91edo: Difference between revisions

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→Regular temperament properties: bring meantone back as the 1/7-comma meantone page mentions 91edo and it's in the category too.
→Music: Add Modern Renderings section, starting with Maretu's ''Aishite ita no ni'' (2023) – microtonal cover in 91edo by Bryan Deister (2026)
 
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{{Infobox ET}}
{{Infobox ET}}
The '''91 equal divisions of the octave''' ('''91edo'''), or '''91-tone equal temperament''' ('''91tet''', '''91et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 91 parts of 13.187 [[cent]]s each.
{{ED intro}}


== Theory ==
== Theory ==
The 3, 5 and 7 for 91 are on the flat side, making this a mostly flat system. It provides the [[optimal patent val]] for 11- and 13-limit [[septimin]] temperament, and the 13-limit rank three [[tripod]] temperament, as well as the 11-limit rank four temperament tempering out [[245/242]] and the 13-limit rank five temperament tempering out [[105/104]], or rank four tempering out 105/104 and [[144/143]], or else 105/104 and [[196/195]] and hence [[225/224]] also. It 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. 91edo supports a variant of [[semaphore]] temperament which tempers out the 4613015762523/4398046511104 comma in the 2.3.7 subgroup, and is produced by a 19\91 generator.  
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 also corresponds to 135/128, the [[eigenmonzo]] (unchanged-interval) of [[1/7-comma meantone]], both being 7 steps. What's also remarkable is that in this instance the chromatic semitone is equal to one step of [[13edo]]. Since 135/128 is also equal to 1/13th of the octave, 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.  


91edo also tempers out the [[Decaononic|devil's tridecalimma]], which assigns 10/9 to 2/13ths of the octave. It also supports the [[trideci]] temperament, which in the 7-limit tempers out 4375/4374 and 83349/81920.  
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 is the second highest it 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]].
=== Odd harmonics ===
{{Harmonics in equal|91}}
 
=== Subsets and supersets ===
91 is the smallest composite number whose composite character is not immediately evident in the decimal system; it is, in fact, the product of 7 and 13. As such, 91edo contains [[7edo]] and [[13edo]] as subsets.  


The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp 91b val fifth of 54\91. From a regular temperament theory perspective, there is more than one way to interpret this, as they're all harmonically not very precise. First, in the 13-limit, is to assume that 27\91 is directly equivalent to 16/13 and set a temperament in the 2.3.5.7.16/13 subgroup, which produces a 27 & 91b temperament with the comma basis 91/90, 6272/6075, {{Monzo|84 13 11 -10}}. Second is to directly take the 27 & 91b val in the 13-limit, which can also be taken using the 27e & 91b.
=== Miscellany ===
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.


In the 13-limit, 91edo is an [[Optimal ET sequence|optimal ET]] for [[vidar]] temperament coming after [[87edo|87d]] val and before [[133edo|133d]] val, although it is no longer consistent in the 11-odd-limit.
== Intervals ==
{{Interval table}}


=== Odd harmonics ===
== Notation ==
{{Harmonics in equal|91}}
=== 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.
{{Sharpness-sharp7a}}


=== Miscellaneous properties ===
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
91 is the smallest composite number whose composite character is not immediately evident in the decimal system; it is, in fact, the product of 7 and 13.
{{Sharpness-sharp7}}


Since the number 7 has historically represented luck, and the number 13 has always stood for bad luck, from an aesthetic standpoint, the factoring of 91 represents a kind of "yin-yang", a combination of opposites.
=== Eliora's notation ===
[[User:Eliora|Eliora]], who believes the diatonic way of naming intervals in 91edo is not useful due to the fact that other temperaments and techniques for 91edo are more prominent, proposes a way of naming that merges the factors 7 and 13—7 equidistant notes are named do, re, mi, and 13 are named by some other virtue. The proposition is to use Old Slavic letter names, since no one uses them for naming or in mathematics. The {{nowrap|7 + 13}} naming convention can be called a duality notation. Intervals can be named through Latin ones for the 7-note scale, and Greek ones for the 13-note.


