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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|32}}
{{ED intro}}
 
== Theory ==
== Theory ==
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.
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}}


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 &amp; 32}} temperament, tempering out [[55/54]], 64/63, and [[245/242]].
=== 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 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.


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 [[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¢ (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.
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.


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 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.
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.  


=== 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 ===
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 18: Line 31:
! Degree
! Degree
! 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 28: Line 41:
| D
| D
| 1/1
| 1/1
|
|-
|-
| 1
| 1
Line 34: Line 48:
| up unison, minor 2nd
| up unison, minor 2nd
| ^D, Eb
| ^D, Eb
| 49/48, 50/49, 45/44
| ''[[25/24]]'', ''[[28/27]]'', [[45/44]], [[49/48]]
| 46/45, 52/51, 51/50
| 46/45, 52/51, 51/50
|-
|-
Line 42: 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 50: Line 64:
| dupminor 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
| 16/15
| ''[[13/12]]'', [[15/14]], [[16/15]]
| 49/46
| 49/46
|-
|-
Line 58: 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 66: 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 74: Line 88:
| major 2nd
| major 2nd
| E
| E
| 8/7, 25/22
| [[8/7]], ''[[9/8]]''
| 57/50
| 57/50
|-
|-
Line 82: Line 96:
| minor 3rd
| minor 3rd
| F
| F
| 7/6, 64/55
| [[7/6]], ''[[13/11]]'', [[15/13]]
| 57/49
| 57/49
|-
|-
Line 90: Line 104:
| upminor 3rd
| upminor 3rd
| ^F
| ^F
| 6/5, 32/27
| [[33/28]]
| 19/16
| 19/16
|-
|-
Line 98: 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 106: 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 114: Line 128:
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| 14/11, 33/26, 80/63
| [[14/11]]
| 19/15
| 19/15
|-
|-
Line 122: 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 130: 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 138: Line 152:
| up 4th
| up 4th
| ^G
| ^G
| 27/20, 110/81
| [[15/11]]
| 19/14, 23/17
| 19/14, 23/17
|-
|-
Line 146: Line 160:
| dup 4th, updim 5th
| dup 4th, updim 5th
| ^^G, ^Ab
| ^^G, ^Ab
| 18/13, 11/8
| [[11/8]], ''[[27/20]]'', [[48/35]]
|
|-
|-
| 16
| 16
Line 153: 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 161: 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 168: Line 184:
| down 5th
| down 5th
| vA
| vA
| 40/27, 81/55
| [[22/15]]
| 28/19, 34/23
| 28/19, 34/23
|-
|-
Line 176: 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 184: 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 192: Line 208:
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| 11/7, 52/33, 63/40
| [[11/7]]
| 30/19
| 30/19
|-
|-
Line 200: 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 208: 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 216: Line 232:
| downmajor 6th
| downmajor 6th
| vB
| vB
| 5/3, 27/16
| [[56/33]]
| 32/19
| 32/19
|-
|-
Line 224: Line 240:
| major 6th
| major 6th
| B
| B
| 12/7, 55/32
| [[12/7]], ''[[22/13]]'', [[26/15]]
| 98/57
| 98/57
|-
|-
Line 232: Line 248:
| minor 7th
| minor 7th
| C
| C
| 7/4, 44/25
| [[7/4]], ''[[16/9]]''
| 100/57
| 100/57
|-
|-
Line 240: Line 256:
| upminor 7th
| upminor 7th
| ^C
| ^C
| 9/5, 70/39
| [[20/11]]
| 34/19
| 34/19
|-
|-
Line 248: 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 256: Line 272:
| dudmajor 7th
| dudmajor 7th
| vvC#
| vvC#
| 15/8
| [[15/8]], ''[[24/13]]'', [[28/15]]
| 92/49
| 92/49
|-
|-
Line 264: 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 272: Line 288:
| major 7th, down 8ve
| major 7th, down 8ve
| C#, vD
| C#, vD
| 96/49, 49/25, 88/45
| ''[[27/14]]'', ''[[48/25]]'', [[88/45]], [[96/49]]
| 45/23, 51/26, 100/51
| 45/23, 51/26, 100/51
|-
|-
Line 280: Line 296:
| 8ve
| 8ve
| D
| D
| 2/1
| [[2/1]]
|  
|  
|}
|}


== JI approximation ==
== Notation ==
=== Zeta function ===
=== Stein–Zimmermann–Gould notation ===
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.
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}
 
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 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).
{{Ups and downs sharpness}}
 
=== Sagittal notation ===
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 ====
<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}}
 
== 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
| [[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)
|}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


[[File:plot32.png|alt=plot32.png|plot32.png]]
== Delta-rational harmony ==
== Delta-rational harmony ==
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.0015.  
