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91edo: Difference between revisions

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The [[harmonic]]s [[3/1|3]], [[5/1|5]] and [[7/1|7]] for 91edo are on the flat side, making this a mostly flat system. It [[tempering out|tempers out]] [[15625/15552]] in the 5-limit, [[225/224]] and [[4375/4374]] in the 7-limit, [[245/242]], [[385/384]] in the 11-limit, and [[105/104]], [[144/143]], [[196/195]] in the 13-limit. It provides the [[optimal patent val]] for 11- and 13-limit [[septimin]] temperament, and the 13-limit rank-3 [[tripod]] temperament, as well as the 11-limit rank-4 temperament tempering out 245/242 and the 13-limit rank-5 temperament tempering out 105/104, or rank-4 tempering out 105/104 and 144/143, or else 105/104 and 196/195 and hence 225/224 also.  
The [[harmonic]]s [[3/1|3]], [[5/1|5]] and [[7/1|7]] for 91edo are on the flat side, making this a mostly flat system. It [[tempering out|tempers out]] [[15625/15552]] in the 5-limit, [[225/224]] and [[4375/4374]] in the 7-limit, [[245/242]], [[385/384]] in the 11-limit, and [[105/104]], [[144/143]], [[196/195]] in the 13-limit. It provides the [[optimal patent val]] for 11- and 13-limit [[septimin]] temperament, and the 13-limit rank-3 [[tripod]] temperament, as well as the 11-limit rank-4 temperament tempering out 245/242 and the 13-limit rank-5 temperament tempering out 105/104, or rank-4 tempering out 105/104 and 144/143, or else 105/104 and 196/195 and hence 225/224 also.  


Using the 91c val, it is audibly indistinguishable from a closed system of [[1/7-comma meantone]], with a 5th only 0.018 cents sharper. The chromatic semitone in this scale corresponds to 135/128, the [[eigenmonzo]] (unchanged-interval) of [[1/7-comma meantone]]. Being 7 steps, what is also remarkable is that in this instance the chromatic semitone is equal to one step of [[13edo]]. Since 135/128 is also equal to 1/13 of the octave, the 91c [[val]] tempers out the [[aluminium comma]] in the 5-limit.  
Using the 91c val, it is audibly indistinguishable from a closed system of [[1/7-comma meantone]], with a 5th only 0.018 cents sharper. The chromatic semitone in this scale corresponds to 135/128, the [[eigenmonzo]] (unchanged interval) of [[1/7-comma meantone]]. Being 7 steps, what is also remarkable is that in this instance the chromatic semitone is equal to one step of [[13edo]]. Since 135/128 is also equal to 1/13 of the octave, the 91c [[val]] tempers out the [[aluminium comma]] in the 5-limit.  


It also tempers out the {{monzo| -11 26 -13 }}, the tridecatonic comma, which assigns [[10/9]] to 2/13 of the octave, and it supports [[trideci]] in the 7-limit, tempering out 4375/4374 and 83349/81920. It supports a variant of [[semaphore]] temperament which tempers out the {{monzo| -42 23 2 }} comma in the 2.3.7 [[subgroup]], and is generated by a 19\91 generator. It is the second highest in a series of four consecutive edos that temper out [[quartisma]] ({{monzo| 24 -6 0 1 -5 }}), and as a corollary it is a tuning for the [[quartkeenlig]] temperament, which can also act as a [[23edo and octave stretching|stretched]] [[23edo]]. In the 13-limit, it supports [[vidar]] and gives a reasonable tuning for its size.
It also tempers out the {{monzo| -11 26 -13 }}, the tridecatonic comma, which assigns [[10/9]] to 2/13 of the octave, and it supports [[trideci]] in the 7-limit, tempering out 4375/4374 and 83349/81920. It supports a variant of [[semaphore]] temperament which tempers out the {{monzo| -42 23 2 }} comma in the 2.3.7 [[subgroup]], and is generated by a 19\91 generator. It is the second highest in a series of four consecutive edos that temper out [[quartisma]] ({{monzo| 24 -6 0 1 -5 }}), and as a corollary it is a tuning for the [[quartkeenlig]] temperament, which can also act as a [[23edo and octave stretching|stretched]] [[23edo]]. In the 13-limit, it supports [[vidar]] and gives a reasonable tuning for its size.
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The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp fifth of 54\91, representing 3/2 in the 91b val.
The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp fifth of 54\91, representing 3/2 in the 91b val.