== Regular temperament properties ==
{| class="wikitable center-1 mw-collapsible mw-collapsed"
{| class="wikitable center-4 center-5 center-6"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 91edo
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! #
! [[TE simple badness|Relative]] (%)
! Eliora's naming system
! Eliora's notation
! Associated ratio
|-
|-
| 2.3
| 0
| {{monzo| -144 91 }}
| unison<br>perfect prime <br>perfect prota
| [{{val| 91 144 }}]
| A <br />Az (А)
| +0.963
| [[1/1]]
| 0.964
| 7.31
|-
|-
| 2.3.5
| 1
| 15625/15552, 43046721/41943040
| major prime <br>major prota
| [{{val| 91 144 211 }}]
| A# <br>Az#
| +1.202
| [[1728/1715]]
| 0.857
| 6.49
|-
|-
| 2.3.5.7
| 2  
| 225/224, 4375/4374, 50421/50000
| augmented prota
| [{{val| 91 144 211 255 }}]
| Az##
| +1.453
|  
| 0.860
| 6.51
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated<br>Ratio
! Temperaments
|-
|-
| 1
| 3
| 2\91
| biaugmented prota
| 26.37
| Az###
| 49/48
|  
| [[Sfourth]]
|-
|-
| 1
| 4
| 4\91
| bidiminished deiteria
| 52.75
| Buki♭♭♭
| 33/32
| [[33/32]]
| [[Quartkeenlig]] (91f)
|-
|-
| 1
|5
| 11\91
| diminished deiteria
| 145.05
| Buki♭♭
| 49/45
|  
| [[Swetneus]] (91ef)
|-
|-
| 1
| 6
| 19\91
| minor deiteria
| 250.55
| Buki♭
| 1240029/1048576
|  
| ''[[Semaphore]] (24 & 91 variant) †''
|-
|-
| 1
| 7
| 20\91
| neutral deiteria
| 263.74
| Buki (Б)
| 7/6
| [[135/128]]
| [[Septimin]] (91)
|-
|-
| 1
| 8
| 24\91
| major deiteria
| 316.48
| Buki#
| 6/5
|  
| [[Catakleismic]] (91f)
|-
|-
| 1
| 9
| 33\91
| augmented deiteria
| 435.16
| Buki##
| 9/7
|  
| [[Supermajor]]
|-
|-
| 1
| 10
| 34\91
| biaugmented deiteria
| 448.35
| Buki###
| 35/27
|  
| [[Semidimfourth]]
|-
|-
| 1
| 11
| 38\91
| bidiminished tritia
| 501.10
| Vedi♭♭♭
| 4/3
| [[13/12]], [[12/11]]
| ''Lalagu (12 & 79) (91) /'' [[meantone]] (91c)
|-
|-
| 1
| 12
| 44\91
| diminished tritia
| 580.22
| Vedi♭♭
| 7/5
|  
| [[Tritonic]]
|-
|-
| 7
| 13
| 38\91<br>(1\91)
| neutral secunde <br>minor tritia
| 501.10<br>(13.19)
| B<br>Vedi♭
| 4/3<br>(81/80)
| [[11/10]]
| [[Absurdity]]
|-
|-
| 13
| 14
| 38\91<br>(1\91)
| neural tritia
| 501.10<br>(13.19)
| Vedi (В)
| 4/3<br>(265/252)
| [[10/9]]
| [[Trideci]] (91) / [[aluminium]] (91c)
|}
† derived from scales in the Scales section, official name not decided upon yet.
 
== Intervals ==
[[User:Eliora|Eliora]], who believes the diatonic way of naming intervals in 91edo is not useful due to the fact that other temperaments and techniques for 91edo are more prominent, proposes a way of naming that merges the factors 7 and 13 - 7 equidistant notes are named do, re, mi, and 13 are named by some other virtue. The proposition is to use Old Slavic letter names, since no one uses them for naming or in mathematics. The 7 + 13 naming convention can be called a duality notation. Intervals can be named through Latin ones for the 7-note scale, and Greek ones for the 13-note.
 
