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The ''32 equal division'' divides the [[octave]] into 32 equal parts of precisely 37.5 [[cent]]s each. While even advocates of less-common [[EDO]]s can struggle to find something about it worth noting, it does provide an excellent tuning for [[Petr Pařízek]]'s [[sixix]] temperament, which tempers out the [[5-limit|5-limit]] sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Pařízek's preferred generator for sixix is (128/15)^(1/11), 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.
{{Infobox ET}}
{{ED intro}}


It also tempers out 2048/2025 in the 5-limit, and [[50/49|50/49]] with [[64/63|64/63]] in the [[7-limit|7-limit]], which means it supports [[Diaschismic_family|pajara temperament]], 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|27edo]]; this fifth is in fact very close to the minimax tuning of the pajara extension [[Diaschismic_family#Pajara-Pajaro|pajaro]], using the 32f val. In the 11-limit it provides the optimal patent val for the 15&32 temperament, tempering out 55/54, 64/63 and 245/242.
== 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.  


Since 32edo is poor at approximating primes and it is a high power of 2, both traditional RTT-based and temperament agnostic MOS theory are of limited usefulness in the system (though it has [[ultrasoft]] [[smitonic]] with L/s = 5/4 and [[supersoft]] [[semiquartal]]). 32edo's 5:2:1 [[blackdye]] scale (1525152515) is notable for having neogothic major thirds, and 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.
=== 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.
 
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 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.
 
=== Odd harmonics ===
{{Harmonics in equal|32}}
 
=== Subsets and supersets ===
Since 32 factors into primes as 2<sup>5</sup>, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.


