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58edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{Wikipedia|58 equal temperament}}
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-06-22 07:32:19 UTC</tt>.<br>
{{ED intro}}
: The original revision id was <tt>238145229</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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. It 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-limit|11]], [[13-limit|13]] and [[17-limit]]s. It is the smallest equal temperament which is consistent through the 17-limit, and is also 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 and buzzard [[temperament]]s.


While the 17th harmonic is a cent and a half cent flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system.
== 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]].  


=Scales=
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.
[[hemif7]]
 
[[hemif10]]
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.
[[hemif17]]</pre></div>
 
<h4>Original HTML content:</h4>
=== Prime harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;58edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;58 equal temperament&lt;/em&gt;, 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. It 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 &lt;a class="wiki_link" href="/11-limit"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/13-limit"&gt;13&lt;/a&gt; and &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt;s. It is the smallest equal temperament which is consistent through the 17-limit, and is also the first et to map the entire 11-limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt; to distinct scale steps, and hence the first et which can define a version of the famous 43-note &lt;a class="wiki_link" href="/Genesis%20scale"&gt;Genesis scale&lt;/a&gt; of &lt;a class="wiki_link" href="/Harry%20Partch"&gt;Harry Partch&lt;/a&gt;. It supports hemififths, myna, diaschismic, harry, mystery and buzzard &lt;a class="wiki_link" href="/temperament"&gt;temperament&lt;/a&gt;s.&lt;br /&gt;
{{Harmonics in equal|58}}
&lt;br /&gt;
 
While the 17th harmonic is a cent and a half cent flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system.&lt;br /&gt;
=== As a tuning of other temperaments ===
&lt;br /&gt;
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]].
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Scales&lt;/h1&gt;
 
&lt;a class="wiki_link" href="/hemif7"&gt;hemif7&lt;/a&gt;&lt;br /&gt;
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.
&lt;a class="wiki_link" href="/hemif10"&gt;hemif10&lt;/a&gt;&lt;br /&gt;
 
&lt;a class="wiki_link" href="/hemif17"&gt;hemif17&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
58edo contains [[2edo]] and [[29edo]] as subsets.
 
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4""
|-
! #
! Cents
! Approximate ratios*
! [[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]]''
| {{UDnote|step=4}}
|-
| 5
| 103.4
| [[16/15]], [[17/16]], [[18/17]]
| {{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]]
| {{UDnote|step=10}}
|-
| 11
| 227.6
| [[8/7]]
| {{UDnote|step=11}}
|-
| 12
| 248.3
| [[15/13]]
| {{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]]
| {{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]]
| {{UDnote|step=19}}
|-
| 20
| 413.8
| [[14/11]]
| {{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]]
| {{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]]
| {{UDnote|step=31}}
|-
| 32
| 662.1
| [[22/15]]
| {{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]]
| {{UDnote|step=38}}
|-
| 39
| 806.9
| [[8/5]]
| {{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]]
| {{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]]
| {{UDnote|step=46}}
|-
| 47
| 972.4
| [[7/4]]
| {{UDnote|step=47}}
|-
| 48
| 993.1
| [[16/9]], [[30/17]]
| {{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]]
| {{UDnote|step=53}}
|-
| 54
| 1117.2
| [[21/11]], [[40/21]], ''[[48/25]]''
| {{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 ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] for 58edo uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, then arrows can be attached to quartertone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
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}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{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 ====
{{Sagittal chart|Evo}}
 
==== Evo-SZ flavor ====
{{Sagittal chart|Evo-SZ}}
 
==== Revo flavor ====
{{Sagittal chart}}
 
=== 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)
|-
! #
! 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
| 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<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 #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
 
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)
 
; [[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:Diaschismic]]
[[Category:Harry]]
[[Category:Hemififths]]
[[Category:Myna]]
[[Category:Mystery]]
[[Category:Harry Partch]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/58edo"