Xenharmonic Wiki

58edo: Difference between revisions

← Older edit
VisualWikitext
→Intervals: reduce cent values to one decimal place, per discussion on Discord. Mark inconsistently mapped intervals italic
→Music: Add Bryan Deister's ''Waltz in 58edo'' (2025)
 
(14 intermediate revisions by 6 users not shown)
Line 4: Line 4:


== Theory ==
== 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]].  
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 fifth with [[29edo]].
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 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 [[support]]s [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]], [[thuja]] [[regular temperament|temperament]]s 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]].
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.
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.
Line 14: Line 14:
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|58}}
{{Harmonics in equal|58}}
=== Octave stretch ===
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 ===
58edo contains [[2edo]] and [[29edo]] as subsets.  
58edo contains [[2edo]] and [[29edo]] as subsets.


== Intervals ==
== Intervals ==
Line 33: Line 36:
| 1
| 1
| 20.7
| 20.7
| [[56/55]], [[64/63]], [[81/80]], ''[[128/125]]''
| [[56/55]], [[64/63]], [[81/80]], [[91/90]], [[105/104]]
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 41.4
| 41.4
| [[36/35]], [[49/48]], [[50/49]], [[55/54]]
| [[36/35]], [[40/39]], [[45/44]], [[49/48]], [[50/49]], [[55/54]]
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|-
|-
Line 58: Line 61:
| 6
| 6
| 124.1
| 124.1
| [[14/13]], [[15/14]], [[27/25]]
| [[14/13]], [[15/14]]
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|-
|-
Line 108: Line 111:
| 16
| 16
| 331.0
| 331.0
| [[17/14]]
| [[17/14]], [[40/33]]
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|-
|-
Line 118: Line 121:
| 18
| 18
| 372.4
| 372.4
| [[21/17]]
| [[21/17]], [[26/21]]
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|-
|-
Line 228: Line 231:
| 40
| 40
| 827.6
| 827.6
| [[34/21]]
| [[21/13]], [[34/21]]
| {{UDnote|step=40}}
| {{UDnote|step=40}}
|-
|-
Line 238: Line 241:
| 42
| 42
| 869.0
| 869.0
| [[28/17]]
| [[28/17]], [[33/20]]
| {{UDnote|step=42}}
| {{UDnote|step=42}}
|-
|-
Line 308: Line 311:
| 56
| 56
| 1158.6
| 1158.6
| [[35/18]], [[49/25]], [[96/49]], [[108/55]]
| [[35/18]], [[39/20]], [[49/25]], [[88/45]], [[96/49]], [[108/55]]
| {{UDnote|step=56}}
| {{UDnote|step=56}}
|-
|-
| 57
| 57
| 1179.3
| 1179.3
| [[55/28]], [[63/32]], ''[[125/64]]'', [[160/81]]
| [[55/28]], [[63/32]], [[160/81]], [[180/91]], [[208/105]]
| {{UDnote|step=57}}
| {{UDnote|step=57}}
|-
|-
Line 321: Line 324:
| {{UDnote|step=58}}
| {{UDnote|step=58}}
|}
|}
<nowiki/> As a 17-limit temperament
<nowiki/>* As a 17-limit temperament, inconsistently mapped intervals in ''italic''


== Notation ==
== Notation ==
=== Ups and downs 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.
58edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}