== Intervals ==
== Intervals ==
{{Odd harmonics in edo|edo=32}}
{| class="wikitable"
{| class="wikitable"
|+
!Degree
!Cents
! colspan="3" |[[Ups and Downs Notation]]
!13-limit Ratios
!Other
|-
|-
|0
! Degree
|0.0
! Cents
|P1
! colspan="3" | [[Ups and downs notation]]
|perfect unison
! 13-limit Ratios
|D
! Other
|1/1
|-
|
| 0
| 0.0
| P1
| perfect unison
| D
| 1/1
|-
|-
|1
| 1
|37.5
| 37.5
|^1, m2
| ^1, m2
|up unison, minor 2nd
| up unison, minor 2nd
|^D, Eb
| ^D, Eb
|49/48, 50/49, 45/44
| 49/48, 50/49, 45/44
|46/45, 52/51, 51/50
| 46/45, 52/51, 51/50
|-
|-
|2
| 2
|75.0
| 75.0
|^m2
| ^m2
|upminor 2nd
| upminor 2nd
|^Eb
| ^Eb
|22/21, 25/24
| 22/21, 25/24
|24/23, 23/22
| 24/23, 23/22
|-
|-
|3
| 3
|112.5
| 112.5
|v~2
| ^^m2
|downmid 2nd
| dupminor 2nd
|^^Eb, vvD#
| ^^Eb
|16/15
| 16/15
|49/46
| 49/46
|-
|-
|4
| 4
|150.0
| 150.0
|^~2
| vvM2
|upmid 2nd
| dudmajor 2nd
|vD#
| vvE
|12/11, 49/45
| 12/11, 49/45
|25/23
| 25/23
|-
|-
|5
| 5
|187.5
| 187.5
|vM2
| A1, vM2
|downmajor 2nd
| aug 1sn, downmajor 2nd
|D#
| D#, vE
|10/9, 39/35
| 10/9, 39/35
|19/17
| 19/17
|-
|-
|6
| 6
|225.0
| 225.0
|M2
| M2
|major 2nd
| major 2nd
|E
| E
|8/7, 25/22
| 8/7, 25/22
|57/50
| 57/50
|-
|-
|7
| 7
|262.5
| 262.5
|m3
| m3
|minor 3rd
| minor 3rd
|F
| F
|7/6, 64/55
| 7/6, 64/55
|57/49
| 57/49
|-
|-
|8
| 8
|300.0
| 300.0
|^m3
| ^m3
|upminor 3rd
| upminor 3rd
|Gb
| ^F
|6/5, 32/27
| 6/5, 32/27
|19/16
| 19/16
|-
|-
|9
| 9
|337.5
| 337.5
|v~3
| ^^m3
|downmid 3rd
| dupminor 3rd
|^Gb
| ^^F
|11/9, 39/32, 63/52
| 11/9, 39/32, 63/52
|17/14, 28/23
| 17/14, 28/23
|-
|-
|10
| 10
|375.0
| 375.0
|^~3
| vvM3
|upmid 3rd
| dudmajor 3rd
|vvF#
| vvF#
|5/4, 26/21, 56/45, 96/77
| 5/4, 26/21, 56/45, 96/77
|36/29
| 36/29
|-
|-
|11
| 11
|412.5
| 412.5
|vM3
| vM3
|downmajor 3rd
| downmajor 3rd
|vF#
| vF#
|14/11, 33/26, 80/63
| 14/11, 33/26, 80/63
|19/15
| 19/15
|-
|-
|12
| 12
|450.0
| 450.0
|M3
| M3
|major 3rd
| major 3rd
|F#
| F#
|13/10, 35/27, 64/49
| 13/10, 35/27, 64/49
|22/17, 57/44
| 22/17, 57/44
|-
|-
|13
| 13
|487.5
| 487.5
|P4
| P4
|perfect 4th
| perfect 4th
|G
| G
|4/3, 33/25, 160/121
| 4/3, 33/25, 160/121
|45/34, 85/64
| 45/34, 85/64
|-
|-
|14
| 14
|525.0
| 525.0
|^4
| ^4
|up 4th
| up 4th
|Ab
| ^G
|27/20, 110/81
| 27/20, 110/81
|19/14, 23/17
| 19/14, 23/17
|-
|-
|15
| 15
|562.5
| 562.5
|v~4
| ^^4, ^d5
|downmid 4th
| dup 4th, updim 5th
|^Ab
| ^^G, ^Ab
|18/13, 11/8
| 18/13, 11/8
|
|-
|-
|16
| 16
|600.0
| 600.0
|^~4,v~5
| vvA4, ^^d5
|upmid 4th, downmid 5th
| dudaug 4th, dupdim 5th
|vvG#, ^^Ab
| vvG#, ^^Ab
|7/5, 10/7, 99/70, 140/99
| 7/5, 10/7, 99/70, 140/99
|17/12, 12/17
| 17/12, 12/17
|-
|-
|17
| 17
|637.5
| 637.5
|^~5
| vA4, vv5
|upmid 5th
| downaug 4th, dud 5th
|vG#
| vG#, vvA
|13/9, 16/11
| 13/9, 16/11
|
|-
|-
|18
| 18
|675.0
| 675.0
|v5
| v5
|down fifth
| down 5th
|G#
| vA
|40/27, 81/55
| 40/27, 81/55
|28/19, 34/23
| 28/19, 34/23
|-
|-
|19
| 19
|712.5
| 712.5
|P5
| P5
|perfect 5th
| perfect 5th
|A
| A
|3/2, 50/33, 121/80
| 3/2, 50/33, 121/80
|68/45, 128/85
| 68/45, 128/85
|-
|-
|20
| 20
|750.0
| 750.0
|m6
| m6
|minor 6th
| minor 6th
|Bb
| Bb
|20/13, 54/35, 49/32
| 20/13, 54/35, 49/32
|17/11, 88/57
| 17/11, 88/57
|-
|-
|21
| 21
|787.5
| 787.5
|^m6
| ^m6
|upminor 6th
| upminor 6th
|^Bb
| ^Bb
|11/7, 52/33, 63/40
| 11/7, 52/33, 63/40
|30/19
| 30/19
|-
|-
|22
| 22
|825.0
| 825.0
|v~6
| ^^m6
|downmid 6th
| dupminor 6th
|^^Bb
| ^^Bb
|8/5, 21/13, 45/28, 77/48
| 8/5, 21/13, 45/28, 77/48
|29/18
| 29/18
|-
|-
|23
| 23
|862.5
| 862.5
|^~6
| vvM6
|upmid 6th
| dudmajor 6th
|vA#
| vvB
|18/11, 64/39, 104/63
| 18/11, 64/39, 104/63
|28/17, 23/14
| 28/17, 23/14
|-
|-
|24
| 24
|900.0
| 900.0
|vM6
| vM6
|downmajor 6th
| downmajor 6th
|A#
| vB
|5/3, 27/16
| 5/3, 27/16
|32/19
| 32/19
|-
|-
|25
| 25
|937.5
| 937.5
|M6
| M6
|major 6th
| major 6th
|B
| B
|12/7, 55/32
| 12/7, 55/32
|98/57
| 98/57
|-
|-
|26
| 26
|975.0
| 975.0
|m7
| m7
|minor 7th
| minor 7th
|C
| C
|7/4, 44/25
| 7/4, 44/25
|100/57
| 100/57
|-
|-
|27
| 27
|1012.5
| 1012.5
|^m7
| ^m7
|upminor 7th
| upminor 7th
|Db
| ^C
|9/5, 70/39
| 9/5, 70/39
|34/19
| 34/19
|-
|-
|28
| 28
|1050.0
| 1050.0
|v~7
| ^^m7
|downmid 7th
| dupminor 7th
|^Db
| ^^C
|11/6, 90/49
| 11/6, 90/49
|46/25
| 46/25
|-
|-
|29
| 29
|1087.5
| 1087.5
|^~7
| vvM7
|upmid 7th
| dudmajor 7th
|vvC#
| vvC#
|15/8
| 15/8
|92/49
| 92/49
|-
|-
|30
| 30
|1125.0
| 1125.0
|vM7
| vM7
|downmajor 7th
| downmajor 7th
|vC#
| vC#
|21/11, 48/25
| 21/11, 48/25
|23/12, 44/23
| 23/12, 44/23
|-
|-
|31
| 31
|1162.5
| 1162.5
|M7
| M7, v8
|major 7th
| major 7th, down 8ve
|C#
| C#, vD
|96/49, 49/25, 88/45
| 96/49, 49/25, 88/45
|45/23, 51/26, 100/51
| 45/23, 51/26, 100/51
|-
|-
|32
| 32
|1200.0
| 1200.0
|P8
| P8
|8ve
| 8ve
|D
| D
|2/1
| 2/1
|
|  
|}
|}