== Regular temperament properties ==
== Intervals ==
{| 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
| {{monzo| -144 91 }}
| {{mapping| 91 144 }}
| +0.963
| 0.964
| 7.31
|-
| 2.3.5
| 15625/15552, 43046721/41943040
| {{mapping| 91 144 211 }}
| +1.202
| 0.857
| 6.49
|-
| 2.3.5.7
| 225/224, 4375/4374, 50421/50000
| {{mapping| 91 144 211 255 }}
| +1.453
| 0.860
| 6.51
|}
 
=== 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
| 2\91
| 26.37
| 49/48
| [[Sfourth]]
|-
| 1
| 4\91
| 52.75
| 33/32
| [[Quartkeenlig]] (91f)
|-
| 1
| 11\91
| 145.05
| 49/45
| [[Swetneus]] (91ef)
|-
| 1
| 19\91
| 250.55
| 1240029/1048576
| ''[[Semaphore]] variant'' (24 &amp; 91)**
|-
| 1
| 20\91
| 263.74
| 7/6
| [[Septimin]] (91)
|-
| 1
| 24\91
| 316.48
| 6/5
| [[Catakleismic]] (91f)
|-
| 1
| 33\91
| 435.16
| 9/7
| [[Supermajor]]
|-
| 1
| 34\91
| 448.35
| 35/27
| [[Semidimfourth]]
|-
| 1
| 38\91
| 501.10
| 4/3
| [[Python]]
|-
| 1
| 44\91
| 580.22
| 7/5
| [[Tritonic]]
|-
| 7
| 38\91<br />(1\91)
| 501.10<br />(13.19)
| 4/3<br />(81/80)
| [[Absurdity]]
|-
| 13
| 38\91<br />(1\91)
| 501.10<br />(13.19)
| 4/3<br />(265/252)
| [[Trideci]] (91)<br />[[Aluminium]] (91c)
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
<nowiki />** Derived from scales in the Scales section, official name not decided upon yet.
 
== Interval table ==
{{Interval table}}
{{Interval table}}


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Ups and downs notation ===
91edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
91edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}
{{Sharpness-sharp7a}}
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| [[2/1]] exact
| [[2/1]] exact
|}
|}
== 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
| {{Monzo| -144 91 }}
| {{Mapping| 91 144 }}
| +0.963
| 0.964
| 7.31
|-
| 2.3.5
| 15625/15552, 43046721/41943040
| {{Mapping| 91 144 211 }}
| +1.202
| 0.857
| 6.49
|-
| 2.3.5.7
| 225/224, 4375/4374, 50421/50000
| {{Mapping| 91 144 211 255 }}
| +1.453
| 0.860
| 6.51
|}
=== 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
| 2\91
| 26.37
| 49/48
| [[Sfourth]]
|-
| 1
| 4\91
| 52.75
| 33/32
| [[Quartkeenlig]] (91f)
|-
| 1
| 11\91
| 145.05
| 49/45
| [[Swetneus]] (91ef)
|-
| 1
| 19\91
| 250.55
| 1240029/1048576
| ''[[Semaphore]] variant'' (24 & 91)**
|-
| 1
| 20\91
| 263.74
| 7/6
| [[Septimin]] (91)
|-
| 1
| 24\91
| 316.48
| 6/5
| [[Catakleismic]] (91f)
|-
| 1
| 33\91
| 435.16
| 9/7
| [[Supermajor (temperament)|Supermajor]]
|-
| 1
| 34\91
| 448.35
| 35/27
| [[Semidimfourth]]
|-
| 1
| 38\91
| 501.10
| 4/3
| [[Python]]
|-
| 1
| 44\91
| 580.22
| 7/5
| [[Tritonic]]
|-
| 7
| 38\91<br>(1\91)
| 501.10<br>(13.19)
| 4/3<br>(81/80)
| [[Absurdity]]
|-
| 13
| 38\91<br>(1\91)
| 501.10<br>(13.19)
| 4/3<br>(265/252)
| [[Trideci]] (91)<br>[[Aluminium]] (91c)
|}
<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
<nowiki/>** Derived from scales in the Scales section, official name not decided upon yet.


== Scales ==
== Scales ==
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* ConcocticSubset[7]: 17 10 17 10 17 10 17
* ConcocticSubset[7]: 17 10 17 10 17 10 17
* ConcocticMaqamSikah: 10 17 17 10 10 17 10
* ConcocticMaqamSikah: 10 17 17 10 10 17 10
== Instruments ==
A [[Lumatone mapping for 91edo]] is available.


== Music ==
== Music ==
=== Modern renderings ===
; {{W|Maretu}}
* [https://www.youtube.com/shorts/7RDvArkSJrk ''Aishite ita no ni''] (2023) – microtonal cover in 91edo by [[Bryan Deister]] (2026)
=== 21st century ===
; [[Mercury Amalgam]]
; [[Mercury Amalgam]]
* ''Sadness - Nope'' (2022) – [https://mercuryamalgam.bandcamp.com/track/sadness-nope-the-molecular-agoge-pt-2 Bandcamp] | [https://www.youtube.com/watch?v=_5WS7AGZxm4 YouTube]
* ''Sadness - Nope'' (2022) – [https://mercuryamalgam.bandcamp.com/track/sadness-nope-the-molecular-agoge-pt-2 Bandcamp] | [https://www.youtube.com/watch?v=_5WS7AGZxm4 YouTube]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/HaYUAg30298 ''microtonal improvisation in 91edo''] (2025)
* [https://www.youtube.com/shorts/z6PeEocYMV8 ''improv 91edo''] (2025)


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
Retrieved from "https://en.xen.wiki/w/91edo"