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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|32}}
{{ED intro}}
 
== 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.
=== 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.
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 15 &amp; 32 temperament, tempering out [[55/54]], 64/63 and [[245/242]].
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 [[subgroup temperaments #oceanfront|oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The "major third" (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[13/10]]~[[9/7]], 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¢ (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.
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.


It is generally the first power-of-2 edo which can be considered to handle [[low-limit JI]] at all. It has unambiguous mappings for primes 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 [[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 ===
=== Harmonics ===
{{Harmonics in equal|32}}
{{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 ==
{| 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
|
|-
|-
|1
| 0
|37.5
| 0.0
|^1, m2
| P1
|up unison, minor 2nd
| perfect unison
|^D, Eb
| D
|49/48, 50/49, 45/44
| 1/1
|46/45, 52/51, 51/50
|-
|-
|2
| 1
|75.0
| 37.5
|^m2
| ^1, m2
|upminor 2nd
| up unison, minor 2nd
|^Eb
| ^D, Eb
|22/21, 25/24
| 49/48, 50/49, 45/44
|24/23, 23/22
| 46/45, 52/51, 51/50
|-
|-
|3
| 2
|112.5
| 75.0
|^^m2
| ^m2
|dupminor 2nd
| upminor 2nd
|^^Eb
| ^Eb
|16/15
| 22/21, 25/24
|49/46
| 24/23, 23/22
|-
|-
|4
| 3
|150.0
| 112.5
|vvM2
| ^^m2
|dudmajor 2nd
| dupminor 2nd
|vvE
| ^^Eb
|12/11, 49/45
| 16/15
|25/23
| 49/46
|-
|-
|5
| 4
|187.5
| 150.0
|A1, vM2
| vvM2
|aug 1sn, downmajor 2nd
| dudmajor 2nd
|D#, vE
| vvE
|10/9, 39/35
| 12/11, 49/45
|19/17
| 25/23
|-
|-
|6
| 5
|225.0
| 187.5
|M2
| A1, vM2
|major 2nd
| aug 1sn, downmajor 2nd
|E
| D#, vE
|8/7, 25/22
| 10/9, 39/35
|57/50
| 19/17
|-
|-
|7
| 6
|262.5
| 225.0
|m3
| M2
|minor 3rd
| major 2nd
|F
| E
|7/6, 64/55
| 8/7, 25/22
|57/49
| 57/50
|-
|-
|8
| 7
|300.0
| 262.5
|^m3
| m3
|upminor 3rd
| minor 3rd
|^F
| F
|6/5, 32/27
| 7/6, 64/55
|19/16
| 57/49
|-
|-
|9
| 8
|337.5
| 300.0
|^^m3
| ^m3
|dupminor 3rd
| upminor 3rd
|^^F
| ^F
|11/9, 39/32, 63/52
| 6/5, 32/27
|17/14, 28/23
| 19/16
|-
|-
|10
| 9
|375.0
| 337.5
|vvM3
| ^^m3
|dudmajor 3rd
| dupminor 3rd
|vvF#
| ^^F
|5/4, 26/21, 56/45, 96/77
| 11/9, 39/32, 63/52
|36/29
| 17/14, 28/23
|-
|-
|11
| 10
|412.5
| 375.0
|vM3
| vvM3
|downmajor 3rd
| dudmajor 3rd
|vF#
| vvF#
|14/11, 33/26, 80/63
| 5/4, 26/21, 56/45, 96/77
|19/15
| 36/29
|-
|-
|12
| 11
|450.0
| 412.5
|M3
| vM3
|major 3rd
| downmajor 3rd
|F#
| vF#
|13/10, 35/27, 64/49
| 14/11, 33/26, 80/63
|22/17, 57/44
| 19/15
|-
|-
|13
| 12
|487.5