{| class="wikitable mw-collapsible"
|+ style=white-space:nowrap | Table of intervals in 91edo
! #
! Eliora's Naming System
! Eliora's Notation
! Associated Ratio
|-
|-
|0
| 15
|unison <br>perfect prime <br>perfect prota
| major tritia
|A <br>Az (А)
| Vedi#
|[[1/1]]
| [[9/8]]
|-
|-
|1
| 16
|major prime <br>major prota
| augmented tritia
|A# <br>Az#
| Vedi##
| [[1728/1715]]
|  
|-
|-
|2
| 17
|augmented prota
| biaugmented tritia
|Az##
| Vedi###
|  
|  
|-
|-
|3
| 18
|biaugmented prota
| bidiminished tesseria
|Az###
| Glagol♭♭♭
| [[8/7]]
|-
| 19
| diminished tesseria
| Glagol♭♭
|  
|  
|-
|-
|4
| 20
|bidiminished deiteria
| minor tesseria
|Buki♭♭♭
| Glagol♭
| [[33/32]]
| [[7/6]]
|-
|-
|5
| 21
|diminished deiteria
| neutral tesseria
|Buki♭♭
| Glagol (Г)
|  
|  
|-
|-
|6
| 22
|minor deiteria
| major tesseria
|Buki♭
| Glagol#
| [[13/11]]
|-
| 23
| augmented tesseria
| Glagol##
|  
|  
|-
|-
|7
| 24
|neutral deiteria
| biaugmented tesseria
|Buki (Б)
| Glagol###
| [[135/128]]
| [[6/5]]
|-
|-
|8
| 25
|major deiteria
| bidiminished pemptia
|Buki#
| Dobro♭♭♭
|  
|  
|-
|-
|9  
| 26
|augmented deiteria
| neutral tertie <br>diminished pemptia
|Buki##
| C <br>Dobro♭♭
|  
| [[11/9]]
|-
| 27
| major tertie <br>minor pemptia
| C# <br />Dobro♭
| [[16/13]], 27/22
|-
|-
|10
| 28
|biaugmented deiteria
| neutral pemptia
|Buki###
| Dobro (Д)
|  
|  
|-
|-
|11
| 29
|bidiminished tritia
| major pemptia
|Vedi♭♭♭
| Dobro#
|[[13/12]], [[12/11]]
| [[5/4]]
|-
|-
|12
| 30
|diminished tritia
| augmented pemptia
|Vedi♭♭
| Dobro##
|  
|  
|-
|-
|13
| 31
|neutral secunde <br>minor tritia
| biaugmented pemptia
|B<br>Vedi♭
| Dobro###
|[[11/10]]
|  
|-
|-
|14
| 32
|neural tritia
| bidiminished hektia
|Vedi (В)
| Yest♭♭♭
|[[10/9]]
| [[14/11]]
|-
|-
|15
| 33
|major tritia
| diminished hektia
|Vedi#
| Yest♭♭
|[[9/8]]
| [[9/7]]
|-
|-
|16
| 34
|augmented tritia
| minor hektia
|Vedi##
| Yest♭
|  
|  
|-
|-
|17
| 35
|biaugmented tritia
| neutral hektia
|Vedi###
| Yest (Е)
|  
|  
|-
|-
|18
| 36
|bidiminished tesseria
| major hektia
|Glagol♭♭♭
| Yest#
|[[8/7]]
|  
|-
|-
|19
| 37
|diminished tesseria
| augmented hektia
|Glagol♭♭
| Yest##
|  
|  
|-
|-
|20
| 38
|minor tesseria
| biaugmented hektia
|Glagol♭
| Yest###
|[[7/6]]
| [[4/3]]
|-
|-
|21
| 39
|neutral tesseria
| neutral quarte <br>bidiminished hebdomia
|Glagol (Г)
| D<br>Zhivete♭♭♭
|  
|  
|-
|-
|22
| 40
|major tesseria
| diminished hebdomia
|Glagol#
| Zhivete♭♭
|[[13/11]]
|  
|-
|-
|23
| 41
|augmented tesseria
| minor hebdomia
|Glagol##
| Zhivete♭
|  
|  
|-
|-
|24
| 42
|biaugmented tesseria
| neutral hebdomia
|Glagol###
| Zhivete (Ж)
|[[6/5]]
| [[11/8]]
|-
|-
|25
| 43
|bidiminished pemptia
| major hebdomia
|Dobro♭♭♭
| Zhivete#
|  
|  
|-
|-
|26
| 44
|neutral tertie <br>diminished pemptia
| augmented hebdomia
|C <br>Dobro♭♭
| Zhivete##
|[[11/9]]
| [[7/5]]
|-
|-
|27
| 45
|major tertie <br>minor pemptia
| biaugmented hebdomia
|C# <br>Dobro♭
| Zhivete###
|[[16/13]], 27/22
|  
|-
|-
|28
| 46
|neutral pemptia
| bidiminished ogdonia
|Dobro (Д)
| Dzelo♭♭♭
|  
|  
|-
|-
|29
| 47
|major pemptia
| diminished ogdonia
|Dobro#
| Dzelo♭♭
|[[5/4]]
| [[10/7]]  
|-
|-
|30
| 48
|augmented pemptia
| minor ogdonia
|Dobro##
| Dzelo♭
|  
|  
|-
|-
|31
| 49
|biaugmented pemptia
| neutral ogdonia
|Dobro###
| Dzelo (Ѕ)
|  
|  
|-
|-
|32
| 52
|bidiminished hektia
| neutral quinte
|Yest♭♭♭
| E
