32edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit

CompactStar (talk | contribs)

1 user group
5,991 edits
VisualWikitext
m →Music: Fix typo
CompactStar (talk | contribs)
autopatrolled
5,991 edits
 
(36 intermediate revisions by 6 users not shown)
Line 3: Line 3:


== Theory ==
== Theory ==
32edo is generally the first power-of-2 edo which can be considered to handle [[limit|low-limit]] just intonation at all. It has unambiguous mappings for [[prime]]s up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the [[patent val]] instead of using [[direct approximation|inconsistent approximations]]. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has an [[ultrasoft]] [[smitonic]] with {{nowrap|L/s {{=}} 5/4}}). 32edo's 5:2:1 [[blackdye]] scale {{nowrap|(1 5 2 5 1 5 2 5 1 5)}}, which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.  
32edo is generally the first power-of-2 edo which can be considered to handle [[limit|low-limit]] just intonation at all. However, its [[3/2|perfect fifth]] is more than 10 cents sharp, and combined with a [[5/4]] more than 10 cents flat, it is [[inconsistent]] to the [[5-odd-limit]] (although interestingly, these errors cancel out to produce a very accurate [[15/8]]). The mapping for [[prime harmonic|prime]] [[5/1|5]] is ambiguous: the [[patent val]] has a lower average error for primes up to [[11/1|11]], but the 32c val, using the sharply tuned [[5/4]] at 412.5 cents, achieves [[11-odd-limit]] [[monotonicity]], so worth considering despite a little bit more error.  
 
Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has an [[ultrasoft]] [[smitonic]] with {{nowrap|L/s {{=}} 5/4}}). 32edo's 5:2:1 [[blackdye]] scale {{nowrap|(1 5 2 5 1 5 2 5 1 5)}}, which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5{{c}} neogothic major thirds and 450{{c}} [[naiadic]]s in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75{{c}} semitone in place of [[16/15]]. The 712.5{{c}} sharp fifth and the 675{{c}} flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.  
 
=== Odd harmonics ===
{{Harmonics in equal|32}}


=== As a tuning of other temperaments ===
=== As a tuning of other temperaments ===
While even advocates of less-common [[edo]]s can struggle to find something about 32edo worth noting, it does provide an excellent tuning for the [[sixix]] temperament, which [[tempering out|tempers out]] the [[5-limit]] sixix comma, [[3125/2916]], using its 9\32 generator of size 337.5 cents. [[Petr Pařízek]]'s preferred generator for sixix is (128/15)<sup>(1/11)</sup>, which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also can be used (with the 9\32 generator) to tune [[mohavila]], an 11-limit temperament which does not temper out sixix.
Although 32edo is generally considered a mediocre system for approximating [[JI]] (at least relative to other EDOs of a similar size), it does still offer several [[regular temperament]]s that could be used for this purpose. Using the patent val, 32edo provides a good tuning for the [[sixix]] temperament, which [[tempering out|tempers out]] the [[5-limit]] sixix comma, [[3125/2916]]. Sixix is generated by an approximate [[6/5]], which is tuned very sharp so that five of them octave-reduce to [[4/3]]. In 32edo this is the 9\32 generator of size 337.5 cents. [[Petr Pařízek]]'s preferred generator for sixix is (128/15)<sup>(1/11)</sup>, which is 337.430{{c}} and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done.


It also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family #Pajaro|pajaro]], using the 32f val. In the 11-limit it provides the [[optimal patent val]] for the {{nowrap| 15 & 17 }} temperament, tempering out [[55/54]], 64/63, and [[245/242]].
The same generator can also be used for the much more inaccurate [[mohavila]] temperament in the 32bde val, which treats the 675{{c}} fifth from [[16edo]] as 3/2 (basically it is [[mavila]]'s response to neutral third temperaments). Since this fifth is much more inaccurate than the sharp fifth, 32edo cannot truly be considered a [[dual-fifth]] system.


The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[9/7]] and [[13/10]], and the minor third is 262.5¢, approximating [[7/6]]. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker [[mode]]s. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675{{c}} (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.
The patent val also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family #Pajaro|pajaro]], using the 32f val.  


=== Odd harmonics ===
The patent fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[chromatic semitone]] of 5 steps and a [[diatonic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450{{c}}, approximating [[9/7]] and [[13/10]], and the minor third is 262.5{{c}}, approximating [[7/6]] and [[15/13]]. In the 11-limit 32edo provides the [[optimal patent val]] for [[kangaroo]] temperament, tempering out [[55/54]], 64/63, and [[245/242]], whereas the 32c val is equally good for [[progress]]. Both temperaments are generated by a flat [[16/11]] such that three of them reduce to [[3/2]] and six of them reduce to [[8/7]]. The 15-note scale could potentially be used as a [[detempering]] of [[15edo]].
{{Harmonics in equal|32}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 32 factors into primes as 2<sup>5</sup>, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.
Since 32 is a power of two and factors as 2<sup>5</sup>, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}. The next power of two edo is [[64edo]], doubling 32edo.
 
[[96edo]] is an important superset of both 32edo and [[12edo]], being known for approximating JI very well.
 
See also [[32nd-octave temperaments]].


