View source for 58edo

{{Infobox ET}}
{{Wikipedia|58 equal temperament}}
{{ED intro}}

== Theory ==
58edo is a strong system in the [[11-limit|11-]], [[13-limit|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 [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]]. 

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. 

=== 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.

=== Subsets and supersets ===
58edo contains [[2edo]] and [[29edo]] as subsets.

== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4""
|-
! #
! Cents
! Approximate ratios*
!Additional ratios of 19 and 23
using the 58hi val
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
|
| {{UDnote|step=0}}
|-
| 1
| 20.7
| [[56/55]], [[64/63]], [[81/80]], [[91/90]], [[105/104]]
|
| {{UDnote|step=1}}
|-
| 2
| 41.4
| [[36/35]], [[40/39]], [[45/44]], [[49/48]], [[50/49]], [[55/54]]
|
| {{UDnote|step=2}}
|-
| 3
| 62.1
| [[26/25]], [[27/26]], [[28/27]], [[33/32]]
|
| {{UDnote|step=3}}
|-
| 4
| 82.8
| [[21/20]], [[22/21]], ''[[25/24]]''
|20/19
| {{UDnote|step=4}}
|-
| 5
| 103.4
| [[16/15]], [[17/16]], [[18/17]]
|19/18
| {{UDnote|step=5}}
|-
| 6
| 124.1
| [[14/13]], [[15/14]]
|
| {{UDnote|step=6}}
|-
| 7
| 144.8
| [[12/11]], [[13/12]]
|
| {{UDnote|step=7}}
|-
| 8
| 165.5
| [[11/10]]
|
| {{UDnote|step=8}}
|-
| 9
| 186.2
| [[10/9]]
|
| {{UDnote|step=9}}
|-
| 10
| 206.9
| [[9/8]], [[17/15]]
|''19/17''
| {{UDnote|step=10}}
|-
| 11
| 227.6
| [[8/7]]
|
| {{UDnote|step=11}}
|-
| 12
| 248.3
| [[15/13]]
|22/19
| {{UDnote|step=12}}
|-
| 13
| 269.0
| [[7/6]]
|
| {{UDnote|step=13}}
|-
| 14
| 289.7
| [[13/11]], [[20/17]]
|
| {{UDnote|step=14}}
|-
| 15
| 310.3
| [[6/5]]
|''19/16''
| {{UDnote|step=15}}
|-
| 16
| 331.0
| [[17/14]], [[40/33]]
|
| {{UDnote|step=16}}
|-
| 17
| 351.7
| [[11/9]], [[16/13]]
|
| {{UDnote|step=17}}
|-
| 18
| 372.4
| [[21/17]], [[26/21]]
|
| {{UDnote|step=18}}
|-
| 19
| 393.1
| [[5/4]]
|''24/19''
| {{UDnote|step=19}}
|-
| 20
| 413.8
| [[14/11]]
|19/15
| {{UDnote|step=20}}
|-
| 21
| 434.5
| [[9/7]]
|
| {{UDnote|step=21}}
|-
| 22
| 455.2
| [[13/10]], [[17/13]], [[22/17]]
|
| {{UDnote|step=22}}
|-
| 23
| 475.9
| [[21/16]]
|
| {{UDnote|step=23}}
|-
| 24
| 496.6
| [[4/3]]
|
| {{UDnote|step=24}}
|-
| 25
| 517.2
| [[27/20]]
|
| {{UDnote|step=25}}
|-
| 26
| 537.9
| [[15/11]]
|19/14, 26/19
| {{UDnote|step=26}}
|-
| 27
| 558.6
| [[11/8]], [[18/13]]
|
| {{UDnote|step=27}}
|-
| 28
| 579.3
| [[7/5]]
|
| {{UDnote|step=28}}
|-
| 29
| 600.0
| [[17/12]], [[24/17]]
|
| {{UDnote|step=29}}
|-
| 30
| 620.7
| [[10/7]]
|
| {{UDnote|step=30}}
|-
| 31
| 641.4
| [[13/9]], [[16/11]]
|''23/16''
| {{UDnote|step=31}}
|-
| 32
| 662.1
| [[22/15]]
|19/13, 28/19
| {{UDnote|step=32}}
|-
| 33
| 682.8
| [[40/27]]
|
| {{UDnote|step=33}}
|-
| 34
| 703.4
| [[3/2]]
|
| {{UDnote|step=34}}
|-
| 35
| 724.1
| [[32/21]]
|
| {{UDnote|step=35}}
|-
| 36
| 744.8
| [[17/11]], [[20/13]], [[26/17]]
|
| {{UDnote|step=36}}
|-
| 37
| 765.5
| [[14/9]]
|
| {{UDnote|step=37}}
|-
| 38
| 786.2
| [[11/7]]
|30/19
| {{UDnote|step=38}}
|-
| 39
| 806.9
| [[8/5]]
|''19/12''
| {{UDnote|step=39}}
|-
| 40
| 827.6
| [[21/13]], [[34/21]]
|
| {{UDnote|step=40}}
|-
| 41
| 848.3
| [[13/8]], [[18/11]]
|
| {{UDnote|step=41}}
|-
| 42
| 869.0
| [[28/17]], [[33/20]]
|
| {{UDnote|step=42}}
|-
| 43
| 889.7
| [[5/3]]
|''32/19''
| {{UDnote|step=43}}
|-
| 44
| 910.3
| [[17/10]], [[22/13]]
|
| {{UDnote|step=44}}
|-
| 45
| 931.0
| [[12/7]]
|
| {{UDnote|step=45}}
|-
| 46
| 951.7
| [[26/15]]
|19/11
| {{UDnote|step=46}}
|-
| 47
| 972.4
| [[7/4]]
|
| {{UDnote|step=47}}
|-
| 48
| 993.1
| [[16/9]], [[30/17]]
|''34/19''
| {{UDnote|step=48}}
|-
| 49
| 1013.8
| [[9/5]]
|
| {{UDnote|step=49}}
|-
| 50
| 1034.5
| [[20/11]]
|
| {{UDnote|step=50}}
|-
| 51
| 1055.2
| [[11/6]], [[24/13]]
|
| {{UDnote|step=51}}
|-
| 52
| 1075.9
| [[13/7]], [[28/15]]
|
| {{UDnote|step=52}}
|-
| 53
| 1096.6
| [[15/8]], [[17/9]], [[32/17]]
|36/19
| {{UDnote|step=53}}
|-
| 54
| 1117.2
| [[21/11]], [[40/21]], ''[[48/25]]''
|19/10
| {{UDnote|step=54}}
|-
| 55
| 1137.9
| [[25/13]], [[27/14]], [[52/27]], [[64/33]]
|
| {{UDnote|step=55}}
|-
| 56
| 1158.6
| [[35/18]], [[39/20]], [[49/25]], [[88/45]], [[96/49]], [[108/55]]
|
| {{UDnote|step=56}}
|-
| 57
| 1179.3
| [[55/28]], [[63/32]], [[160/81]], [[180/91]], [[208/105]]
|
| {{UDnote|step=57}}
|-
| 58
| 1200.0
| [[2/1]]
|
| {{UDnote|step=58}}
|}
<nowiki/>* As a 17-limit temperament, inconsistently mapped intervals in ''italic''