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 ===
 
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
{| class="wikitable mw-collapsible mw-collapsed sortable"
|+ style="font-size: 105%; white-space: nowrap;" | Fully delta-rational triads
|-
|-
! Steps
! Steps
Line 299: Line 463:
|-
|-
| 0,1,2
| 0,1,2
| +1 +1
| +1+1
| 0.00021
| 0.00023
|-
|-
| 0,1,3
| 0,1,3
| +1 +2
| +1+2
| 0.00046
| 0.00051
|-
|-
| 0,1,4
| 0,1,4
| +1 +3
| +1+3
| 0.00071
| 0.00083
|-
|-
| 0,2,3
| 0,2,3
| +2 +1
| +2+1
| 0.00040
| 0.00041
|-
|-
| 0,2,4
| 0,2,4
| +1 +1
| +1+1
| 0.00088
| 0.00092
|-
| 0,2,5
| +2 +3
| 0.00137
|-
|-
| 0,3,4
| 0,3,4
| +3 +1
| +3+1
| 0.00059
| 0.00060
|-
| 0,3,5
| +3 +2
| 0.00128
|-
|-
| 0,3,11
| 0,3,11
| +1 +3
| +1+3
| 0.00012
| 0.00014
|-
|-
| 0,4,11
| 0,4,11
| +1 +2
| +1+2
| 0.00078
| 0.00087
|-
|-
| 0,5,8
| 0,5,8
| +3 +2
| +3+2
| 0.00074
| 0.00076
|-
|-
| 0,6,16
| 0,6,16
| +1 +2
| +1+2
| 0.00068
| 0.00076
|-
| 0,7,13
| +1 +1
| 0.00099
|-
| 0,8,18
| +2 +3
| 0.00141
|-
|-
| 0,8,26
| 0,8,26
| +1 +3
| +1+3
| 0.00014
| 0.00016
|-
| 0,9,13
| +2 +1
| 0.00131
|-
|-
| 0,9,23
| 0,9,23
| +1 +2
| +1+2
| 0.00000
| 0.00000
|-
|-
| 0,12,17
| 0,12,17
| +2 +1
| +2+1
| 0.00004
| 0.00004
|-
|-
| 0,13,20
| 0,13,20
| +3 +2
| +3+2
| 0.00008
| 0.00008
|-
|-
| 0,15,21
| 0,15,21
| +2 +1
| +2+1
| 0.00006
| 0.00007
|-
| 0,15,31
| +2 +3
| 0.00098
|-
|-
| 0,18,27
| 0,18,27
| +3 +2
| +3+2
| 0.00000
| 0.00000
|-
| 0,21,31
| +3 +2
| 0.00145
|-
|-
| 0,22,30
| 0,22,30
| +2 +1
| +2+1
| 0.00029
| 0.00030
|-
|-
| 0,25,31
| 0,25,31
| +3 +1
| +3+1
| 0.00061
| 0.00062
|}
|}


=== Partially delta-rational tetrads ===
{| class="wikitable mw-collapsible mw-collapsed sortable"
{| class="mw-collapsible mw-collapsed class="wikitable sortable"
|+ style="font-size: 105%; white-space: nowrap;" | Partially delta-rational tetrads
|-
|-
! Steps
! Steps
Line 407: Line 543:
|-
|-
| 0,1,2,3
| 0,1,2,3
| +1 +? +1
| +1+?+1
| 0.00056
| 0.00056
|-
|-
| 0,1,2,4
| 0,1,2,4
| +1 +? +2
| +1+?+2
| 0.00100
| 0.00100
|-
| 0,1,2,5
| +1 +? +3
| 0.00133
|-
|-
| 0,1,3,4
| 0,1,3,4
| +1 +? +1
| +1+?+1
| 0.00085
| 0.00085
|-
| 0,1,3,5
| +1 +? +2
| 0.00141
|-
| 0,1,4,5
| +1 +? +1
| 0.00114
|-
| 0,1,5,6
| +1 +? +1
| 0.00145
|-
| 0,1,15,16
| +2 +? +3
| 0.00123
|-
| 0,1,15,17
| +1 +? +3
| 0.00133
|-
|-
| 0,1,16,17
| 0,1,16,17
| +2 +? +3
| +2+?+3
| 0.00091
| 0.00091
|-
|-
| 0,1,16,18
| 0,1,16,18
| +1 +? +3
| +1+?+3
| 0.00093
| 0.00093
|-
|-
| 0,1,17,18
| 0,1,17,18
| +2 +? +3
| +2+?+3
| 0.00058
| 0.00058
|-
|-
| 0,1,17,19