== Intervals ==
== Intervals ==
Line 24: Line 32:
! Cents
! Cents
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
! 13-limit Ratios
! Approximate ratios<br>(13-limit [[patent val]])<ref group="note">Inconsistent intervals are in ''italics''.</ref>
! Other
! Other
|-
|-
Line 33: Line 41:
| D
| D
| 1/1
| 1/1
|
|-
|-
| 1
| 1
Line 39: Line 48:
| up unison, minor 2nd
| up unison, minor 2nd
| ^D, Eb
| ^D, Eb
| 49/48, 50/49, 45/44
| ''[[25/24]]'', [[45/44]], [[49/48]]
| 46/45, 52/51, 51/50
| 46/45, 52/51, 51/50
|-
|-
Line 47: Line 56:
| upminor 2nd
| upminor 2nd
| ^Eb
| ^Eb
| 22/21, 25/24
| [[22/21]], [[26/25]], ''[[33/32]]'', ''[[36/35]]'', ''[[81/80]]''
| 24/23, 23/22
| 24/23, 23/22
|-
|-
Line 55: Line 64:
| dupminor 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
| 16/15
| ''[[13/12]]'', [[15/14]], [[16/15]]
| 49/46
| 49/46
|-
|-
Line 63: Line 72:
| dudmajor 2nd
| dudmajor 2nd
| vvE
| vvE
| 12/11, 49/45
| ''[[10/9]]'', [[12/11]], ''[[14/13]]'', [[35/32]]
| 25/23
| 25/23
|-
|-
Line 71: Line 80:
| aug 1sn, downmajor 2nd
| aug 1sn, downmajor 2nd
| D#, vE
| D#, vE
| 10/9, 39/35
| ''[[11/10]]''
| 19/17
| 19/17
|-
|-
Line 79: Line 88:
| major 2nd
| major 2nd
| E
| E
| 8/7, 25/22
| [[8/7]], ''[[9/8]]''
| 57/50
| 57/50
|-
|-
Line 87: Line 96:
| minor 3rd
| minor 3rd
| F
| F
| 7/6, 64/55
| [[7/6]], ''[[13/11]]'', [[15/13]]
| 57/49
| 57/49
|-
|-
Line 95: Line 104:
| upminor 3rd
| upminor 3rd
| ^F
| ^F
| 6/5, 32/27
| [[33/28]]
| 19/16
| 19/16
|-
|-
Line 103: Line 112:
| dupminor 3rd
| dupminor 3rd
| ^^F
| ^^F
| 11/9, 39/32, 63/52
| ''[[6/5]]'', [[11/9]], [[39/32]]
| 17/14, 28/23
| 17/14, 28/23
|-
|-
Line 111: Line 120:
| dudmajor 3rd
| dudmajor 3rd
| vvF#
| vvF#
| 5/4, 26/21, 56/45, 96/77
| [[5/4]], [[16/13]], [[27/22]]
| 36/29
| 36/29
|-
|-
Line 119: Line 128:
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| 14/11, 33/26, 80/63
| [[14/11]]
| 19/15
| 19/15
|-
|-
Line 127: Line 136:
| major 3rd
| major 3rd
| F#
| F#
| 13/10, 35/27, 64/49
| [[9/7]], [[13/10]], ''[[32/25]]'', [[64/49]]
| 22/17, 57/44
| 22/17, 57/44
|-
|-
Line 135: Line 144:
| perfect 4th
| perfect 4th
| G
| G
| 4/3, 33/25, 160/121
| [[4/3]], [[21/16]]
| 45/34, 85/64
| 45/34, 85/64
|-
|-
Line 143: Line 152:
| up 4th
| up 4th
| ^G
| ^G
| 27/20, 110/81
| [[15/11]]
| 19/14, 23/17
| 19/14, 23/17
|-
|-
Line 151: Line 160:
| dup 4th, updim 5th
| dup 4th, updim 5th
| ^^G, ^Ab
| ^^G, ^Ab
| 18/13, 11/8
| [[11/8]], [[48/35]], ''[[27/20]]''
|
|-
|-
| 16
| 16
Line 158: Line 168:
| dudaug 4th, dupdim 5th
| dudaug 4th, dupdim 5th
| vvG#, ^^Ab
| vvG#, ^^Ab
| 7/5, 10/7, 99/70, 140/99
| [[7/5]], [[10/7]], [[45/32]], [[64/45]]
| 17/12, 12/17
| 17/12, 12/17
|-
|-
Line 166: Line 176:
| downaug 4th, dud 5th
| downaug 4th, dud 5th
| vG#, vvA
| vG#, vvA
| 13/9, 16/11
| [[16/11]], [[35/24]], ''[[40/27]]''
|
|-
|-
| 18
| 18
Line 173: Line 184:
| down 5th
| down 5th
| vA
| vA
| 40/27, 81/55
| [[22/15]]
| 28/19, 34/23
| 28/19, 34/23
|-
|-
Line 181: Line 192:
| perfect 5th
| perfect 5th
| A
| A
| 3/2, 50/33, 121/80
| [[3/2]], [[32/21]]
| 68/45, 128/85
| 68/45, 128/85
|-
|-
Line 189: Line 200:
| minor 6th
| minor 6th
| Bb
| Bb
| 20/13, 54/35, 49/32
| [[14/9]], [[20/13]], ''[[25/16]]'', [[49/32]]
| 17/11, 88/57
| 17/11, 88/57
|-
|-
Line 197: Line 208:
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| 11/7, 52/33, 63/40
| [[11/7]]
| 30/19
| 30/19
|-
|-
Line 205: Line 216:
| dupminor 6th
| dupminor 6th
| ^^Bb
| ^^Bb
| 8/5, 21/13, 45/28, 77/48
| [[8/5]], [[13/8]], [[44/27]]
| 29/18
| 29/18
|-
|-
Line 213: Line 224:
| dudmajor 6th
| dudmajor 6th
| vvB
| vvB
| 18/11, 64/39, 104/63
| ''[[5/3]]'', [[18/11]], [[64/39]]
| 28/17, 23/14
| 28/17, 23/14
|-
|-
Line 221: Line 232:
| downmajor 6th
| downmajor 6th
| vB
| vB
| 5/3, 27/16
| [[56/33]]
| 32/19
| 32/19
|-
|-
Line 229: Line 240:
| major 6th
| major 6th
| B
| B
| 12/7, 55/32
| [[12/7]], ''[[22/13]]'', [[26/15]]
| 98/57
| 98/57
|-
|-
Line 237: Line 248:
| minor 7th
| minor 7th
| C
| C
| 7/4, 44/25
| [[7/4]], ''[[16/9]]''
| 100/57
| 100/57
|-
|-
Line 245: Line 256:
| upminor 7th
| upminor 7th
| ^C
| ^C
| 9/5, 70/39
| [[20/11]]
| 34/19
| 34/19
|-
|-
Line 253: Line 264:
| dupminor 7th
| dupminor 7th
| ^^C
| ^^C
| 11/6, 90/49
| ''[[9/5]]'', [[11/6]], ''[[13/7]]'', [[64/35]]
| 46/25
| 46/25
|-
|-
Line 261: Line 272:
| dudmajor 7th
| dudmajor 7th
| vvC#
| vvC#
| 15/8
| [[15/8]], ''[[24/13]]'', [[28/15]]
| 92/49
| 92/49
|-
|-
Line 269: Line 280:
| downmajor 7th
| downmajor 7th
| vC#
| vC#
| 21/11, 48/25
| [[21/11]], [[25/13]], ''[[64/33]]'', ''[[35/18]]'', ''[[160/81]]''
| 23/12, 44/23
| 23/12, 44/23
|-
|-
Line 277: Line 288:
| major 7th, down 8ve
| major 7th, down 8ve
| C#, vD
| C#, vD
| 96/49, 49/25, 88/45
| ''[[48/25]]'', [[88/45]], [[96/49]]
| 45/23, 51/26, 100/51
| 45/23, 51/26, 100/51
|-
|-
Line 285: Line 296:
| 8ve
| 8ve
| D
| D
| 2/1
| [[2/1]]
|  
|  
|}
|}