|[[14/11]]
| 121/81
|-
|-
|33
| 53
|diminished hektia
| major quinte
|Yest♭♭
| E#
|[[9/7]]
| [[3/2]]
|-
|-
|34
| 54
|minor hektia
| augmented quinte <br>diminished ennatia
|Yest♭
| E## <br>Zemle♭♭
| [[256/169]]
|-
| 55
| minor ennatia
| Zemle♭
|  
|  
|-
|-
|35
| 56
|neutral hektia
| neutral ennatia
|Yest (Е)
| Zemle (З)
|  
|  
|-
|-
|36
| 63
|major hektia
| neutral decatia
|Yest#
| Izhe (И)
|  
|  
|-
|-
|37
| 64
|augmented hektia
| major decatia <br>minor sexte
|Yest##
| Izhe# <br>F♭
|  
|  
|-
|-
|38
| 65
|biaugmented hektia
| neutral sexte
|Yest###
| F
|[[4/3]]
|  
|-
|-
|39
| 70
|neutral quarte <br>bidiminished hebdomia
| neutral hendecatia
|D<br>Zhivete♭♭♭
| Jerve (Ђ)
|  
|  
|-
|-
|40
| 77
|diminished hebdomia
| neutral dodecatia
|Zhivete♭♭
| Kako (К)
|  
|  
|-
|-
|41
| 78
|minor hebdomia
| neutral septime
|Zhivete♭
| G
|  
|  
|-
|-
|42
| 84
|neutral hebdomia
| neutral decatotritia
|Zhivete (Ж)
| Ludi (Л)
|[[11/8]]
|-
|43
|major hebdomia
|Zhivete#
|  
|  
|-
|-
|44
| 91
|augmented hebdomia
| perfect octave <br>perfect decatotetartia
|Zhivete##
| A<br>Az (А)
|[[7/5]]
| [[2/1]] exact
|}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
|45
! rowspan="2" | [[Subgroup]]
|biaugmented hebdomia
! rowspan="2" | [[Comma list]]
|Zhivete###
! rowspan="2" | [[Mapping]]
|
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|46
! [[TE error|Absolute]] (¢)
|bidiminished ogdonia
! [[TE simple badness|Relative]] (%)
|Dzelo♭♭♭
|  
|-
|-
|47
| 2.3
|diminished ogdonia
| {{Monzo| -144 91 }}
|Dzelo♭♭
| {{Mapping| 91 144 }}
|[[10/7]]
| +0.963
| 0.964
| 7.31
|-
|-
|48
| 2.3.5
|minor ogdonia
| 15625/15552, 43046721/41943040
|Dzelo♭
| {{Mapping| 91 144 211 }}
|  
| +1.202
| 0.857
| 6.49
|-
|-
|49
| 2.3.5.7
|neutral ogdonia
| 225/224, 4375/4374, 50421/50000
|Dzelo (Ѕ)
| {{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
|-
|-
|52
! Periods<br>per 8ve
|neutral quinte
! Generator*
|E
! Cents*
|121/81
! Associated<br>ratio*
! Temperament
|-
|-
|53
| 1
|major quinte
| 2\91
|E#
| 26.37
|[[3/2]]
| 49/48
| [[Sfourth]]
|-
|-
|54
| 1
|augmented quinte <br>diminished ennatia
| 4\91
|E## <br>Zemle♭♭
| 52.75
|[[256/169]]
| 33/32
| [[Quartkeenlig]] (91f)
|-
|-
|55
| 1
|minor ennatia
| 11\91
|Zemle♭
| 145.05
|
| 49/45
| [[Swetneus]] (91ef)
|-
|-
|56
| 1
|neutral ennatia
| 19\91
|Zemle (З)
| 250.55
|
| 1240029/1048576
| ''[[Semaphore]] variant'' (24 & 91)**
|-
|-
|63
| 1
|neutral decatia
| 20\91
|Izhe (И)
| 263.74
|
| 7/6
| [[Septimin]] (91)
|-
|-
|64
| 1
|major decatia <br>minor sexte
| 24\91
|Izhe# <br>F♭
| 316.48
|
| 6/5
| [[Catakleismic]] (91f)
|-
|-
|65
| 1
|neutral sexte
| 33\91
|F
| 435.16
|
| 9/7
| [[Supermajor (temperament)|Supermajor]]
|-
|-
|70
| 1
|neutral hendecatia
| 34\91
|Jerve (Ђ)
| 448.35
|
| 35/27
| [[Semidimfourth]]
|-
|-
|77
| 1
|neutral dodecatia
| 38\91
|Kako (К)
| 501.10
|
| 4/3
| [[Python]]
|-
|-
|78
| 1
|neutral septime
| 44\91
|G
| 580.22
|
| 7/5
| [[Tritonic]]
|-
|-
|84
| 7
|neutral decatotritia
| 38\91<br>(1\91)
|Ludi (Л)
| 501.10<br>(13.19)
|
| 4/3<br>(81/80)
| [[Absurdity]]
|-
|-
|91
| 13
|perfect octave <br>perfect decatotetartia
| 38\91<br>(1\91)
|A<br>Az (А)
| 501.10<br>(13.19)
|[[2/1]] exact
| 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 ==
''See also: [[5- to 8-tone scales in 91edo]]''
{{See also| 5- to 10-tone scales in 91edo }}
*[[Semaphore5]]: 19 15 19 19 19
 