| 0,1,17,19
| +1 +? +3
| +1+?+3
| 0.00051
| 0.00051
|-
|-
| 0,1,18,19
| 0,1,18,19
| +2 +? +3
| +2+?+3
| 0.00025
| 0.00025
|-
|-
| 0,1,18,20
| 0,1,18,20
| +1 +? +3
| +1+?+3
| 0.00009
| 0.00009
|-
|-
| 0,1,19,20
| 0,1,19,20
| +2 +? +3
| +2+?+3
| 0.00010
| 0.00010
|-
|-
| 0,1,19,21
| 0,1,19,21
| +1 +? +3
| +1+?+3
| 0.00034
| 0.00034
|-
|-
| 0,1,20,21
| 0,1,20,21
| +2 +? +3
| +2+?+3
| 0.00045
| 0.00045
|-
|-
| 0,1,20,22
| 0,1,20,22
| +1 +? +3
| +1+?+3
| 0.00078
| 0.00078
|-
|-
| 0,1,21,22
| 0,1,21,22
| +2 +? +3
| +2+?+3
| 0.00081
| 0.00081
|-
| 0,1,21,23
| +1 +? +3
| 0.00123
|-
| 0,1,22,23
| +2 +? +3
| 0.00117
|-
| 0,1,28,29
| +1 +? +2
| 0.00148
|-
| 0,1,29,30
| +1 +? +2
| 0.00112
|-
|-
| 0,1,30,31
| 0,1,30,31
| +1 +? +2
| +1+?+2
| 0.00076
| 0.00076
|-
|-
| 0,2,3,4
| 0,2,3,4
| +2 +? +1
| +2+?+1
| 0.00082
| 0.00082
|-
| 0,2,4,5
| +2 +? +1
| 0.00116
|-
|-
| 0,2,6,11
| 0,2,6,11
| +1 +? +3
| +1+?+3
| 0.00077
| 0.00077
|-
|-
| 0,2,7,12
| 0,2,7,12
| +1 +? +3
| +1+?+3
| 0.00009
| 0.00009
|-
|-
| 0,2,8,13
| 0,2,8,13
| +1 +? +3
| +1+?+3
| 0.00097
| 0.00097
|-
| 0,2,10,11
| +3 +? +2
| 0.00149
|-
| 0,2,11,12
| +3 +? +2
| 0.00110
|-
| 0,2,11,14
| +1 +? +2
| 0.00136
|-
|-
| 0,2,12,13
| 0,2,12,13
| +3 +? +2
| +3+?+2
| 0.00072
| 0.00072
|-
|-
| 0,2,12,15
| 0,2,12,15
| +1 +? +2
| +1+?+2
| 0.00060
| 0.00060
|-
|-
| 0,2,13,14
| 0,2,13,14
| +3 +? +2
| +3+?+2
| 0.00032
| 0.00032
|-
|-
| 0,2,13,16
| 0,2,13,16
| +1 +? +2
| +1+?+2
| 0.00018
| 0.00018
|-
|-
| 0,2,14,15
| 0,2,14,15
| +3 +? +2
| +3+?+2
| 0.00009
| 0.00009
|-
|-
| 0,2,14,17
| 0,2,14,17
| +1 +? +2
| +1+?+2
| 0.00097
| 0.00097
|-
|-
| 0,2,15,16
| 0,2,15,16
| +3 +? +2
| +3+?+2
| 0.00050
| 0.00050
|-
|-
| 0,2,16,17
| 0,2,16,17
| +3 +? +2
| +3+?+2
| 0.00093
| 0.00093
|-
| 0,2,16,20
| +1 +? +3
| 0.00145
|-
| 0,2,17,18
| +3 +? +2
| 0.00136
|-
| 0,2,17,19
| +2 +? +3
| 0.00118
|-
|-
| 0,2,17,21
| 0,2,17,21
| +1 +? +3
| +1+?+3
| 0.00061
| 0.00061
|-
|-
| 0,2,18,20
| 0,2,18,20
| +2 +? +3
| +2+?+3
| 0.00050
| 0.00050
|-
|-
| 0,2,18,22
| 0,2,18,22
| +1 +? +3
| +1+?+3
| 0.00025
| 0.00025
|-
|-
| 0,2,19,21
| 0,2,19,21
| +2 +? +3
| +2+?+3
| 0.00020
| 0.00020
|-
| 0,2,19,23
| +1 +? +3
| 0.00114
|-
|-
| 0,2,20,22
| 0,2,20,22
| +2 +? +3
| +2+?+3
| 0.00091
| 0.00091
|-
| 0,2,30,31
| +1 +? +1
| 0.00135
|-
| 0,3,4,5
| +3 +? +1
| 0.00103
|-
|-
| 0,3,4,8
| 0,3,4,8
| +2 +? +3
| +2+?+3
| 0.00098
| 0.00098
|-
| 0,3,4,12
| +1 +? +3
| 0.00148
|-
| 0,3,5,6
| +3 +? +1
| 0.00139
|-
|-
| 0,3,5,9
| 0,3,5,9
| +2 +? +3
| +2+?+3
| 0.00007
| 0.00007
|-