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
32edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Ups and downs sharpness}}
{{Sharpness-sharp5-szg}}


Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.


{{Sharpness-sharp5}}
=== Kite's ups and downs notation ===
 
32edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.
{{Ups and downs sharpness}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[25edo#Sagittal notation|25-EDO]], and is a subset of the notation for [[64edo#Second-best fifth notation|64b]].
This notation uses the same sagittal sequence as [[25edo #Sagittal notation|25edo]], and is a subset of the notation for [[64edo #Second-best fifth notation|64b-edo]].


==== Evo flavor ====
==== Evo flavor ====
Line 395: Line 406:
| 37.5
| 37.5
| 49/48
| 49/48
| [[Slender]] (32)
| [[Diesic]] (32cf) / [[slender]] (32)
|-
|-
| 1
| 1
Line 418: Line 429:
| 15\32
| 15\32
| 562.5
| 562.5
| 7/5
| 16/11
| [[Progress]] (32cf)
| [[Progress]] (32cf) / [[kangaroo]] (32)
|-
|-
| 2
| 2
Line 428: Line 439:
|-
|-
| 8
| 8
| 14\33<br>(1\32)
| 1\32
| 487.5<br>(37.5)
| 37.5
| 4/3<br>(36/35)
| 36/35
| [[Octonion]] (32cf)
| [[Octonion]] (32cf)
|-
|-
| 16
| 16
| 14\33<br>(1\32)
| 1\32
| 487.5<br>(37.5)
| 37.5
| 4/3<br>(45/44)
| 45/44
| [[Sedecic]] (32)
| [[Sedecic]] (32)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Delta-rational harmony ==
== Delta-rational harmony ==
Line 1,270: Line 1,281:


[[File:plot32.png|alt=plot32.png|plot32.png]]
[[File:plot32.png|alt=plot32.png|plot32.png]]
== Scales ==
A list of useful scales known so far in 32edo.
'''Named'''
* [[Antipental blues]]: 9 5 1 4 9 4
'''Unnamed'''
* Bluesy spy music-type scale no. 1: 7 3 9 9 4
* Bluesy spy music-type scale no. 2: 9 6 1 3 13
* Nocturnal fay-type scale: 5 14 3 3 7
* Pseudo-kokinjoshi no. 1: 6 2 11 6 7
* Pseudo-kokinjoshi no. 2: 3 10 5 9 5
* Pseudo-kokinjoshi no. 3: 5 9 5 3 10
* Pseudo-pelog #m3 #m6: 3 5 10 4 10
* Pseudo-pelog ##m3 #P5 #m6: 3 6 10 3 10
* Robotic-type scale, good for contemporary classical: 16 3 4 3 6
* Sunny crystalline-type scale no. 1: 4 4 6 5 8 5
* Sunny crystalline-type scale no. 2: 9 6 4 4 4 5


== Instruments ==
== Instruments ==
Line 1,275: Line 1,305:


== Music ==
== Music ==
; [[Brody Bigwood]]
=== Modern renderings ===
* [https://www.youtube.com/watch?v=yMokW3-0vIs ''Beyond the Grid''] (2024)
; {{W|Koji Kondo}}
* [https://www.youtube.com/shorts/OUlNwN-bAsc "Lost Woods" from ''The Legend of Zelda: Ocarina of Time OST''] (1998) – covered by [[Bryan Deister]] (2025)


=== 21st century ===
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/nTQfjPjeee8 ''32edo improv''] (2025)
* [https://www.youtube.com/shorts/nTQfjPjeee8 ''32edo improv''] (2025)
* [https://www.youtube.com/shorts/zFgw-AfGEcQ ''<nowiki>Licorice Hearted - 32edo [short 1]</nowiki>]'' (2026)
* ''Licorice Hearted'' (2026)
** [https://www.youtube.com/shorts/zFgw-AfGEcQ short 1] · [https://www.youtube.com/shorts/ocgMIf4xopo short 2]
 
; [[groundfault]]
* "Winter's Mortal Hope", from ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/track/winters-mortal-hope-32edo Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo&t=1357 YouTube (22:37–26:00)]


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=y2G6Fs2HMUs ''Canon on Twinkle Twinkle Little Star'', for organ] (2023) – ([https://www.youtube.com/watch?v=JWRGLa59ZwY for Baroque Oboe & Viola])
* ''Canon on Twinkle Twinkle Little Star''
** [https://www.youtube.com/watch?v=y2G6Fs2HMUs for organ] (2023) · [https://www.youtube.com/watch?v=JWRGLa59ZwY for baroque oboe & viola] (2024)


; [[Petr Pařízek]]
; [[Petr Pařízek]]
* [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']
* [https://web.archive.org/web/20201127014118/http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']


; [[Billy Stiltner]]
; [[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/album/1332 ''1332'']
* [https://billystiltner.bandcamp.com/album/1332 ''1332''] (2019)


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 ''32 32 32 Nothing Less Will Do'']
* [https://web.archive.org/web/20201127013223/http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 ''32 32 32 Nothing Less Will Do'']
 
; [[Anne Veinberg]]
* [https://www.youtube.com/watch?v=fm32IYUtITU ''8 vs 16 vs 32 Notes Per Octave — The 96-Tone Piano in Its Subdivision Glory''] (2026, performed on a digital recreation of [[Julián Carrillo]]'s 96-tone metamorphoser piano, using subsets [[8edo]], [[16edo]], 32edo, and then glissandos in full [[96edo]])


; [[Stephen Weigel]]
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=00kH3CqSgMY "Zinnia Riplet" (32-EDO)] (featured in [https://spectropolrecords.bandcamp.com/album/possible-worlds-vol-4 ''Possible Worlds Vol. 4''] of Spectropol Records)
* [https://www.youtube.com/watch?v=00kH3CqSgMY "Zinnia Riplet"], featured in [https://spectropolrecords.bandcamp.com/album/possible-worlds-vol-4 ''Possible Worlds Vol. 4''] (2019) of Spectropol Records
* [https://soundcloud.com/overtoneshock/admins-hot-tub-32-edo ''Admin's Hot Tub'']
* [https://soundcloud.com/overtoneshock/admins-hot-tub-32-edo ''Admin's Hot Tub''] (2019)
 
; [[x2st]]
* [https://www.youtube.com/watch?v=yMokW3-0vIs ''Beyond the Grid''] (2024)
* [https://www.youtube.com/watch?v=SIvKCy4kLNE ''Slither''] (2025)
* [https://www.youtube.com/watch?v=BzJlIV8pcKg ''Cipher''] (2025)
* [https://www.youtube.com/watch?v=LI-lMuAYws8 ''Revelation''] (2026)


== Notes ==
<references group="note"/>
[[Category:Listen]]
[[Category:Listen]]
[[Category:Sixix]]
[[Category:Sixix]]
{{Todo|add scales list}}
{{Todo|add scales list}}

Latest revision as of 10:32, 6 November 2026

← 31edo 32edo 33edo →
Prime factorization 25
Step size 37.5 ¢ 
Fifth 19\32 (712.5 ¢)
Semitones (A1:m2) 5:1 (187.5 ¢ : 37.5 ¢)
Consistency limit 3
Distinct consistency limit 3

32 equal divisions of the octave (abbreviated 32edo or 32ed2), also called 32-tone equal temperament (32tet) or 32 equal temperament (32et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 32 equal parts of exactly 37.5 ¢ each. Each step represents a frequency ratio of 21/32, or the 32nd root of 2.

Theory

32edo is generally the first power-of-2 edo which can be considered to handle low-limit just intonation at all. However, its perfect fifth is more than 10 cents sharp, and combined with a 5/4 more than 10 cents flat, it is inconsistent to the 5-odd-limit (although interestingly, these errors cancel out to produce a very accurate 15/8). The mapping for prime 5 is ambiguous: the patent val has a lower average error for primes up to 11, but the 32c val, using the sharply tuned 5/4 at 412.5 cents, achieves 11-odd-limit monotonicity, so worth considering despite a little bit more error.

Since 32edo is poor at approximating primes and it is a high power of 2, both traditional RTT and temperament-agnostic mos theory are of limited usefulness in the system (though it has an ultrasoft smitonic with L/s = 5/4). 32edo's 5:2:1 blackdye scale (1 5 2 5 1 5 2 5 1 5), which is melodically comparable to 31edo's 5:2:1 diasem, is notable for having 412.5 ¢ neogothic major thirds and 450 ¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75 ¢ semitone in place of 16/15. The 712.5 ¢ sharp fifth and the 675 ¢ flat fifth correspond to 3/2 and 40/27 in 5-limit blackdye, making 5:2:1 blackdye a dual-fifth scale.

