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→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}}
| Prime factorization = 7 × 13
{{ED intro}}
| Step size = 13.1868¢
| Fifth = 53\91 (698.9¢)
| Semitones = 7:8 (92.3¢ : 105.5¢)
| Consistency = 9
}}
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.


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. From an aesthetic standpoint, the factoring of 91 represents a kind of "yin-yang" since historically, the number 7 symbolizes luck and 13 misfortune.  
== Theory ==
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.  


== Theory ==
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.
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. It is the second highest it a series of four consecutive edos that temper out [[quartisma]] ({{monzo| 24 -6 0 1 -5 }}). 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.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|91}}
{{Harmonics in equal|91}}


== Regular temperament properties ==
=== Subsets and supersets ===
{| class="wikitable center-4 center-5 center-6"
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.
! rowspan="2" | Subgroup
 
! rowspan="2" | [[Comma list]]
=== Miscellany ===
! rowspan="2" | [[Mapping]]
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.
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -144 91 }}
| [{{val| 91 144 }}]
| +0.963
| 0.964
| 7.31
|-
| 2.3.5
| 15625/15552, 43046721/41943040
| [{{val| 91 144 211 }}]
| +1.202
| 0.857
| 6.49
|-
| 2.3.5.7
| 225/224, 4375/4374, 50421/50000
| [{{val| 91 144 211 255 }}]
| +1.453
| 0.860
| 6.51
|}


== Intervals ==
== Intervals ==
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.
{{Interval table}}


{| class="wikitable mw-collapsible"
== Notation ==
|+ style=white-space:nowrap | Table of intervals in 91edo
=== 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}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}
 
=== 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.
 