| 0,3,6,10
| +2 +? +3
| 0.00114
|-
|-
| 0,3,7,12
| 0,3,7,12
| +1 +? +2
| +1+?+2
| 0.00048
| 0.00048
|-
|-
| 0,3,8,13
| 0,3,8,13
| +1 +? +2
| +1+?+2
| 0.00071
| 0.00071
|-
|-
| 0,3,9,16
| 0,3,9,16
| +1 +? +3
| +1+?+3
| 0.00074
| 0.00074
|-
|-
| 0,3,10,17
| 0,3,10,17
| +1 +? +3
| +1+?+3
| 0.00057
| 0.00057
|-
| 0,3,17,18
| +2 +? +1
| 0.00128
|-
| 0,3,17,21
| +1 +? +2
| 0.00142
|-
|-
| 0,3,17,23
| 0,3,17,23
| +1 +? +3
| +1+?+3
| 0.00026
| 0.00026
|-
|-
| 0,3,18,19
| 0,3,18,19
| +2 +? +1
| +2+?+1
| 0.00082
| 0.00082
|-
| 0,3,18,20
| +1 +? +1
| 0.00101
|-
|-
| 0,3,18,21
| 0,3,18,21
| +2 +? +3
| +2+?+3
| 0.00075
| 0.00075
|-
|-
| 0,3,18,22
| 0,3,18,22
| +1 +? +2
| +1+?+2
| 0.00025
| 0.00025
|-
| 0,3,18,24
| +1 +? +3
| 0.00107
|-
|-
| 0,3,19,20
| 0,3,19,20
| +2 +? +1
| +2+?+1
| 0.00035
| 0.00035
|-
|-
| 0,3,19,21
| 0,3,19,21
| +1 +? +1
| +1+?+1
| 0.00019
| 0.00019
|-
|-
| 0,3,19,22
| 0,3,19,22
| +2 +? +3
| +2+?+3
| 0.00030
| 0.00030
|-
|-
| 0,3,19,23
| 0,3,19,23
| +1 +? +2
| +1+?+2
| 0.00094
| 0.00094
|-
|-
| 0,3,20,21
| 0,3,20,21
| +2 +? +1
| +2+?+1
| 0.00013
| 0.00013
|-
|-
| 0,3,20,22
| 0,3,20,22
| +1 +? +1
| +1+?+1
| 0.00066
| 0.00066
|-
| 0,3,20,23
| +2 +? +3
| 0.00137
|-
|-
| 0,3,21,22
| 0,3,21,22
| +2 +? +1
| +2+?+1
| 0.00063
| 0.00063
|-
| 0,3,22,23
| +2 +? +1
| 0.00113
|-
| 0,3,25,30
| +1 +? +3
| 0.00146
|-
|-
| 0,3,26,31
| 0,3,26,31
| +1 +? +3
| +1+?+3
| 0.00016
| 0.00016
|-
|-
| 0,4,5,12
| 0,4,5,12
| +1 +? +2
| +1+?+2
| 0.00059
| 0.00059
|-
|-
| 0,4,5,15
| 0,4,5,15
| +1 +? +3
| +1+?+3
| 0.00060
| 0.00060
|-
| 0,4,6,16
| +1 +? +3
| 0.00118
|-
| 0,4,7,12
| +2 +? +3
| 0.00128
|-
|-
| 0,4,8,13
| 0,4,8,13
| +2 +? +3
| +2+?+3
| 0.00013
| 0.00013
|-
| 0,4,10,19
| +1 +? +3
| 0.00127
|-
|-
| 0,4,11,20
| 0,4,11,20
| +1 +? +3
| +1+?+3
| 0.00049
| 0.00049
|-
| 0,4,12,13
| +3 +? +1
| 0.00122
|-
|-
| 0,4,12,18
| 0,4,12,18
| +1 +? +2
| +1+?+2
| 0.00042
| 0.00042
|-
|-
| 0,4,13,14
| 0,4,13,14
| +3 +? +1
| +3+?+1
| 0.00079
| 0.00079
|-
| 0,4,13,15
| +3 +? +2
| 0.00107
|-
|-
| 0,4,13,16
| 0,4,13,16
| +1 +? +1
| +1+?+1
| 0.00088
| 0.00088
|-
| 0,4,13,19
| +1 +? +2
| 0.00119
|-
|-
| 0,4,14,15
| 0,4,14,15
| +3 +? +1
| +3+?+1
| 0.00035
| 0.00035
|-
|-
| 0,4,14,16
| 0,4,14,16
| +3 +? +2
| +3+?+2
| 0.00024
| 0.00024
|-
|-
| 0,4,14,17
| 0,4,14,17
| +1 +? +1
| +1+?+1
| 0.00024
| 0.00024
|-
|-
| 0,4,15,16
| 0,4,15,16
| +3 +? +1
| +3+?+1
| 0.00009
| 0.00009
|-
|-
| 0,4,15,17
| 0,4,15,17
| +3 +? +2
| +3+?+2
| 0.00060
| 0.00060
|-
| 0,4,15,18
| +1 +? +1
| 0.00139
|-
|-
| 0,4,16,17