Odd harmonics

Approximation of odd harmonics in 32edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +10.5 -11.3 +6.2 -16.4 +11.2 -15.5 -0.8 +7.5 +2.5 +16.7 +9.2
Relative (%) +28.1 -30.2 +16.5 -43.8 +29.8 -41.4 -2.0 +20.1 +6.6 +44.6 +24.6
Steps
(reduced) 		51
(19) 		74
(10) 		90
(26) 		101
(5) 		111
(15) 		118
(22) 		125
(29) 		131
(3) 		136
(8) 		141
(13) 		145
(17)

As a tuning of other temperaments

Although 32edo is generally considered a mediocre system for approximating JI (at least relative to other EDOs of a similar size), it does still offer several regular temperaments that could be used for this purpose. Using the patent val, 32edo provides a good tuning for the sixix temperament, which tempers out the 5-limit sixix comma, 3125/2916. Sixix is generated by an approximate 6/5, which is tuned very sharp so that five of them octave-reduce to 4/3. In 32edo this is the 9\32 generator of size 337.5 cents. Petr Pařízek's preferred generator for sixix is (128/15)(1/11), which is 337.430 ¢ and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done.

The same generator can also be used for the much more inaccurate mohavila temperament in the 32bde val, which treats the 675 ¢ fifth from 16edo as 3/2 (basically it is mavila's response to neutral third temperaments). Since this fifth is much more inaccurate than the sharp fifth, 32edo cannot truly be considered a dual-fifth system.

The patent val also tempers out 2048/2025 in the 5-limit, and 50/49 with 64/63 in the 7-limit, which means it supports pajara, with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of 27edo; this fifth is in fact very close to the minimax tuning of the pajara extension pajaro, using the 32f val.

The patent fifth of 32edo can be used to generate a very unequal archy (specifically oceanfront) diatonic scale, with a chromatic semitone of 5 steps and a diatonic semitone of only 1. The diatonic major third (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450 ¢, approximating 9/7 and 13/10, and the minor third is 262.5 ¢, approximating 7/6 and 15/13. In the 11-limit 32edo provides the optimal patent val for kangaroo temperament, tempering out 55/54, 64/63, and 245/242, whereas the 32c val is equally good for progress. Both temperaments are generated by a flat 16/11 such that three of them reduce to 3/2 and six of them reduce to 8/7. The 15-note scale could potentially be used as a detempering of 15edo.

Subsets and supersets

Since 32 is a power of two and factors as 25, 32edo contains subset edos 2, 4, 8, and 16. The next power of two edo is 64edo, doubling 32edo.

96edo is an important superset of both 32edo and 12edo, being known for approximating JI very well.

See also 32nd-octave temperaments.

Intervals

Degree Cents Ups and downs notation Approximate ratios
(13-limit patent val)[note 1] 		Other
0 0.0 P1 perfect unison D 1/1
1 37.5 ^1, m2 up unison, minor 2nd ^D, Eb 25/24, 45/44, 49/48 46/45, 52/51, 51/50
2 75.0 ^m2 upminor 2nd ^Eb 22/21, 26/25, 33/32, 36/35, 81/80 24/23, 23/22
3 112.5 ^^m2 dupminor 2nd ^^Eb 13/12, 15/14, 16/15 49/46
4 150.0 vvM2 dudmajor 2nd vvE 10/9, 12/11, 14/13, 35/32 25/23
5 187.5 A1, vM2 aug 1sn, downmajor 2nd D#, vE 11/10 19/17
6 225.0 M2 major 2nd E 8/7, 9/8 57/50
7 262.5 m3 minor 3rd F 7/6, 13/11, 15/13 57/49
8 300.0 ^m3 upminor 3rd ^F 33/28 19/16
9 337.5 ^^m3 dupminor 3rd ^^F 6/5, 11/9, 39/32 17/14, 28/23
10 375.0 vvM3 dudmajor 3rd vvF# 5/4, 16/13, 27/22 36/29
11 412.5 vM3 downmajor 3rd vF# 14/11 19/15
12 450.0 M3 major 3rd F# 9/7, 13/10, 32/25, 64/49 22/17, 57/44
13 487.5 P4 perfect 4th G 4/3, 21/16 45/34, 85/64
14 525.0 ^4 up 4th ^G 15/11 19/14, 23/17
15 562.5 ^^4, ^d5 dup 4th, updim 5th ^^G, ^Ab 11/8, 48/35, 27/20
16 600.0 vvA4, ^^d5 dudaug 4th, dupdim 5th vvG#, ^^Ab 7/5, 10/7, 45/32, 64/45 17/12, 12/17
17 637.5 vA4, vv5 downaug 4th, dud 5th vG#, vvA 16/11, 35/24, 40/27
18 675.0 v5 down 5th vA 22/15 28/19, 34/23
19 712.5 P5 perfect 5th A 3/2, 32/21 68/45, 128/85
20 750.0 m6 minor 6th Bb 14/9, 20/13, 25/16, 49/32 17/11, 88/57
21 787.5 ^m6 upminor 6th ^Bb 11/7 30/19
22 825.0 ^^m6 dupminor 6th ^^Bb 8/5, 13/8, 44/27 29/18
23 862.5 vvM6 dudmajor 6th vvB 5/3, 18/11, 64/39 28/17, 23/14
24 900.0 vM6 downmajor 6th vB 56/33 32/19
25 937.5 M6 major 6th B 12/7, 22/13, 26/15 98/57
26 975.0 m7 minor 7th C 7/4, 16/9 100/57
27 1012.5 ^m7 upminor 7th ^C 20/11 34/19
28 1050.0 ^^m7 dupminor 7th ^^C 9/5, 11/6, 13/7, 64/35 46/25
29 1087.5 vvM7 dudmajor 7th vvC# 15/8, 24/13, 28/15 92/49
30 1125.0 vM7 downmajor 7th vC# 21/11, 25/13, 64/33, 35/18, 160/81 23/12, 44/23
31 1162.5 M7, v8 major 7th, down 8ve C#, vD 48/25, 88/45, 96/49 45/23, 51/26, 100/51
32 1200.0 P8 8ve D 2/1