{| class="wikitable center-1 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 91edo
|-
! #
! #
! Eliora's Naming System
! Eliora's naming system
! Eliora's Notation
! Eliora's notation
! Associated Ratio
! Associated ratio
|-
| 0
| unison<br>perfect prime <br>perfect prota
| A <br />Az (А)
| [[1/1]]
|-
| 1
| major prime <br>major prota
| A# <br>Az#
| [[1728/1715]]
|-
|-
|0
| 2
|unison <br>perfect prime <br>perfect prota
| augmented prota
|C <br>Az (А)
| Az##
|[[1/1]]
|  
|-
|-
|1
| 3
|major prime <br>major prota
| biaugmented prota
|C# <br>Az#
| Az###
|  
|  
|-
|-
|2
| 4
|augmented prota
| bidiminished deiteria
|Az##
| Buki♭♭♭
| [[33/32]]
|-
|5
| diminished deiteria
| Buki♭♭
|  
|  
|-
|-
|3
| 6
|biaugmented prota
| minor deiteria
|Az###
| Buki♭
|  
|  
|-
|-
|4
| 7
|bidiminished deiteria
| neutral deiteria
|Buki♭♭♭
| Buki (Б)
| [[135/128]]
|-
| 8
| major deiteria
| Buki#
|  
|  
|-
|-
|5
| 9
|diminished deiteria
| augmented deiteria
|Buki♭♭
| Buki##
|  
|  
|-
|-
|6
| 10
|minor deiteria
| biaugmented deiteria
|Buki♭
| Buki###
|  
|  
|-
|-
|7
| 11
|neutral deiteria
| bidiminished tritia
|Buki (Б)
| Vedi♭♭♭
| [[13/12]], [[12/11]]
|-
| 12
| diminished tritia
| Vedi♭♭
|  
|  
|-
|-
|8  
| 13
|major deiteria
| neutral secunde <br>minor tritia
|Buki#
| B<br>Vedi♭
| [[11/10]]
|-
| 14
| neural tritia
| Vedi (В)
| [[10/9]]
|-
| 15
| major tritia
| Vedi#
| [[9/8]]
|-
| 16
| augmented tritia
| Vedi##
|  
|  
|-
|-
|9
| 17
|augmented deiteria
| biaugmented tritia
|Buki##
| Vedi###
|  
|  
|-
|-
|10
| 18
|biaugmented deiteria
| bidiminished tesseria
|Buki###
| Glagol♭♭♭
| [[8/7]]
|-
| 19
| diminished tesseria
| Glagol♭♭
|  
|  
|-
|-
|11
| 20
|bidiminished tritia
| minor tesseria
|Vedi♭♭♭
| Glagol♭
|[[13/12]], [[12/11]]
| [[7/6]]
|-
|-
|12
| 21
|diminished tritia
| neutral tesseria
|Vedi♭♭
| Glagol (Г)
|  
|  
|-
|-
|13
| 22
|neutral secunde <br>minor tritia
| major tesseria
|D <br>Vedi♭
| Glagol#
|[[11/10]]
| [[13/11]]
|-
|-
|14
| 23
|neural tritia
| augmented tesseria
|Vedi (В)
| Glagol##
|[[10/9]]
|  
|-
|-
|15
| 24
|major tritia
| biaugmented tesseria
|Vedi#
| Glagol###
|[[9/8]]
| [[6/5]]
|-
|-
|16
| 25
|augmented tritia
| bidiminished pemptia
|Vedi##
| Dobro♭♭♭
|  
|  
|-
|-
|17
| 26
|biaugmented tritia
| neutral tertie <br>diminished pemptia
|Vedi###
| C <br>Dobro♭♭
| [[11/9]]
|-
| 27
| major tertie <br>minor pemptia
| C# <br />Dobro♭
| [[16/13]], 27/22
|-
| 28
| neutral pemptia
| Dobro (Д)
|  
|  
|-
|-
|18
| 29
|bidiminished tesseria
| major pemptia
|Glagol♭♭♭
| Dobro#
|[[8/7]]
| [[5/4]]
|-
| 30
| augmented pemptia
| Dobro##
|
|-
|-
|19
| 31
|diminished tesseria
| biaugmented pemptia
|Glagol♭♭
| Dobro###
|  
|  
|-
|-
|20
| 32
|minor tesseria
| bidiminished hektia
|Glagol♭
| Yest♭♭♭
|[[7/6]]
| [[14/11]]
|-
| 33
| diminished hektia
| Yest♭♭
| [[9/7]]
|-
| 34
| minor hektia
| Yest♭
|
|-
|-
|21
| 35
|neutral tesseria
| neutral hektia
|Glagol (Г)
| Yest (Е)
|  
|  
|-
|-
|22
| 36
|major tesseria
| major hektia
|Glagol#
| Yest#
|[[13/11]]
|  
|-
|-
|23
| 37
|augmented tesseria
| augmented hektia
|Glagol##
| Yest##
|  
|  
|-
|-
|24
| 38
|biaugmented tesseria
| biaugmented hektia
|Glagol###
| Yest###
|[[6/5]]
| [[4/3]]
|-
|-
|25
| 39
|bidiminished pemptia
| neutral quarte <br>bidiminished hebdomia
|Dobro♭♭♭
| D<br>Zhivete♭♭♭
|  
|  
|-
|-
|26
| 40
|neutral tertie <br>diminished pemptia
| diminished hebdomia
|E <br>Dobro♭♭
| Zhivete♭♭
|[[11/9]]
|  
|-
|-
|27
| 41
|major tertie <br>minor pemptia
| minor hebdomia
|E# <br>Dobro♭
| Zhivete♭
|[[16/13]], 27/22
|  
|-
|-
|28
| 42
|neutral pemptia
| neutral hebdomia
|Dobro (Д)
| Zhivete (Ж)
| [[11/8]]
|-
| 43
| major hebdomia
| Zhivete#
|  
|  
|-
|-
|29
| 44
|major pemptia
| augmented hebdomia
|Dobro#
| Zhivete##
|[[5/4]]
| [[7/5]]
|-
|-
|30
| 45
|augmented pemptia
| biaugmented hebdomia
|Dobro##
| Zhivete###
|  
|  
|-
|-
|31
| 46
|biaugmented pemptia
| bidiminished ogdonia
|Dobro###
| Dzelo♭♭♭
|  
|  
|-
|-
|32
| 47
|bidiminished hektia
| diminished ogdonia
|Yest♭♭♭
| Dzelo♭♭
|[[14/11]]
| [[10/7]]  
|-
|-
|33
| 48
|diminished hektia
| minor ogdonia
|Yest♭♭
| Dzelo♭
|[[9/7]]
|  
|-
|-
|34
| 49
|minor hektia
| neutral ogdonia
|Yest♭
| Dzelo (Ѕ)
|  
|  
|-
|-
|35
| 52
|neutral hektia
| neutral quinte
|Yest (Е)
| E
| 121/81
|-
| 53
| major quinte
| E#
| [[3/2]]
|-
| 54
| augmented quinte <br>diminished ennatia
| E## <br>Zemle♭♭
| [[256/169]]
|-
| 55
| minor ennatia
| Zemle♭
|  
|  
|-
|-
|36
| 56
|major hektia
| neutral ennatia
|Yest#
| Zemle (З)
|  
|  
|-
|-
|37
| 63
|augmented hektia
| neutral decatia
|Yest##
| Izhe (И)
|  
|  
|-
|-
|38
| 64
|biaugmented hektia
| major decatia <br>minor sexte
|Yest###
| Izhe# <br>F♭
|[[4/3]]
|  
|-
|-
|39
| 65
|neutral quarte <br>bidiminished hebdomia
| neutral sexte
|F <br>Zhivete♭♭♭
| F
|  
|  
|-
|-
|40
| 70
|diminished hebdomia
| neutral hendecatia
|Zhivete♭♭
| Jerve (Ђ)
|  
|  
|-
|-
|41
| 77
|minor hebdomia
| neutral dodecatia
|Zhivete♭
| Kako (К)
|  
|  
|-
|-
|42
| 78
|neutral hebdomia
| neutral septime
|Zhivete (Ж)
| G
|[[11/8]]
|  
|-
|-
|43
| 84
|major hebdomia
| neutral decatotritia
|Zhivete#
| Ludi (Л)
|  
|  
|-
|-
|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*
|G
! Cents*
|121/81
! Associated<br>ratio*
! Temperament
|-
|-
|53
| 1
|major quinte
| 2\91
|G#
| 26.37
|[[3/2]]
| 49/48
| [[Sfourth]]
|-
|-
|54
| 1
|augmented quinte <br>diminished ennatia
| 4\91
|G## <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>A♭
| 316.48
|
| 6/5
| [[Catakleismic]] (91f)
|-
|-
|65
| 1
|neutral sexte
| 33\91
|A
| 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
|B
| 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)
|C <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 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
* 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
* 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
* HungarianNaiveSurorwell[13]: 7 7 8 6 11 5 5 7 10 4 4 13 4
* Septimin[9]: 11 9 11 9 11 9 11 9 11
* Quartkeenlig[23]: 44444444444444444444443
* ArabicSeptimin[9]: 5 15 11 9 11 9 5 15 11
* ConcocticSubset[7]: 17 10 17 10 17 10 17
* [[Semaphore5]]: 19 19 19 19 15
* ConcocticMaqamSikah: 10 17 17 10 10 17 10
* [[Semaphore9]]
 
* [[Semaphore14]]
== 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:Equal divisions of the octave]]
=== 21st century ===
[[Category:91edo| ]] <!-- main article -->
; [[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 }}
Retrieved from "https://en.xen.wiki/w/91edo"