| 0,4,16,17
| +3 +? +1
| +3+?+1
| 0.00055
| 0.00055
|-
| 0,4,16,18
| +3 +? +2
| 0.00145
|-
| 0,4,16,24
| +1 +? +3
| 0.00119
|-
| 0,4,17,18
| +3 +? +1
| 0.00102
|-
|-
| 0,4,17,25
| 0,4,17,25
| +1 +? +3
| +1+?+3
| 0.00058
| 0.00058
|-
| 0,4,18,19
| +3 +? +1
| 0.00149
|-
| 0,4,18,22
| +2 +? +3
| 0.00102
|-
|-
| 0,4,19,23
| 0,4,19,23
| +2 +? +3
| +2+?+3
| 0.00040
| 0.00040
|-
|-
| 0,4,21,26
| 0,4,21,26
| +1 +? +2
| +1+?+2
| 0.00030
| 0.00030
|-
| 0,4,22,27
| +1 +? +2
| 0.00131
|-
|-
| 0,4,23,30
| 0,4,23,30
| +1 +? +3
| +1+?+3
| 0.00062
| 0.00062
|-
| 0,4,24,31
| +1 +? +3
| 0.00115
|-
|-
| 0,5,6,9
| 0,5,6,9
| +3 +? +2
| +3+?+2
| 0.00013
| 0.00013
|-
| 0,5,7,10
| +3 +? +2
| 0.00120
|-
|-
| 0,5,7,19
| 0,5,7,19
| +1 +? +3
| +1+?+3
| 0.00069
| 0.00069
|-
| 0,5,9,15
| +2 +? +3
| 0.00142
|-
|-
| 0,5,9,17
| 0,5,9,17
| +1 +? +2
| +1+?+2
| 0.00047
| 0.00047
|-
| 0,5,10,12
| +2 +? +1
| 0.00147
|-
| 0,5,10,14
| +1 +? +1
| 0.00115
|-
|-
| 0,5,10,16
| 0,5,10,16
| +2 +? +3
| +2+?+3
| 0.00038
| 0.00038
|-
|-
| 0,5,11,13
| 0,5,11,13
| +2 +? +1
| +2+?+1
| 0.00067
| 0.00067
|-
|-
| 0,5,11,15
| 0,5,11,15
| +1 +? +1
| +1+?+1
| 0.00027
| 0.00027
|-
|-
| 0,5,11,22
| 0,5,11,22
| +1 +? +3
| +1+?+3
| 0.00052
| 0.00052
|-
|-
| 0,5,12,14
| 0,5,12,14
| +2 +? +1
| +2+?+1
| 0.00015
| 0.00015
|-
|-
| 0,5,13,15
| 0,5,13,15
| +2 +? +1
| +2+?+1
| 0.00099
| 0.00099
|-
|-
| 0,5,15,22
| 0,5,15,22
| +1 +? +2
| +1+?+2
| 0.00090
| 0.00090
|-
| 0,5,16,23
| +1 +? +2
| 0.00113
|-
|-
| 0,5,16,26
| 0,5,16,26
| +1 +? +3
| +1+?+3
| 0.00034
| 0.00034
|-
| 0,5,18,23
| +2 +? +3
| 0.00129
|-
|-
| 0,5,19,24
| 0,5,19,24
| +2 +? +3
| +2+?+3
| 0.00051
| 0.00051
|-
| 0,5,21,30
| +1 +? +3
| 0.00119
|-
| 0,5,22,31
| +1 +? +3
| 0.00105
|-
| 0,5,23,24
| +3 +? +1
| 0.00144
|-
|-
| 0,5,23,29
| 0,5,23,29
| +1 +? +2
| +1+?+2
| 0.00015
| 0.00015
|-
|-
| 0,5,24,25
| 0,5,24,25
| +3 +? +1
| +3+?+1
| 0.00090
| 0.00090
|-
| 0,5,24,26
| +3 +? +2
| 0.00115
|-
|-
| 0,5,24,27
| 0,5,24,27
| +1 +? +1
| +1+?+1
| 0.00085
| 0.00085
|-
|-
| 0,5,25,26
| 0,5,25,26
| +3 +? +1
| +3+?+1
| 0.00034
| 0.00034
|-
|-
| 0,5,25,27
| 0,5,25,27
| +3 +? +2
| +3+?+2
| 0.00011
| 0.00011
|-
|-
| 0,5,25,28
| 0,5,25,28
| +1 +? +1
| +1+?+1
| 0.00058
| 0.00058
|-
|-
| 0,5,26,27
| 0,5,26,27
| +3 +? +1
| +3+?+1
| 0.00023
| 0.00023
|-
|-
| 0,5,26,28
| 0,5,26,28
| +3 +? +2
| +3+?+2
| 0.00096
| 0.00096
|-
|-
| 0,5,27,28
| 0,5,27,28
| +3 +? +1
| +3+?+1
| 0.00081
| 0.00081
|-
| 0,5,28,29
| +3 +? +1
| 0.00140
|-
| 0,6,7,21
| +1 +? +3
| 0.00124
|-
| 0,6,8,22
