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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.  
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.  


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]].
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.  


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|columns=11}}
{{Harmonics in equal|58|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 58edo (continued)}}


=== Prime harmonics ===
=== As a tuning of other temperaments ===
{{Harmonics in equal|58}}
As an equal temperament, 58et [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]) and 1594323/1562500 ([[unicorn comma]]) 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]].


=== Octave stretch ===
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.
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]] or [[150ed6]].  


=== Subsets and supersets ===
=== Subsets and supersets ===
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== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould 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.
[[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}}
58edo 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 ===
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
{{Sharpness-sharp6-iw}}
{{Sharpness-sharp6-iw}}


=== Sagittal notation ===
=== Sagittal notation ===
==== Evo flavor ====
==== 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 ====
==== 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 ===
=== Hemipyth notation ===
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|+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)
|+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)
|-
|-
! &#35;
! #
! Cents
! Cents
! Note names<br />on D
! Note names<br>on D
|-
|-
| 0
| 0
Line 516: Line 490:
|-
|-
| 2.3.5
| 2.3.5
| 2048/2025, [[1594323/1562500]]
| 2048/2025, 1594323/1562500
| {{Mapping| 58 92 135 }}
| {{Mapping| 58 92 135 }}
| −1.29
| −1.29
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| 28/27
| 28/27
| [[Unicorn]] / alicorn / qilin
| [[Unicorn]] / alicorn / qilin
|-
| 1
| 9\58
| 186.2
| 10/9
| [[Mintone]]
|-
|-
| 1
| 1
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| 49/40
| 49/40
| [[Hemififths]]
| [[Hemififths]]
|-
| 1
| 19\58
| 393.1
| 64/51
| [[Emmthird]]
|-
|-
| 1
| 1
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| 21/16
| 21/16
| [[Buzzard]] / [[subfourth]]
| [[Buzzard]] / [[subfourth]]
|-
| 1
| 25\58
| 517.2
| 27/20
| [[Gravity]] / [[abergravity]] / [[gravid]]
|-
|-
| 1
| 1
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| 11/8
| 11/8
| [[Thuja]]
| [[Thuja]]
|-
| 2
| 1\58
| 20.7
| 81/80
| [[Bicommatic]]
|-
|-
| 2
| 2
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|-
|-
| 2
| 2
| 1\58
| 4\58
| 20.7
| 82.8
| 81/80
| 21/20
| [[Bicommatic]]
| [[Harry]]
|-
|-
| 2
| 2
| 9\58
| 5\58
| 186.2
| 103.4
| 10/9
| 17/16
| [[Secant]]
| [[Diaschismic]]
|-
|-
| 2
| 2
| 17\58<br>(12\58)
| 7\58
| 351.7<br>(248.3)
| 144.8
| 11/9<br>(15/13)
| 13/12
| [[Sruti]]
| [[Bisemidim]]
|-
|-
| 2
| 2
| 21\58<br>(8\58)
| 8\58
| 434.5<br>(165.5)
| 165.5
| 9/7<br>(11/10)
| 11/10
| [[Echidna]]
| [[Echidna]] / [[supers]]
|-
|-
| 2
| 2
| 24\58<br>(5\58)
| 9\58
| 496.6<br>(103.4)
| 186.2
| 4/3<br>(17/16)
| 10/9
| [[Diaschismic]]
| [[Secant]]
|-
|-
| 2
| 2
| 25\58<br>(4\58)
| 12\58
| 517.2<br>(82.8)
| 248.3
| 27/20<br>(21/20)
| 15/13
| [[Harry]]
| [[Sruti]]
|-
|-
| 29
| 29
| 19\58<br>(1\58)
| 1\58
| 393.1<br>(20.7)
| 20.7
| 5/4<br>(91/90)
| 91/90
| [[Mystery]]
| [[Mystery]]
|}
|}
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


58et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).
58et can also be detempered to [[emmthird]] ({{nowrap| 58 & 113 }}), [[semihemi]] ({{nowrap| 58 & 140 }}), [[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 ==
== Scales ==
* Baobab{{idio}}: 15 9 10 9 8 7 (approximated from [[30afdo]])
* [[Compdye]]
* [[Compdye]]
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]]{{idio}}
* [[Hemif7]]
* [[Hemif7]]
* [[Hemif10]]
* [[Hemif10]]
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* [https://www.youtube.com/shorts/4J4MNno-4PA ''58edo improv''] (2025)
* [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/7gkRyld5OU8 ''Waltz in 58edo''] (2025)
* [https://www.youtube.com/shorts/H5XG4lrvrP8 ''58edo groove''] (2025)


; [[Francium]]
; [[Francium]]
Line 687: Line 679:
; [[Cam Taylor]]
; [[Cam Taylor]]
* [https://www.youtube.com/watch?v=Keclakcqie8 ''58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone''] (2021)
* [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]]
[[Category:Buzzard]]
Retrieved from "https://en.xen.wiki/w/58edo"