| 450.0
|P4
| M3
|perfect 4th
| major 3rd
|G
| F#
|4/3, 33/25, 160/121
| 13/10, 35/27, 64/49
|45/34, 85/64
| 22/17, 57/44
|-
|-
|14
| 13
|525.0
| 487.5
|^4
| P4
|up 4th
| perfect 4th
|^G
| G
|27/20, 110/81
| 4/3, 33/25, 160/121
|19/14, 23/17
| 45/34, 85/64
|-
|-
|15
| 14
|562.5
| 525.0
|^^4, ^d5
| ^4
|dup 4th, updim 5th
| up 4th
|^^G, ^Ab
| ^G
|18/13, 11/8
| 27/20, 110/81
|
| 19/14, 23/17
|-
|-
|16
| 15
|600.0
| 562.5
|vvA4, ^^d5
| ^^4, ^d5
|dudaug 4th, dupdim 5th
| dup 4th, updim 5th
|vvG#, ^^Ab
| ^^G, ^Ab
|7/5, 10/7, 99/70, 140/99
| 18/13, 11/8
|17/12, 12/17
|-
|-
|17
| 16
|637.5
| 600.0
|vA4, vv5
| vvA4, ^^d5
|downaug 4th, dud 5th
| dudaug 4th, dupdim 5th
|vG#, vvA
| vvG#, ^^Ab
|13/9, 16/11
| 7/5, 10/7, 99/70, 140/99
|
| 17/12, 12/17
|-
|-
|18
| 17
|675.0
| 637.5
|v5
| vA4, vv5
|down 5th
| downaug 4th, dud 5th
|vA
| vG#, vvA
|40/27, 81/55
| 13/9, 16/11
|28/19, 34/23
|-
|-
|19
| 18
|712.5
| 675.0
|P5
| v5
|perfect 5th
| down 5th
|A
| vA
|3/2, 50/33, 121/80
| 40/27, 81/55
|68/45, 128/85
| 28/19, 34/23
|-
|-
|20
| 19
|750.0
| 712.5
|m6
| P5
|minor 6th
| perfect 5th
|Bb
| A
|20/13, 54/35, 49/32
| 3/2, 50/33, 121/80
|17/11, 88/57
| 68/45, 128/85
|-
|-
|21
| 20
|787.5
| 750.0
|^m6
| m6
|upminor 6th
| minor 6th
|^Bb
| Bb
|11/7, 52/33, 63/40
| 20/13, 54/35, 49/32
|30/19
| 17/11, 88/57
|-
|-
|22
| 21
|825.0
| 787.5
|^^m6
| ^m6
|dupminor 6th
| upminor 6th
|^^Bb
| ^Bb
|8/5, 21/13, 45/28, 77/48
| 11/7, 52/33, 63/40
|29/18
| 30/19
|-
|-
|23
| 22
|862.5
| 825.0
|vvM6
| ^^m6
|dudmajor 6th
| dupminor 6th
|vvB
| ^^Bb
|18/11, 64/39, 104/63
| 8/5, 21/13, 45/28, 77/48
|28/17, 23/14
| 29/18
|-
|-
|24
| 23
|900.0
| 862.5
|vM6
| vvM6
|downmajor 6th
| dudmajor 6th
|vB
| vvB
|5/3, 27/16
| 18/11, 64/39, 104/63
|32/19
| 28/17, 23/14
|-
|-
|25
| 24
|937.5
| 900.0
|M6
| vM6
|major 6th
| downmajor 6th
|B
| vB
|12/7, 55/32
| 5/3, 27/16
|98/57
| 32/19
|-
|-
|26
| 25
|975.0
| 937.5
|m7
| M6
|minor 7th
| major 6th
|C
| B
|7/4, 44/25
| 12/7, 55/32
|100/57
| 98/57
|-
|-
|27
| 26
|1012.5
| 975.0
|^m7
| m7
|upminor 7th
| minor 7th
|^C
| C
|9/5, 70/39
| 7/4, 44/25
|34/19
| 100/57
|-
|-
|28
| 27
|1050.0
| 1012.5
|^^m7
| ^m7
|dupminor 7th
| upminor 7th
|^^C
| ^C
|11/6, 90/49
| 9/5, 70/39
|46/25
| 34/19
|-
|-
|29
| 28
|1087.5
| 1050.0
|vvM7
| ^^m7
|dudmajor 7th
| dupminor 7th
|vvC#
| ^^C
|15/8
| 11/6, 90/49
|92/49
| 46/25
|-
|-
|30
| 29
|1125.0
| 1087.5
|vM7
| vvM7
|downmajor 7th
| dudmajor 7th
|vC#
| vvC#
|21/11, 48/25
| 15/8
|23/12, 44/23
| 92/49
|-
|-
|31
| 30
|1162.5
| 1125.0
|M7, v8
| vM7
|major 7th, down 8ve
| downmajor 7th
|C#, vD
| vC#
|96/49, 49/25, 88/45
| 21/11, 48/25
|45/23, 51/26, 100/51
| 23/12, 44/23
|-
|-
|32
| 31
|1200.0
| 1162.5
|P8
| M7, v8
|8ve
| major 7th, down 8ve
|D
| C#, vD
|2/1
| 96/49, 49/25, 88/45
|
| 45/23, 51/26, 100/51
|-
| 32
| 1200.0
| P8
| 8ve
| D
| 2/1
|  
|}
|}