*[[Semaphore9]]: 15 4 15 4 15 4 15 15 4
* [[Semaphore5]]: 19 15 19 19 19
*[[Semaphore14]]: 4 11 4 4 11 4 4 11 4 11 4 4 11 4  
* [[Semaphore9]]: 15 4 15 4 15 4 15 15 4
* [[Semaphore14]]: 4 11 4 4 11 4 4 11 4 11 4 4 11 4  
* NaiveMajor[7]: 13 16 10 13 16 13 10
* NaiveMajor[7]: 13 16 10 13 16 13 10
* NaiveMinor[7]: 13 10 16 13 10 13 16
* NaiveMinor[7]: 13 10 16 13 10 13 16
Line 474: Line 487:
* SeptiminHijaz[9]: 5 15 11 9 11 9 5 15 11
* SeptiminHijaz[9]: 5 15 11 9 11 9 5 15 11
* Meantone[12]: 878787887878
* Meantone[12]: 878787887878
* [[Meantone43 in 91edo]]
* [[Meantone55 in 91edo]]
* NaiveOrwell[13]: 5795797597579
* NaiveOrwell[13]: 5795797597579
* ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9
* ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9
Line 480: Line 495:
* 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 ==
* [http://chrisvaisvil.com/dprk-ison-chase-12-of-91-edo-ambient/ DPRK ISON CHASE] by [[Chris Vaisvil]]
=== Modern renderings ===
* [https://www.youtube.com/watch?v=StCR6hcm5tM DPRK ISON CHASE - YouTube]
; {{W|Maretu}}
* [https://www.youtube.com/watch?v=_5WS7AGZxm4 Sadness - Nope] by Mercury Amalgam
* [https://www.youtube.com/shorts/7RDvArkSJrk ''Aishite ita no ni''] (2023) – microtonal cover in 91edo by [[Bryan Deister]] (2026)