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats with arrows:

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

If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.

Kite's ups and downs notation

32edo can also be notated with Kite's ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

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

Sagittal notation

This notation uses the same sagittal sequence as 25edo, and is a subset of the notation for 64b-edo.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Approximation to JI

The following tables show how 15-odd-limit intervals are represented in 32edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 32edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.637 1.7
15/8, 16/15 0.769 2.0
13/9, 18/13 0.882 2.4
13/10, 20/13 4.214 11.2
7/6, 12/7 4.371 11.7
11/7, 14/11 5.008 13.4
9/5, 10/9 5.096 13.6
7/4, 8/7 6.174 16.5
15/14, 28/15 6.943 18.5
11/9, 18/11 9.908 26.4
3/2, 4/3 10.545 28.1
13/11, 22/13 10.790 28.8
11/8, 16/11 11.182 29.8
5/4, 8/5 11.314 30.2
13/12, 24/13 11.427 30.5
15/11, 22/15 11.951 31.9
15/13, 26/15 14.759 39.4
9/7, 14/9 14.916 39.8
11/10, 20/11 15.004 40.0
13/8, 16/13 15.528 41.4
5/3, 6/5 15.641 41.7
13/7, 14/13 15.798 42.1
9/8, 16/9 16.410 43.8
7/5, 10/7 17.488 46.6
15-odd-limit intervals in 32edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.637 1.7
15/8, 16/15 0.769 2.0
13/10, 20/13 4.214 11.2
7/6, 12/7 4.371 11.7
11/7, 14/11 5.008 13.4
7/4, 8/7 6.174 16.5
15/14, 28/15 6.943 18.5
11/9, 18/11 9.908 26.4
3/2, 4/3 10.545 28.1
11/8, 16/11 11.182 29.8
5/4, 8/5 11.314 30.2
15/11, 22/15 11.951 31.9
15/13, 26/15 14.759 39.4
9/7, 14/9 14.916 39.8
13/8, 16/13 15.528 41.4
7/5, 10/7 17.488 46.6
9/8, 16/9 21.090 56.2
13/7, 14/13 21.702 57.9
5/3, 6/5 21.859 58.3
11/10, 20/11 22.496 60.0
13/12, 24/13 26.073 69.5
13/11, 22/13 26.710 71.2
9/5, 10/9 32.404 86.4
13/9, 18/13 36.618 97.6

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [51 -32⟩ [⟨32 51]] -3.327 3.32 8.87
2.3.7 64/63, 46118408/43046721 [⟨32 51 90]] -2.950 2.76 7.38
2.3.5 648/625, 20480/19683 [⟨32 51 75]] (32c) -5.965 4.61 12.3
2.3.5.7 64/63, 245/243, 392/375 [⟨32 51 75 90]] (32c) -5.027 4.31 11.5
2.3.5 2048/2025, 3125/2916 [⟨32 51 74]] (32) +0.177 4.72 12.6
2.3.5.7 50/49, 64/63, 3125/2916 [⟨32 51 75 90]] (32) -1.008 4.15 11.1

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 1\32 37.5 49/48 Diesic (32cf) / slender (32)
1 9\32 262.5 7/6 Septimin (32f)
1 9\32 337.5 6/5 Sixix (32f)
1 13\32 487.5 4/3 Superpyth (32c, 7-limit) / ultrapyth (32) / quasiultra (32)
1 15\32 562.5 16/11 Progress (32cf) / kangaroo (32)
2 13\32 487.5 4/3 Pajara (32, 7-limit)
8 1\32 37.5 36/35 Octonion (32cf)
16 1\32 37.5 45/44 Sedecic (32)

* In minimal-generator form

Delta-rational harmony

The tables below show chords that approximate 3-integer-limit delta-rational chords with least-squares error less than 0.0015.

