/27	Monocerus
2	1\58	20.7	81/80	Bicommatic
2	9\58	186.2	10/9	Secant
2	17\58 (12\58)	351.7 (248.3)	11/9 (15/13)	Sruti
2	21\58 (8\58)	434.5 (165.5)	9/7 (11/10)	Echidna
2	24\58 (5\58)	496.6 (103.4)	4/3 (17/16)	Diaschismic
2	25\58 (4\58)	517.2 (82.8)	27/20 (21/20)	Harry
29	19\58 (1\58)	393.1 (20.7)	5/4 (91/90)	Mystery

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

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

Octave stretch or compression

58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as 92edt, 150ed6 or 289zpi.

Scales

Instruments

Music

Jeff Brown

Fruitbats in Formation (2023)

Bryan Deister

Francium

We Wish You A Larry Christmas (2024) – in larry, 58edo tuning

Cam Taylor

58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone (2021)

Xotla

"Wormhole Shmurmhole", from Just Another Microtonal Music Album (2025–2026) – in part, the rest being in 31edo

@@ Line 8: / Line 8: @@
 While the [[17/1|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. Since {{nowrap|58 {{=}} 2 × 29}}, 58edo shares the same excellent perfect fifth with [[29edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth and no [[5edo]] or [[7edo]] fifths.
+The [[19/1|19th]] and [[23/1|23rd]] harmonics are very flat, so it makes sense to use their second-best approximations in the 58hi val. This val is, in fact, one of the first to be [[diamond monotone]] in the 23-odd-limit, past the idiosyncratic 53e val. However, its accuracy is questionable, with primes 19 and 23 being about 13 cents sharp, so one may want to use a larger system like [[62edo]] for the 23-limit instead, which has the added benefit of being meantone.
+=== Prime harmonics ===
+{{Harmonics in equal|58}}
+=== As a tuning of other temperaments ===
 As an equal temperament, 58et [[tempering out|tempers out]] [[2048/2025]] in the [[5-limit]]; [[126/125]], [[1728/1715]], and [[5120/5103]] in the [[7-limit]]; [[176/175]], [[243/242]], [[441/440]], [[540/539]], and [[896/891]] in the 11-limit; [[144/143]], [[351/350]], [[364/363]] in the 13-limit. It [[support]]s [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]], [[thuja]] [[regular temperament|temperament]]s plus a number of [[gravity family]] [[extension]]s, 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{{cent}} narrower than the just interval.
-=== Prime harmonics ===
-{{Harmonics in equal|58}}
 === Subsets and supersets ===
@@ Line 324: / Line 327: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
+[[Stein–Zimmermann–Gould notation]] for 58edo uses sharps and flats combined with quartertone accidentals and arrows:
-{{Sharpness-sharp6a}}
+{{Sharpness-sharp6-szg}}
-Half-sharps and half-flats can be used to avoid triple arrows:
+If double arrows are not desirable, then arrows can be attached to quartertone accidentals:
-{{Sharpness-sharp6b}}
+{{Sharpness-sharp6-qt-szg}}
-Alternatively, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.
+=== Kite's ups and downs notation ===
-{{Sharpness-sharp6}}
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
+{{Ups and downs sharpness}}
-If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
+Half-sharps and half-flats can be used to avoid triple arrows:
-{{Sharpness-sharp6-qt}}
+{{Ups and downs sharpness|58|true}}
 === Ivan Wyschnegradsky's notation ===
 Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
 {{Sharpness-sharp6-iw}}
 === Sagittal notation ===
 ==== Evo flavor ====
-<imagemap>
+{{Sagittal chart|Evo}}
-File:58-EDO_Evo_Sagittal.svg
-desc none
+==== Evo-SZ flavor ====
-rect 80 0 300 50 [[Sagittal_notation]]
+{{Sagittal chart|Evo-SZ}}
-rect 300 0 662 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 120 106 [[81/80]]
-rect 120 80 230 106 [[55/54]]
-rect 230 80 350 106 [[33/32]]
-default [[File:58-EDO_Evo_Sagittal.svg]]
-</imagemap>
 ==== Revo flavor ====
-<imagemap>
+{{Sagittal chart}}
-File:58-EDO_Revo_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 300 0 662 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 120 106 [[81/80]]
-rect 120 80 230 106 [[55/54]]
-rect 230 80 350 106 [[33/32]]
-default [[File:58-EDO_Revo_Sagittal.svg]]
-</imagemap>
-==== Evo-SZ flavor ====
-<imagemap>
-File:58-EDO_Evo-SZ_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 120 106 [[81/80]]
-rect 120 80 230 106 [[55/54]]
-rect 230 80 350 106 [[33/32]]
-default [[File:58-EDO_Evo-SZ_Sagittal.svg]]
-</imagemap>
 === Hemipyth notation ===
@@ Line 383: / Line 359: @@
 |+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)
 |-
-! &#35;
+! #
 ! Cents
-! Note names<br />on D
+! Note names<br>on D
 |-
 | 0
@@ Line 600: / Line 576: @@
 | 21/16
 | [[Buzzard]] / [[subfourth]]
+|-
+| 1
+| 25\58
+| 517.2
+| 27/20
+| [[Gravity]] / [[abergravity]] / [[gravid]]
 |-
 | 1
@@ Line 655: / Line 637: @@
 | [[Mystery]]
 |}
-<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).
+== Octave stretch or compression ==
+edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]], [[150ed6]] or [[zpi|289zpi]].
 == Scales ==
 * [[Compdye]]
+* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]]
 * [[Hemif7]]
 * [[Hemif10]]
@@ Line 678: / Line 664: @@
 * [https://www.youtube.com/shorts/4J4MNno-4PA ''58edo improv''] (2025)
 * [https://www.youtube.com/shorts/7gkRyld5OU8 ''Waltz in 58edo''] (2025)
+* [https://www.youtube.com/shorts/H5XG4lrvrP8 ''58edo groove''] (2025)
 ; [[Francium]]
@@ Line 684: / Line 671: @@
 ; [[Cam Taylor]]
 * [https://www.youtube.com/watch?v=Keclakcqie8 ''58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone''] (2021)
+; [[Xotla]]
+* [https://www.youtube.com/watch?v=yTkPhjTEQMw "Wormhole Shmurmhole"], from [https://www.youtube.com/playlist?list=PL4HmfPDldHXueRjmTV-iJN3fsLn_O9jvT ''Just Another Microtonal Music Album''] (2025–2026) – in part, the rest being in 31edo
 [[Category:Buzzard]]

58edo: Difference between revisions

Latest revision as of 07:42, 26 May 2026

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