| +1 +? +3
| 0.00148
|-
|-
| 0,6,9,14
| 0,6,9,14
| +1 +? +1
| +1+?+1
| 0.00013
| 0.00013
|-
|-
| 0,6,11,18
| 0,6,11,18
| +2 +? +3
| +2+?+3
| 0.00020
| 0.00020
|-
| 0,6,11,24
| +1 +? +3
| 0.00109
|-
|-
| 0,6,12,21
| 0,6,12,21
| +1 +? +2
| +1+?+2
| 0.00064
| 0.00064
|-
| 0,6,14,17
| +3 +? +2
| 0.00102
|-
|-
| 0,6,15,18
| 0,6,15,18
| +3 +? +2
| +3+?+2
| 0.00025
| 0.00025
|-
| 0,6,16,28
| +1 +? +3
| 0.00103
|-
|-
| 0,6,18,26
| 0,6,18,26
| +1 +? +2
| +1+?+2
| 0.00075
| 0.00075
|-
| 0,6,19,23
| +1 +? +1
| 0.00127
|-
|-
| 0,6,19,25
| 0,6,19,25
| +2 +? +3
| +2+?+3
| 0.00062
| 0.00062
|-
|-
| 0,6,20,22
| 0,6,20,22
| +2 +? +1
| +2+?+1
| 0.00074
| 0.00074
|-
|-
| 0,6,20,24
| 0,6,20,24
| +1 +? +1
| +1+?+1
| 0.00046
| 0.00046
|-
|-
| 0,6,20,31
| 0,6,20,31
| +1 +? +3
| +1+?+3
| 0.00043
| 0.00043
|-
|-
| 0,6,21,23
| 0,6,21,23
| +2 +? +1
| +2+?+1
| 0.00025
| 0.00025
|-
| 0,6,22,24
| +2 +? +1
| 0.00127
|-
|-
| 0,6,24,31
| 0,6,24,31
| +1 +? +2
| +1+?+2
| 0.00091
| 0.00091
|-
| 0,7,8,10
| +3 +? +1
| 0.00132
|-
| 0,7,8,11
| +2 +? +1
| 0.00135
|-
|-
| 0,7,8,12
| 0,7,8,12
| +3 +? +2
| +3+?+2
| 0.00097
| 0.00097
|-
|-
| 0,7,8,14
| 0,7,8,14
| +1 +? +1
| +1+?+1
| 0.00076
| 0.00076
|-
|-
| 0,7,8,24
| 0,7,8,24
| +1 +? +3
| +1+?+3
| 0.00043
| 0.00043
|-
|-
| 0,7,9,11
| 0,7,9,11
| +3 +? +1
| +3+?+1
| 0.00053
| 0.00053
|-
|-
| 0,7,9,12
| 0,7,9,12
| +2 +? +1
| +2+?+1
| 0.00018
| 0.00018
|-
|-
| 0,7,9,13
| 0,7,9,13
| +3 +? +2
| +3+?+2
| 0.00054
| 0.00054
|-
|-
| 0,7,9,20
| 0,7,9,20
| +1 +? +2
| +1+?+2
| 0.00020
| 0.00020
|-
|-
| 0,7,10,12
| 0,7,10,12
| +3 +? +1
| +3+?+1
| 0.00028
| 0.00028
|-
| 0,7,10,13
| +2 +? +1
| 0.00101
|-
| 0,7,11,13
| +3 +? +1
| 0.00111
|-
| 0,7,11,26
| +1 +? +3
| 0.00120
|-
|-
| 0,7,12,20
| 0,7,12,20
| +2 +? +3
| +2+?+3
| 0.00010
| 0.00010
|-
|-
| 0,7,14,24
| 0,7,14,24
| +1 +? +2
| +1+?+2
| 0.00004
| 0.00004
|-
|-
| 0,7,15,29
| 0,7,15,29
| +1 +? +3
| +1+?+3
| 0.00028
| 0.00028
|-
| 0,7,16,21
| +1 +? +1
| 0.00113
|-
|-
| 0,7,17,22
| 0,7,17,22
| +1 +? +1
| +1+?+1
| 0.00091
| 0.00091
|-
|-
| 0,7,19,26
| 0,7,19,26
| +2 +? +3
| +2+?+3
| 0.00073
| 0.00073
|-
| 0,7,19,28
| +1 +? +2
| 0.00106
|-
|-
| 0,7,22,25
| 0,7,22,25
| +3 +? +2
| +3+?+2
| 0.00065
| 0.00065
|-
|-
| 0,7,23,26
| 0,7,23,26
| +3 +? +2
| +3+?+2
| 0.00086
| 0.00086
|-
|-
| 0,7,27,31
| 0,7,27,31
| +1 +? +1
| +1+?+1
| 0.00074
| 0.00074
|-
|-
| 0,7,28,30
| 0,7,28,30
| +2 +? +1
| +2+?+1
| 0.00044
| 0.00044
|-
|-
| 0,7,29,31
| 0,7,29,31
| +2 +? +1
| +2+?+1
| 0.00074
| 0.00074
|-
|-
| 0,8,11,23
| 0,8,11,23