Fully delta-rational triads
Steps Delta signature Least-squares error
0,1,2 +1+1 0.00023
0,1,3 +1+2 0.00051
0,1,4 +1+3 0.00083
0,2,3 +2+1 0.00041
0,2,4 +1+1 0.00092
0,3,4 +3+1 0.00060
0,3,11 +1+3 0.00014
0,4,11 +1+2 0.00087
0,5,8 +3+2 0.00076
0,6,16 +1+2 0.00076
0,8,26 +1+3 0.00016
0,9,23 +1+2 0.00000
0,12,17 +2+1 0.00004
0,13,20 +3+2 0.00008
0,15,21 +2+1 0.00007
0,18,27 +3+2 0.00000
0,22,30 +2+1 0.00030
0,25,31 +3+1 0.00062
Partially delta-rational tetrads
Steps Delta signature Least-squares error
0,1,2,3 +1+?+1 0.00056
0,1,2,4 +1+?+2 0.00100
0,1,3,4 +1+?+1 0.00085
0,1,16,17 +2+?+3 0.00091
0,1,16,18 +1+?+3 0.00093
0,1,17,18 +2+?+3 0.00058
0,1,17,19 +1+?+3 0.00051
0,1,18,19 +2+?+3 0.00025
0,1,18,20 +1+?+3 0.00009
0,1,19,20 +2+?+3 0.00010
0,1,19,21 +1+?+3 0.00034
0,1,20,21 +2+?+3 0.00045
0,1,20,22 +1+?+3 0.00078
0,1,21,22 +2+?+3 0.00081
0,1,30,31 +1+?+2 0.00076
0,2,3,4 +2+?+1 0.00082
0,2,6,11 +1+?+3 0.00077
0,2,7,12 +1+?+3 0.00009
0,2,8,13 +1+?+3 0.00097
0,2,12,13 +3+?+2 0.00072
0,2,12,15 +1+?+2 0.00060
0,2,13,14 +3+?+2 0.00032
0,2,13,16 +1+?+2 0.00018
0,2,14,15 +3+?+2 0.00009
0,2,14,17 +1+?+2 0.00097
0,2,15,16 +3+?+2 0.00050
0,2,16,17 +3+?+2 0.00093
0,2,17,21 +1+?+3 0.00061
0,2,18,20 +2+?+3 0.00050
0,2,18,22 +1+?+3 0.00025
0,2,19,21 +2+?+3 0.00020
0,2,20,22 +2+?+3 0.00091
0,3,4,8 +2+?+3 0.00098
0,3,5,9 +2+?+3 0.00007
0,3,7,12 +1+?+2 0.00048
0,3,8,13 +1+?+2 0.00071
0,3,9,16 +1+?+3 0.00074
0,3,10,17 +1+?+3 0.00057
0,3,17,23 +1+?+3 0.00026
0,3,18,19 +2+?+1 0.00082
0,3,18,21 +2+?+3 0.00075
0,3,18,22 +1+?+2 0.00025
0,3,19,20 +2+?+1 0.00035
0,3,19,21 +1+?+1 0.00019
0,3,19,22 +2+?+3 0.00030
0,3,19,23 +1+?+2 0.00094
0,3,20,21 +2+?+1 0.00013
0,3,20,22 +1+?+1 0.00066
0,3,21,22 +2+?+1 0.00063
0,3,26,31 +1+?+3 0.00016
0,4,5,12 +1+?+2 0.00059
0,4,5,15 +1+?+3 0.00060
0,4,8,13 +2+?+3 0.00013
0,4,11,20 +1+?+3 0.00049
0,4,12,18 +1+?+2 0.00042
0,4,13,14 +3+?+1 0.00079
0,4,13,16 +1+?+1 0.00088
0,4,14,15 +3+?+1 0.00035
0,4,14,16 +3+?+2 0.00024
0,4,14,17 +1+?+1 0.00024
0,4,15,16 +3+?+1 0.00009
0,4,15,17 +3+?+2 0.00060
0,4,16,17 +3+?+1 0.00055
0,4,17,25 +1+?+3 0.00058
0,4,19,23 +2+?+3 0.00040
0,4,21,26 +1+?+2 0.00030
0,4,23,30 +1+?+3 0.00062
0,5,6,9 +3+?+2 0.00013
0,5,7,19 +1+?+3 0.00069
0,5,9,17 +1+?+2 0.00047
0,5,10,16 +2+?+3 0.00038
0,5,11,13 +2+?+1 0.00067
0,5,11,15 +1+?+1 0.00027
0,5,11,22 +1+?+3 0.00052
0,5,12,14 +2+?+1 0.00015
0,5,13,15 +2+?+1 0.00099
0,5,15,22 +1+?+2 0.00090
0,5,16,26 +1+?+3 0.00034
0,5,19,24 +2+?+3 0.00051
0,5,23,29 +1+?+2 0.00015
0,5,24,25 +3+?+1 0.00090
0,5,24,27 +1+?+1 0.00085
0,5,25,26 +3+?+1 0.00034
0,5,25,27 +3+?+2 0.00011
0,5,25,28 +1+?+1 0.00058
0,5,26,27 +3+?+1 0.00023
0,5,26,28 +3+?+2 0.00096
0,5,27,28 +3+?+1 0.00081
0,6,9,14 +1+?+1 0.00013
0,6,11,18 +2+?+3 0.00020
0,6,12,21 +1+?+2 0.00064
0,6,15,18 +3+?+2 0.00025
0,6,18,26 +1+?+2 0.00075
0,6,19,25 +2+?+3 0.00062
0,6,20,22 +2+?+1 0.00074
0,6,20,24 +1+?+1 0.00046
0,6,20,31 +1+?+3 0.00043
0,6,21,23 +2+?+1 0.00025
0,6,24,31 +1+?+2 0.00091
0,7,8,12 +3+?+2 0.00097
0,7,8,14 +1+?+1 0.00076
0,7,8,24 +1+?+3 0.00043
0,7,9,11 +3+?+1 0.00053
0,7,9,12 +2+?+1 0.00018
0,7,9,13 +3+?+2 0.00054
0,7,9,20 +1+?+2 0.00020
0,7,10,12 +3+?+1 0.00028
0,7,12,20 +2+?+3 0.00010
0,7,14,24 +1+?+2 0.00004
0,7,15,29 +1+?+3 0.00028
0,7,17,22 +1+?+1 0.00091
0,7,19,26 +2+?+3 0.00073
0,7,22,25 +3+?+2 0.00065
0,7,23,26 +3+?+2 0.00086
0,7,27,31 +1+?+1 0.00074
0,7,28,30 +2+?+1 0.00044
0,7,29,31 +2+?+1 0.00074
0,8,11,23 +1+?+2 0.00070
0,8,11,28 +1+?+3 0.00080
0,8,13,22 +2+?+3 0.00070
0,8,14,20 +1+?+1 0.00072
0,8,15,19 +3+?+2 0.00057
0,8,16,18 +3+?+1 0.00031
0,8,16,19 +2+?+1 0.00023
0,8,16,27 +1+?+2 0.00085
0,8,17,19 +3+?+1 0.00063
0,8,19,27 +2+?+3 0.00084
0,8,23,28 +1+?+1 0.00055
0,9,10,15 +3+?+2 0.00092
0,9,11,30 +1+?+3 0.00012
0,9,13,20 +1+?+1 0.00100
0,9,13,26 +1+?+2 0.00021
0,9,17,29 +1+?+2 0.00062
0,9,19,28 +2+?+3 0.00096
0,9,20,26 +1+?+1 0.00070
0,9,21,25 +3+?+2 0.00055
0,9,22,24 +3+?+1 0.00031
0,9,22,25 +2+?+1 0.00034
0,9,23,25 +3+?+1 0.00077
0,10,13,17 +2+?+1 0.00066
0,10,14,25 +2+?+3 0.00076
0,10,16,21 +3+?+2 0.00034
0,10,18,25 +1+?+1 0.00004
0,10,27,29 +3+?+1 0.00080
0,10,27,30 +2+?+1 0.00029
0,10,27,31 +3+?+2 0.00077
0,10,28,30 +3+?+1 0.00040
0,11,12,18 +3+?+2 0.00040
0,11,12,28 +1+?+2 0.00038
0,11,13,16 +3+?+1 0.00049
0,11,14,17 +3+?+1 0.00085
0,11,14,26 +2+?+3 0.00077
0,11,16,24 +1+?+1 0.00085
0,11,18,22 +2+?+1 0.00057
0,11,21,26 +3+?+2 0.00058
0,11,23,30 +1+?+1 0.00023
0,12,15,24 +1+?+1 0.00060
0,12,18,21 +3+?+1 0.00014
0,12,21,29 +1+?+1 0.00078
0,12,23,27 +2+?+1 0.00036
0,12,25,30 +3+?+2 0.00084
0,13,16,21 +2+?+1 0.00057
0,13,19,28 +1+?+1 0.00023
0,13,22,25 +3+?+1 0.00019
0,13,27,31 +2+?+1 0.00012
0,14,15,30 +2+?+3 0.00004
0,14,17,24 +3+?+2 0.00028
0,14,20,25 +2+?+1 0.00048
0,14,26,29 +3+?+1 0.00012
0,15,16,20 +3+?+1 0.00002
0,15,24,29 +2+?+1 0.00028
0,16,20,31 +1+?+1 0.00042
0,16,24,31 +3+?+2 0.00051
0,17,21,29 +3+?+2 0.00090
0,17,22,28 +2+?+1 0.00062
0,17,23,27 +3+?+1 0.00039
0,18,25,31 +2+?+1 0.00007
0,18,26,30 +3+?+1 0.00001
0,19,21,30 +3+?+2 0.00014
0,20,21,26 +3+?+1 0.00032
0,21,24,29 +3+?+1 0.00026