[[Category:91edo| ]] <!-- main article -->
=== 21st century ===
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
; [[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]
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/HaYUAg30298 ''microtonal improvisation in 91edo''] (2025)
* [https://www.youtube.com/shorts/z6PeEocYMV8 ''improv 91edo''] (2025)
 
; [[Chris Vaisvil]]
* ''DPRK ISON CHASE'' (2014) – [http://chrisvaisvil.com/dprk-ison-chase-12-of-91-edo-ambient/ blog] | [https://www.youtube.com/watch?v=StCR6hcm5tM YouTube]
 
[[Category:Animist]]
[[Category:Frostmic]]
[[Category:Listen]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Septimin]]
[[Category:Septimin]]
[[Category:Tripod]]
[[Category:Tripod]]
[[Category:Cassacot]]
[[Category:Animist]]
{{Todo| cleanup }}

Latest revision as of 08:01, 14 May 2026

← 90edo 91edo 92edo →
Prime factorization 7 × 13
Step size 13.1868 ¢ 
Fifth 53\91 (698.901 ¢)
Semitones (A1:m2) 7:8 (92.31 ¢ : 105.5 ¢)
Consistency limit 9
Distinct consistency limit 9

91 equal divisions of the octave (abbreviated 91edo or 91ed2), also called 91-tone equal temperament (91tet) or 91 equal temperament (91et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 91 equal parts of about 13.2 ¢ each. Each step represents a frequency ratio of 21/91, or the 91st root of 2.

Theory

The harmonics 3, 5 and 7 for 91edo are on the flat side, making this a mostly flat system. It 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.

It also tempers out the [-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 [-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 ([24 -6 0 1 -5⟩), and as a corollary it is a tuning for the quartkeenlig temperament, which can also act as a stretched 23edo. In the 13-limit, it supports vidar and gives a reasonable tuning for its size.

Odd harmonics

Approximation of odd harmonics in 91edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -3.05 -3.90 -6.19 -6.11 +2.53 +3.43 +6.24 +0.54 +5.78 +3.94 +4.69
Relative (%) -23.2 -29.5 -46.9 -46.3 +19.2 +26.0 +47.3 +4.1 +43.9 +29.9 +35.6
Steps
(reduced) 		144
(53) 		211
(29) 		255
(73) 		288
(15) 		315
(42) 		337
(64) 		356
(83) 		372
(8) 		387
(23) 		400
(36) 		412
(48)

Subsets and supersets

91 is the smallest composite number whose composite character is not immediately evident in the decimal system; it is, in fact, the product of 7 and 13. As such, 91edo contains 7edo and 13edo as subsets.

