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'''91edo''', the '''91 equal division''' divides the octave into 91 parts of 13.187 cents each.
{{Infobox ET}}
{{ED intro}}


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


== Theory ==
== Notation ==
{{primes in edo|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}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}


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]] (117440512/117406179). 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.
=== 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.


=== Naive modes ===
{| class="wikitable center-1 mw-collapsible mw-collapsed"
91edo possesses naive versions of heptatonic and tridecatonic scales.
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 91edo
|-
! #
! Eliora's naming system
! Eliora's notation
! 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]]
|-
| 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 <br>minor tritia
| B<br>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 <br>diminished pemptia
| C <br>Dobro♭♭
| [[11/9]]
|-
| 27
| major tertie <br>minor pemptia
| C# <br />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 <br>bidiminished hebdomia
| D<br>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 <br>diminished ennatia
| E## <br>Zemle♭♭
| [[256/169]]
|-
| 55
| minor ennatia
| Zemle♭
|
|-
| 56
| neutral ennatia
| Zemle (З)
|
|-
| 63
| neutral decatia
| Izhe (И)
|
|-
| 64
| major decatia <br>minor sexte
| Izhe# <br>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 <br>perfect decatotetartia
| A<br>Az (А)
| [[2/1]] exact
|}


For example, it can recreate diatonic major by first making an equiheptatonic scale with step size 13, and then raising III, VI and VII by a desired amount. Similar approximation can be done in [[28edo]] or any edo divisible by 7.
== 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
|}


Likewise, it can also recreate Orwells from tridecatonic scale. However, there is not a "trivial" way to do it due to larger amount of notes. The "correct" way to recreate a mode in this fashion would be using the differences of [[Irvic scale|Irvian mode]] and principal mode - that is by applying the generator or construction at the original tonic. For example, [[22edo]]'s Irvian mode for Orwell[13] is 2212212221221, while for 7/6 generator from the tonic is 1221221221222, and applying the differences results in 5795795777779 scale. This scale is quite bulky for musical performance since it contains an even row of 5x7 steps. A more vibrant possible variant is 5795797597579, which is derived from differences of 2212212212221 and 1221222122122 22edo Orwells.
=== 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


Since 7 and 13 are the only factors of 91, these numbers are the only ones which can produce naive scales.  
<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
* NaiveMajor[7]: 13 16 10 15 14 13 10 - fifth adjusted to match with NaiveOrwell[13]
* 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
* NaiveOrwell[13]: 5795797597579
* ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9 - above scale with 2, 6, and 12 degrees lowered 4 steps
* ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9
* [[Semaphore5]]
* HungarianNaiveSurorwell[13]: 7 7 8 6 11 5 5 7 10 4 4 13 4
* [[Semaphore9]]
* Quartkeenlig[23]: 44444444444444444444443
* [[Semaphore14]]
* 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 ==
== 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/shorts/7RDvArkSJrk ''Aishite ita no ni''] (2023) – microtonal cover in 91edo by [[Bryan Deister]] (2026)


== See also ==
=== 21st century ===
* [[Wikipedia: 91 (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]


[[Category:Equal divisions of the octave]]
; [[Bryan Deister]]
[[Category:91edo| ]] <!-- main article -->
* [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]]
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