Octave stretch or compression

Whether octave stretch, shrink or neither is advised for 32edo depends on which vals one wishes to use.

For 32, pure-octaves, or slight compression (~0.5 ¢), works well.

For 32f, moderate compression (~2 ¢) works well. This is close to 133zpi (32.07edo).

For 32c or 32cf, substantial compression (3-4 ¢) is well suited.

For 32be, substantial stretch works (~5 ¢). This is close to 132zpi (31.86edo).

The graph shows zeta near 32edo.

plot32.png

Scales

A list of useful scales known so far in 32edo.

Named

Unnamed

  • Bluesy spy music-type scale no. 1: 7 3 9 9 4
  • Bluesy spy music-type scale no. 2: 9 6 1 3 13
  • Nocturnal fay-type scale: 5 14 3 3 7
  • Pseudo-kokinjoshi no. 1: 6 2 11 6 7
  • Pseudo-kokinjoshi no. 2: 3 10 5 9 5
  • Pseudo-kokinjoshi no. 3: 5 9 5 3 10
  • Pseudo-pelog #m3 #m6: 3 5 10 4 10
  • Pseudo-pelog ##m3 #P5 #m6: 3 6 10 3 10
  • Robotic-type scale, good for contemporary classical: 16 3 4 3 6
  • Sunny crystalline-type scale no. 1: 4 4 6 5 8 5
  • Sunny crystalline-type scale no. 2: 9 6 4 4 4 5

Instruments

Music

Modern renderings

Koji Kondo

21st century

Bryan Deister
groundfault
Claudi Meneghin
Petr Pařízek
Billy Stiltner
Chris Vaisvil
Anne Veinberg
Stephen Weigel
x2st

Notes

  1. ↑ Inconsistent intervals are in italics.
Retrieved from "https://en.xen.wiki/index.php?title=32edo&oldid=241377"