== JI approximation ==
== Notation ==
=== Zeta function ===
=== 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]]
== 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| 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 ==
== Delta-rational harmony ==
=== Fully delta-rational triads ===
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.0015.
{| class="wikitable"
 
|Steps
{| class="wikitable mw-collapsible mw-collapsed sortable"
|Signature
|+ style="font-size: 105%; white-space: nowrap;" | Fully delta-rational triads
|Least-squares error
|-
! 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
|-
|-
|(0,1,2)
| 0,12,15,24
|[1, 1]
| +1+?+1
|0.00021
| 0.00060
|-
|-
|(0,1,3)
| 0,12,18,21
|[1, 2]
| +3+?+1
|0.00046
| 0.00014
|-
|-
|(0,1,4)
| 0,12,21,29
|[1, 3]
| +1+?+1
|0.00071
| 0.00078
|-
|-
|(0,2,3)
| 0,12,23,27
|[2, 1]
| +2+?+1
|0.00040
| 0.00036
|-
|-
|(0,2,4)
| 0,12,25,30
|[1, 1]
| +3+?+2
|0.00088
| 0.00084
|-
|-
|(0,2,5)
| 0,13,16,21
|[2, 3]
| +2+?+1
|0.00137
| 0.00057
|-
|-
|(0,3,4)
| 0,13,19,28
|[3, 1]
| +1+?+1
|0.00059
| 0.00023
|-
|-
|(0,3,5)
| 0,13,22,25
|[3, 2]
| +3+?+1
|0.00128
| 0.00019
|-
|-
|(0,3,11)
| 0,13,27,31
|[1, 3]
| +2+?+1
|0.00012
| 0.00012
|-
|-
|(0,4,11)
| 0,14,15,30
|[1, 2]
| +2+?+3
|0.00078
| 0.00004
|-
|-
|(0,5,8)
| 0,14,17,24
|[3, 2]
| +3+?+2
|0.00074
| 0.00028
|-
|-
|(0,6,16)
| 0,14,20,25
|[1, 2]
| +2+?+1
|0.00068
| 0.00048
|-
|-
|(0,7,13)
| 0,14,26,29
|[1, 1]
| +3+?+1
|0.00099
| 0.00012
|-
|-
|(0,8,18)
| 0,15,16,20
|[2, 3]
| +3+?+1
|0.00141
| 0.00002
|-
|-
|(0,8,26)
| 0,15,24,29
|[1, 3]
| +2+?+1
|0.00014
| 0.00028
|-
|-
|(0,9,13)
| 0,16,20,31
|[2, 1]
| +1+?+1
|0.00131
| 0.00042
|-
|-
|(0,9,23)
| 0,16,24,31
|[1, 2]
| +3+?+2
|0.00000
| 0.00051
|-
|-
|(0,12,17)
| 0,17,21,29
|[2, 1]
| +3+?+2
|0.00004
| 0.00090
|-
|-
|(0,13,20)
| 0,17,22,28
|[3, 2]
| +2+?+1
|0.00008
| 0.00062
|-
|-
|(0,15,21)
| 0,17,23,27
|[2, 1]
| +3+?+1
|0.00006
| 0.00039
|-
|-
|(0,15,31)
| 0,18,25,31
|[2, 3]
| +2+?+1
|0.00098
| 0.00007
|-
|-
|(0,18,27)
| 0,18,26,30
|[3, 2]
| +3+?+1
|0.00000
| 0.00001
|-
|-
|(0,21,31)
| 0,19,21,30
|[3, 2]
| +3+?+2
|0.00145
| 0.00014
|-
|-
|(0,22,30)
| 0,20,21,26
|[2, 1]
| +3+?+1
|0.00029
| 0.00032
|-
|-
|(0,25,31)
| 0,21,24,29
|[3, 1]
| +3+?+1
|0.00061
| 0.00026
|}
|}
== Instruments ==
[[Lumatone mapping for 32edo]]


== Music ==
== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=yMokW3-0vIs ''Beyond the Grid''] (2024)
; [[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])
* [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])
Line 405: Line 1,273:
; [[Petr Pařízek]]
; [[Petr Pařízek]]
* [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']
* [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']
; [[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/album/1332 ''1332'']


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
Line 415: Line 1,286:
[[Category:Listen]]
[[Category:Listen]]
[[Category:Sixix]]
[[Category:Sixix]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/32edo"