Miscellany

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.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 13.2 ^D, E♭♭
2 26.4 ^^D, ^E♭♭
3 39.6 ^3D, ^^E♭♭
4 52.7 32/31, 33/32, 34/33 v3D♯, ^3E♭♭
5 65.9 vvD♯, v3E♭
6 79.1 23/22 vD♯, vvE♭
7 92.3 39/37 D♯, vE♭
8 105.5 17/16, 33/31 ^D♯, E♭
9 118.7 15/14 ^^D♯, ^E♭
10 131.9 27/25, 41/38 ^3D♯, ^^E♭
11 145.1 37/34 v3D𝄪, ^3E♭
12 158.2 34/31 vvD𝄪, v3E
13 171.4 32/29 vD𝄪, vvE
14 184.6 10/9 D𝄪, vE
15 197.8 28/25, 37/33 E
16 211 26/23 ^E, F♭
17 224.2 33/29 ^^E, ^F♭
18 237.4 39/34 ^3E, ^^F♭
19 250.5 37/32 v3E♯, ^3F♭
20 263.7 vvE♯, v3F
21 276.9 34/29 vE♯, vvF
22 290.1 13/11 E♯, vF
23 303.3 25/21, 31/26 F
24 316.5 6/5 ^F, G♭♭
25 329.7 23/19, 29/24 ^^F, ^G♭♭
26 342.9 39/32 ^3F, ^^G♭♭
27 356 v3F♯, ^3G♭♭
28 369.2 vvF♯, v3G♭
29 382.4 vF♯, vvG♭
30 395.6 39/31 F♯, vG♭
31 408.8 ^F♯, G♭
32 422 37/29 ^^F♯, ^G♭
33 435.2 9/7 ^3F♯, ^^G♭
34 448.4 22/17, 35/27 v3F𝄪, ^3G♭
35 461.5 vvF𝄪, v3G
36 474.7 vF𝄪, vvG
37 487.9 F𝄪, vG
38 501.1 G
39 514.3 31/23, 39/29 ^G, A♭♭
40 527.5 ^^G, ^A♭♭
41 540.7 26/19 ^3G, ^^A♭♭
42 553.8 11/8 v3G♯, ^3A♭♭
43 567 25/18 vvG♯, v3A♭
44 580.2 7/5 vG♯, vvA♭
45 593.4 31/22 G♯, vA♭
46 606.6 ^G♯, A♭
47 619.8 10/7 ^^G♯, ^A♭
48 633 36/25 ^3G♯, ^^A♭
49 646.2 16/11 v3G𝄪, ^3A♭
50 659.3 19/13 vvG𝄪, v3A
51 672.5 vG𝄪, vvA
52 685.7 G𝄪, vA
53 698.9 A
54 712.1 ^A, B♭♭
55 725.3 ^^A, ^B♭♭
56 738.5 ^3A, ^^B♭♭
57 751.6 17/11, 37/24 v3A♯, ^3B♭♭
58 764.8 14/9 vvA♯, v3B♭
59 778 vA♯, vvB♭
60 791.2 41/26 A♯, vB♭
61 804.4 ^A♯, B♭
62 817.6 ^^A♯, ^B♭
63 830.8 ^3A♯, ^^B♭
64 844 v3A𝄪, ^3B♭
65 857.1 vvA𝄪, v3B
66 870.3 38/23 vA𝄪, vvB
67 883.5 5/3 A𝄪, vB
68 896.7 42/25 B
69 909.9 22/13 ^B, C♭
70 923.1 29/17 ^^B, ^C♭
71 936.3 ^3B, ^^C♭
72 949.5 v3B♯, ^3C♭
73 962.6 vvB♯, v3C
74 975.8 vB♯, vvC
75 989 23/13, 39/22 B♯, vC
76 1002.2 25/14, 41/23 C
77 1015.4 9/5 ^C, D♭♭
78 1028.6 29/16 ^^C, ^D♭♭
79 1041.8 31/17 ^3C, ^^D♭♭
80 1054.9 v3C♯, ^3D♭♭
81 1068.1 vvC♯, v3D♭
82 1081.3 28/15 vC♯, vvD♭
83 1094.5 32/17 C♯, vD♭
84 1107.7 ^C♯, D♭
85 1120.9 ^^C♯, ^D♭
86 1134.1 ^3C♯, ^^D♭
87 1147.3 31/16, 33/17 v3C𝄪, ^3D♭
88 1160.4 vvC𝄪, v3D
89 1173.6 vC𝄪, vvD
90 1186.8 C𝄪, vD
91 1200 2/1 D

Notation

Ups and downs notation

91edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol                
                 
Flat symbol              
                 

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Eliora's notation

Eliora, who believes the diatonic way of naming intervals in 91edo is not useful due to the fact that other temperaments and techniques for 91edo are more prominent, proposes a way of naming that merges the factors 7 and 13—7 equidistant notes are named do, re, mi, and 13 are named by some other virtue. The proposition is to use Old Slavic letter names, since no one uses them for naming or in mathematics. The 7 + 13 naming convention can be called a duality notation. Intervals can be named through Latin ones for the 7-note scale, and Greek ones for the 13-note.