| +1 +? +2
| +1+?+2
| 0.00070
| 0.00070
|-
|-
| 0,8,11,28
| 0,8,11,28
| +1 +? +3
| +1+?+3
| 0.00080
| 0.00080
|-
|-
| 0,8,13,22
| 0,8,13,22
| +2 +? +3
| +2+?+3
| 0.00070
| 0.00070
|-
|-
| 0,8,14,20
| 0,8,14,20
| +1 +? +1
| +1+?+1
| 0.00072
| 0.00072
|-
| 0,8,15,17
| +3 +? +1
| 0.00124
|-
| 0,8,15,18
| +2 +? +1
| 0.00112
|-
|-
| 0,8,15,19
| 0,8,15,19
| +3 +? +2
| +3+?+2
| 0.00057
| 0.00057
|-
|-
| 0,8,16,18
| 0,8,16,18
| +3 +? +1
| +3+?+1
| 0.00031
| 0.00031
|-
|-
| 0,8,16,19
| 0,8,16,19
| +2 +? +1
| +2+?+1
| 0.00023
| 0.00023
|-
| 0,8,16,20
| +3 +? +2
| 0.00119
|-
|-
| 0,8,16,27
| 0,8,16,27
| +1 +? +2
| +1+?+2
| 0.00085
| 0.00085
|-
|-
| 0,8,17,19
| 0,8,17,19
| +3 +? +1
| +3+?+1
| 0.00063
| 0.00063
|-
|-
| 0,8,19,27
| 0,8,19,27
| +2 +? +3
| +2+?+3
| 0.00084
| 0.00084
|-
| 0,8,21,31
| +1 +? +2
| 0.00112
|-
|-
| 0,8,23,28
| 0,8,23,28
| +1 +? +1
| +1+?+1
| 0.00055
| 0.00055
|-
|-
| 0,9,10,15
| 0,9,10,15
| +3 +? +2
| +3+?+2
| 0.00092
| 0.00092
|-
| 0,9,11,16
| +3 +? +2
| 0.00106
|-
|-
| 0,9,11,30
| 0,9,11,30
| +1 +? +3
| +1+?+3
| 0.00012
| 0.00012
|-
|-
| 0,9,13,20
| 0,9,13,20
| +1 +? +1
| +1+?+1
| 0.00100
| 0.00100
|-
| 0,9,13,23
| +2 +? +3
| 0.00114
|-
|-
| 0,9,13,26
| 0,9,13,26
| +1 +? +2
| +1+?+2
| 0.00021
| 0.00021
|-
|-
| 0,9,17,29
| 0,9,17,29
| +1 +? +2
| +1+?+2
| 0.00062
| 0.00062
|-
|-
| 0,9,19,28
| 0,9,19,28
| +2 +? +3
| +2+?+3
| 0.00096
| 0.00096
|-
|-
| 0,9,20,26
| 0,9,20,26
| +1 +? +1
| +1+?+1
| 0.00070
| 0.00070
|-
| 0,9,21,23
| +3 +? +1
| 0.00136
|-
| 0,9,21,24
| +2 +? +1
| 0.00121
|-
|-
| 0,9,21,25
| 0,9,21,25
| +3 +? +2
| +3+?+2
| 0.00055
| 0.00055
|-
|-
| 0,9,22,24
| 0,9,22,24
| +3 +? +1
| +3+?+1
| 0.00031
| 0.00031
|-
|-
| 0,9,22,25
| 0,9,22,25
| +2 +? +1
| +2+?+1
| 0.00034
| 0.00034
|-
| 0,9,22,26
| +3 +? +2
| 0.00145
|-
|-
| 0,9,23,25
| 0,9,23,25
| +3 +? +1
| +3+?+1
| 0.00077
| 0.00077
|-
| 0,10,11,19
| +1 +? +1
| 0.00101
|-
| 0,10,11,26
| +1 +? +2
| 0.00142
|-
|-
| 0,10,13,17
| 0,10,13,17
| +2 +? +1
| +2+?+1
| 0.00066
| 0.00066
|-
| 0,10,14,18
| +2 +? +1
| 0.00109
|-
|-
| 0,10,14,25
| 0,10,14,25
| +2 +? +3
| +2+?+3
| 0.00076
| 0.00076
|-
|-
| 0,10,16,21
| 0,10,16,21
| +3 +? +2
| +3+?+2
| 0.00034
| 0.00034
|-
|-
| 0,10,18,25
| 0,10,18,25
| +1 +? +1
| +1+?+1
| 0.00004
| 0.00004
|-
| 0,10,18,31
| +1 +? +2
| 0.00139
|-
| 0,10,19,29
| +2 +? +3
| 0.00108
|-
| 0,10,26,30
| +3 +? +2
| 0.00146
|-
|-
| 0,10,27,29
| 0,10,27,29
| +3 +? +1
| +3+?+1
| 0.00080
| 0.00080
|-
|-
| 0,10,27,30
| 0,10,27,30
| +2 +? +1
| +2+?+1
| 0.00029
| 0.00029
|-
|-