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


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


Line 495: Line 502:
=== Interval mappings ===
=== Interval mappings ===
{{15-odd-limit|58}}
{{15-odd-limit|58}}
=== Zeta peak index ===
{| class="wikitable center-all"
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[289zpi]]
| 58.0667185533159
| 20.6658827964969
| 7.814035
| 1.358357
| 18.056292
| 58edo
| 1198.62120219682
| 16
| 12
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 540: Line 517:
| 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
| 1.22
| 1.22
Line 547: Line 524:
| 2.3.5.7
| 2.3.5.7
| 126/125, 1728/1715, 2048/2025
| 126/125, 1728/1715, 2048/2025
| {{mapping| 58 92 135 163 }}
| {{Mapping| 58 92 135 163 }}
| −1.29
| −1.29
| 1.05
| 1.05
Line 554: Line 531:
| 2.3.5.7.11
| 2.3.5.7.11
| 126/125, 176/175, 243/242, 896/891
| 126/125, 176/175, 243/242, 896/891
| {{mapping| 58 92 135 163 201 }}
| {{Mapping| 58 92 135 163 201 }}
| −1.45
| −1.45
| 1.00
| 1.00
Line 561: Line 538:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 126/125, 144/143, 176/175, 196/195, 364/363
| 126/125, 144/143, 176/175, 196/195, 364/363
| {{mapping| 58 92 135 163 201 215 }}
| {{Mapping| 58 92 135 163 201 215 }}
| −1.56
| −1.56
| 0.94
| 0.94
Line 568: Line 545:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 126/125, 136/135, 144/143, 176/175, 196/195, 364/363
| 126/125, 136/135, 144/143, 176/175, 196/195, 364/363
| {{mapping| 58 92 135 163 201 215 237 }}
| {{Mapping| 58 92 135 163 201 215 237 }}
| −1.28
| −1.28
| 1.10
| 1.10
Line 587: Line 564:
| 1
| 1
| 3\58
| 3\58
| 62.07
| 62.1
| 28/27
| 28/27
| [[Unicorn]] / alicorn / qilin
| [[Unicorn]] / alicorn / qilin
Line 593: Line 570:
| 1
| 1
| 11\58
| 11\58
| 227.59
| 227.6
| 8/7
| 8/7
| [[Gorgik]]
| [[Gorgik]]
Line 599: Line 576:
| 1
| 1
| 13\58
| 13\58
| 268.97
| 269.0
| 7/6
| 7/6
| [[Infraorwell]]
| [[Infraorwell]]
Line 605: Line 582:
| 1
| 1
| 15\58
| 15\58
| 310.34
| 310.3
| 6/5
| 6/5
| [[Myna]]
| [[Myna]]
Line 611: Line 588:
| 1
| 1
| 17\58
| 17\58
| 351.72
| 351.7
| 49/40
| 49/40
| [[Hemififths]]
| [[Hemififths]]
Line 617: Line 594:
| 1
| 1
| 19\58
| 19\58
| 393.10
| 393.1
| 64/51
| 64/51
| [[Emmthird]]
| [[Emmthird]]
Line 623: Line 600:
| 1
| 1
| 23\58
| 23\58
| 475.86
| 475.9
| 21/16
| 21/16
| [[Buzzard]] / [[subfourth]]
| [[Buzzard]] / [[subfourth]]
Line 629: Line 606:
| 1
| 1
| 27\58
| 27\58
| 558.62
| 558.6
| 11/8
| 11/8
| [[Thuja]]
| [[Thuja]]
Line 635: Line 612:
| 2
| 2
| 3\58
| 3\58
| 62.07
| 62.1
| 28/27
| 28/27
| [[Monocerus]]
| [[Monocerus]]
Line 641: Line 618:
| 2
| 2
| 1\58
| 1\58
| 20.69
| 20.7
| 81/80
| 81/80
| [[Bicommatic]]
| [[Bicommatic]]
Line 647: Line 624:
| 2
| 2
| 9\58
| 9\58
| 186.21
| 186.2
| 10/9
| 10/9
| [[Secant]]
| [[Secant]]
Line 653: Line 630:
| 2
| 2
| 17\58<br>(12\58)
| 17\58<br>(12\58)
| 351.72<br>(248.28)
| 351.7<br>(248.3)
| 11/9<br>(15/13)
| 11/9<br>(15/13)
| [[Sruti]]
| [[Sruti]]
Line 659: Line 636:
| 2
| 2
| 21\58<br>(8\58)
| 21\58<br>(8\58)
| 434.48<br>(165.52)
| 434.5<br>(165.5)
| 9/7<br>(11/10)
| 9/7<br>(11/10)
| [[Echidna]]
| [[Echidna]]
Line 665: Line 642:
| 2
| 2
| 24\58<br>(5\58)
| 24\58<br>(5\58)
| 496.55<br>(103.45)
| 496.6<br>(103.4)
| 4/3<br>(17/16)
| 4/3<br>(17/16)
| [[Diaschismic]]
| [[Diaschismic]]
Line 671: Line 648:
| 2
| 2
| 25\58<br>(4\58)
| 25\58<br>(4\58)
| 517.24<br>(82.76)
| 517.2<br>(82.8)
| 27/20<br>(21/20)
| 27/20<br>(21/20)
| [[Harry]]
| [[Harry]]
Line 677: Line 654:
| 29
| 29
| 19\58<br>(1\58)
| 19\58<br>(1\58)
| 393.10<br>(20.69)
| 393.1<br>(20.7)
| 5/4<br>(91/90)
| 5/4<br>(91/90)
| [[Mystery]]
| [[Mystery]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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 &amp; 140}}), [[supers]] ({{nowrap|58 &amp; 152}}), [[condor]] ({{nowrap|58 &amp; 159}}), and [[eagle]] ({{nowrap|58 &amp; 212}}).
58et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).


== Scales ==
== Scales ==
Line 699: Line 676:
== Music ==
== Music ==
; [[Jeff Brown]]
; [[Jeff Brown]]
* [https://www.youtube.com/watch?v=0373hBH87LY ''Fruitbats in Formation'']
* [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)


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


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


[[Category:Buzzard]]
[[Category:Buzzard]]
[[Category:Diaschismic]]
[[Category:Diaschismic]]
[[Category:Harry]]
[[Category:Harry]]
[[Category:Harry Partch]]
[[Category:Hemififths]]
[[Category:Hemififths]]
[[Category:Listen]]
[[Category:Myna]]
[[Category:Myna]]
[[Category:Mystery]]
[[Category:Mystery]]
[[Category:Harry Partch]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/58edo"