Table of intervals in 91edo
# Eliora's naming system Eliora's notation Associated ratio
0 unison
perfect prime
perfect prota 		A
Az (А) 		1/1
1 major prime
major prota 		A#
Az# 		1728/1715
2 augmented prota Az##
3 biaugmented prota Az###
4 bidiminished deiteria Buki♭♭♭ 33/32
5 diminished deiteria Buki♭♭
6 minor deiteria Buki♭
7 neutral deiteria Buki (Б) 135/128
8 major deiteria Buki#
9 augmented deiteria Buki##
10 biaugmented deiteria Buki###
11 bidiminished tritia Vedi♭♭♭ 13/12, 12/11
12 diminished tritia Vedi♭♭
13 neutral secunde
minor tritia 		B
Vedi♭ 		11/10
14 neural tritia Vedi (В) 10/9
15 major tritia Vedi# 9/8
16 augmented tritia Vedi##
17 biaugmented tritia Vedi###
18 bidiminished tesseria Glagol♭♭♭ 8/7
19 diminished tesseria Glagol♭♭
20 minor tesseria Glagol♭ 7/6
21 neutral tesseria Glagol (Г)
22 major tesseria Glagol# 13/11
23 augmented tesseria Glagol##
24 biaugmented tesseria Glagol### 6/5
25 bidiminished pemptia Dobro♭♭♭
26 neutral tertie
diminished pemptia 		C
Dobro♭♭ 		11/9
27 major tertie
minor pemptia 		C#
Dobro♭ 		16/13, 27/22
28 neutral pemptia Dobro (Д)
29 major pemptia Dobro# 5/4
30 augmented pemptia Dobro##
31 biaugmented pemptia Dobro###
32 bidiminished hektia Yest♭♭♭ 14/11
33 diminished hektia Yest♭♭ 9/7
34 minor hektia Yest♭
35 neutral hektia Yest (Е)
36 major hektia Yest#
37 augmented hektia Yest##
38 biaugmented hektia Yest### 4/3
39 neutral quarte
bidiminished hebdomia 		D
Zhivete♭♭♭
40 diminished hebdomia Zhivete♭♭
41 minor hebdomia Zhivete♭
42 neutral hebdomia Zhivete (Ж) 11/8
43 major hebdomia Zhivete#
44 augmented hebdomia Zhivete## 7/5
45 biaugmented hebdomia Zhivete###
46 bidiminished ogdonia Dzelo♭♭♭
47 diminished ogdonia Dzelo♭♭ 10/7
48 minor ogdonia Dzelo♭
49 neutral ogdonia Dzelo (Ѕ)
52 neutral quinte E 121/81
53 major quinte E# 3/2
54 augmented quinte
diminished ennatia 		E##
Zemle♭♭ 		256/169
55 minor ennatia Zemle♭
56 neutral ennatia Zemle (З)
63 neutral decatia Izhe (И)
64 major decatia
minor sexte 		Izhe#
F♭
65 neutral sexte F
70 neutral hendecatia Jerve (Ђ)
77 neutral dodecatia Kako (К)
78 neutral septime G
84 neutral decatotritia Ludi (Л)
91 perfect octave
perfect decatotetartia 		A
Az (А) 		2/1 exact

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-144 91⟩ [⟨91 144]] +0.963 0.964 7.31
2.3.5 15625/15552, 43046721/41943040 [⟨91 144 211]] +1.202 0.857 6.49
2.3.5.7 225/224, 4375/4374, 50421/50000 [⟨91 144 211 255]] +1.453 0.860 6.51

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
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
(1\91) 		501.10
(13.19) 		4/3
(81/80) 		Absurdity
13 38\91
(1\91) 		501.10
(13.19) 		4/3
(265/252) 		Trideci (91)
Aluminium (91c)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

** Derived from scales in the Scales section, official name not decided upon yet.

Scales

See also: 5- to 10-tone scales in 91edo
  • Semaphore5: 19 15 19 19 19
  • Semaphore9: 15 4 15 4 15 4 15 15 4
  • Semaphore14: 4 11 4 4 11 4 4 11 4 11 4 4 11 4
  • NaiveMajor[7]: 13 16 10 13 16 13 10
  • NaiveMinor[7]: 13 10 16 13 10 13 16
  • Septimin[9]: 11 9 11 9 11 9 11 9 11
  • SeptiminHijaz[9]: 5 15 11 9 11 9 5 15 11
  • Meantone[12]: 878787887878
  • Meantone43 in 91edo
  • Meantone55 in 91edo
  • NaiveOrwell[13]: 5795797597579
  • ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9
  • HungarianNaiveSurorwell[13]: 7 7 8 6 11 5 5 7 10 4 4 13 4
  • Quartkeenlig[23]: 44444444444444444444443
  • ConcocticSubset[7]: 17 10 17 10 17 10 17
  • ConcocticMaqamSikah: 10 17 17 10 10 17 10

Instruments

A Lumatone mapping for 91edo is available.

Music

Modern renderings

Maretu

21st century

Mercury Amalgam
Bryan Deister
Chris Vaisvil
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