=Z function=
== Notation ==
=== Ups and downs 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).
{{Sharpness-sharp5a}}
 
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]]:
 
{{Sharpness-sharp5}}
 
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.
 
=== 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]].
 
==== Evo flavor ====
<imagemap>
File:32-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 407 0 567 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 407 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:32-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:32-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 367 0 527 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 3067 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:32-EDO_Revo_Sagittal.svg]]
</imagemap>
 
== Approximation to JI ==
{{Q-odd-limit intervals|32}}
 
=== Zeta properties ===
Below is a plot of the [[Zeta]] function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.
Below is a plot of the [[Zeta]] function, showing how its peak (ie most negative) value is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.


[[File:plot32.png|alt=plot32.png|plot32.png]]
[[File:plot32.png|alt=plot32.png|plot32.png]]


=Music=
== Regular temperament properties ==
* [https://www.youtube.com/watch?v=00kH3CqSgMY Zinnia Riplet (32-EDO)] by [[Stephen Weigel]] (featured in [https://spectropolrecords.bandcamp.com/album/possible-worlds-vol-4 Possible Worlds Vol. 4] of Spectropol Records)
{| class="wikitable center-4 center-5 center-6"
* [https://soundcloud.com/overtoneshock/admins-hot-tub-32-edo Admin's Hot Tub] by [[Stephen Weigel]]
|-
* [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg Sixix] by Petr Pařízek
! rowspan="2" | [[Subgroup]]
* [http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 32 32 32 Nothing Less Will Do] by [[Chris Vaisvil]]
! 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| 51 -32 }}
| {{Mapping| 32 51 }}
| -3.327
| 3.32
| 8.87
|-
| 2.3.7
| 64/63, 46118408/43046721
| {{Mapping| 32 51 90 }}
| -2.950
| 2.76
| 7.38
|- style="border-top: double;"
| 2.3.5
| 648/625, 20480/19683
| {{Mapping| 32 51 75 }} (32c)
| -5.965
| 4.61
| 12.3
|-
| 2.3.5.7
| 64/63, 245/243, 392/375
| {{Mapping| 32 51 75 90 }} (32c)
| -5.027
| 4.31
| 11.5
|- style="border-top: double;"
| 2.3.5
| 2048/2025, 3125/2916
| {{Mapping| 32 51 74 }} (32)
| +0.177
| 4.72
| 12.6
|-
| 2.3.5.7
| 50/49, 64/63, 3125/2916
| {{Mapping| 32 51 75 90 }} (32)
| -1.008
| 4.15
| 11.1
|}
 
=== 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*
! Temperaments
|-
| 1
| 1\32
| 37.5
| 49/48
| [[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
| 7/5
| [[Progress]] (32cf)
|-
| 2
| 13\32
| 487.5
| 4/3
| [[Pajara]] (32, 7-limit)
|-
| 8
| 14\33<br>(1\32)
| 487.5<br>(37.5)
| 4/3<br>(36/35)
| [[Octonion]] (32cf)
|-
| 16
| 14\33<br>(1\32)
| 487.5<br>(37.5)
| 4/3<br>(45/44)
| [[Sedecic]] (32)
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Delta-rational harmony ==
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.0015.
 
{| class="wikitable mw-collapsible mw-collapsed sortable"
|+ style="font-size: 105%; white-space: nowrap;" | 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
|}
 
{| class="wikitable mw-collapsible mw-collapsed sortable"
|+ style="font-size: 105%; white-space: nowrap;" | 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
|}
 
== Instruments ==
[[Lumatone mapping for 32edo]]
 
== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=yMokW3-0vIs ''Beyond the Grid''] (2024)
 
; [[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])
 
; [[Petr Pařízek]]
* [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']
 
; [[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/album/1332 ''1332'']
 
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 ''32 32 32 Nothing Less Will Do'']
 
; [[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://soundcloud.com/overtoneshock/admins-hot-tub-32-edo ''Admin's Hot Tub'']


[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:listen]]
[[Category:Sixix]]
[[Category:sixix]]
[[Category:zeta]]
Retrieved from "https://en.xen.wiki/w/32edo"