== Notation ==
=== Ups and downs notation ===
58edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. 
{{Ups and downs sharpness}}

Half-sharps and half-flats can be used to avoid triple arrows:
{{Ups and downs sharpness|58|true}}

Alternatively, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.
{{Sharpness-sharp6}}

If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}

=== 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>
File:58-EDO_Evo_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_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
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 ===
{| class="wikitable center-all right-2 center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)
|-
! &#35;
! Cents
! Note names<br />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 ===
{{15-odd-limit|58}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 2048/2025, [[1594323/1562500]]
| {{Mapping| 58 92 135 }}
| −1.29
| 1.22
| 5.89
|-
| 2.3.5.7
| 126/125, 1728/1715, 2048/2025
| {{Mapping| 58 92 135 163 }}
| −1.29
| 1.05
| 5.10
|-
| 2.3.5.7.11
| 126/125, 176/175, 243/242, 896/891
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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 [[72edo|72]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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
| 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<br>(12\58)
| 351.7<br>(248.3)
| 11/9<br>(15/13)
| [[Sruti]]
|-
| 2
| 21\58<br>(8\58)
| 434.5<br>(165.5)
| 9/7<br>(11/10)
| [[Echidna]]
|-
| 2
| 24\58<br>(5\58)
| 496.6<br>(103.4)
| 4/3<br>(17/16)
| [[Diaschismic]]
|-
| 2
| 25\58<br>(4\58)
| 517.2<br>(82.8)
| 27/20<br>(21/20)
| [[Harry]]
|-
| 29
| 19\58<br>(1\58)
| 393.1<br>(20.7)
| 5/4<br>(91/90)
| [[Mystery]]
|}
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct

58et 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 ==
58edo'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]]
* [[Hemif17]]

== Instruments ==
* [[Lumatone mapping for 58edo]]
* [[Skip fretting system 58 2 15|15\58 × 2\58 isomorphic instrument layout]]
* [[Skip fretting system 58 4 15|15\58 × 4\58 isomorphic instrument layout]]
* [[Skip fretting system 58 2 17|17\58 × 2\58 isomorphic instrument layout]]

== Music ==
; [[Jeff Brown]]
* [https://www.youtube.com/watch?v=0373hBH87LY ''Fruitbats in Formation''] (2023)

; [[Bryan Deister]]
* [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]]
* [https://www.youtube.com/watch?v=XXMjoUxVfLs ''We Wish You A Larry Christmas''] (2024) – in larry, 58edo tuning

; [[Cam Taylor]]
* [https://www.youtube.com/watch?v=Keclakcqie8 ''58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone''] (2021)

[[Category:Buzzard]]
[[Category:Diaschismic]]
[[Category:Harry]]
[[Category:Hemififths]]
[[Category:Myna]]
[[Category:Mystery]]
[[Category:Harry Partch]]
[[Category:Listen]]