| 0,10,27,31
| 0,10,27,31
| +3 +? +2
| +3+?+2
| 0.00077
| 0.00077
|-
|-
| 0,10,28,30
| 0,10,28,30
| +3 +? +1
| +3+?+1
| 0.00040
| 0.00040
|-
| 0,10,28,31
| +2 +? +1
| 0.00147
|-
|-
| 0,11,12,18
| 0,11,12,18
| +3 +? +2
| +3+?+2
| 0.00040
| 0.00040
|-
|-
| 0,11,12,28
| 0,11,12,28
| +1 +? +2
| +1+?+2
| 0.00038
| 0.00038
|-
|-
| 0,11,13,16
| 0,11,13,16
| +3 +? +1
| +3+?+1
| 0.00049
| 0.00049
|-
|-
| 0,11,14,17
| 0,11,14,17
| +3 +? +1
| +3+?+1
| 0.00085
| 0.00085
|-
|-
| 0,11,14,26
| 0,11,14,26
| +2 +? +3
| +2+?+3
| 0.00077
| 0.00077
|-
|-
| 0,11,16,24
| 0,11,16,24
| +1 +? +1
| +1+?+1
| 0.00085
| 0.00085
|-
|-
| 0,11,18,22
| 0,11,18,22
| +2 +? +1
| +2+?+1
| 0.00057
| 0.00057
|-
| 0,11,19,23
| +2 +? +1
| 0.00138
|-
| 0,11,19,30
| +2 +? +3
| 0.00120
|-
|-
| 0,11,21,26
| 0,11,21,26
| +3 +? +2
| +3+?+2
| 0.00058
| 0.00058
|-
|-
| 0,11,23,30
| 0,11,23,30
| +1 +? +1
| +1+?+1
| 0.00023
| 0.00023
|-
|-
| 0,12,15,24
| 0,12,15,24
| +1 +? +1
| +1+?+1
| 0.00060
| 0.00060
|-
| 0,12,16,22
| +3 +? +2
| 0.00103
|-
| 0,12,17,20
| +3 +? +1
| 0.00132
|-
|-
| 0,12,18,21
| 0,12,18,21
| +3 +? +1
| +3+?+1
| 0.00014
| 0.00014
|-
| 0,12,19,31
| +2 +? +3
| 0.00132
|-
|-
| 0,12,21,29
| 0,12,21,29
| +1 +? +1
| +1+?+1
| 0.00078
| 0.00078
|-
|-
| 0,12,23,27
| 0,12,23,27
| +2 +? +1
| +2+?+1
| 0.00036
| 0.00036
|-
|-
| 0,12,25,30
| 0,12,25,30
| +3 +? +2
| +3+?+2
| 0.00084
| 0.00084
|-
| 0,13,14,24
| +1 +? +1
| 0.00133
|-
| 0,13,15,29
| +2 +? +3
| 0.00117
|-
|-
| 0,13,16,21
| 0,13,16,21
| +2 +? +1
| +2+?+1
| 0.00057
| 0.00057
|-
|-
| 0,13,19,28
| 0,13,19,28
| +1 +? +1
| +1+?+1
| 0.00023
| 0.00023
|-
| 0,13,21,27
| +3 +? +2
| 0.00122
|-
|-
| 0,13,22,25
| 0,13,22,25
| +3 +? +1
| +3+?+1
| 0.00019
| 0.00019
|-
| 0,13,23,26
| +3 +? +1
| 0.00143
|-
|-
| 0,13,27,31
| 0,13,27,31
| +2 +? +1
| +2+?+1
| 0.00012
| 0.00012
|-
|-
| 0,14,15,30
| 0,14,15,30
| +2 +? +3
| +2+?+3
| 0.00004
| 0.00004
|-
|-
| 0,14,17,24
| 0,14,17,24
| +3 +? +2
| +3+?+2
| 0.00028
| 0.00028
|-
|-
| 0,14,20,25
| 0,14,20,25
| +2 +? +1
| +2+?+1
| 0.00048
| 0.00048
|-
|-
| 0,14,26,29
| 0,14,26,29
| +3 +? +1
| +3+?+1
| 0.00012
| 0.00012
|-
|-
| 0,15,16,20
| 0,15,16,20
| +3 +? +1
| +3+?+1
| 0.00002
| 0.00002
|-
| 0,15,21,28
| +3 +? +2
| 0.00134
|-
|-
| 0,15,24,29
| 0,15,24,29
| +2 +? +1
| +2+?+1
| 0.00028
| 0.00028
|-
| 0,16,18,24
| +2 +? +1
| 0.00143
|-
| 0,16,19,23
| +3 +? +1
| 0.00102
|-
| 0,16,20,24
| +3 +? +1
| 0.00104
|-
|-
| 0,16,20,31
| 0,16,20,31
| +1 +? +1
| +1+?+1
| 0.00042
| 0.00042
|-
|-
| 0,16,24,31
| 0,16,24,31
| +3 +? +2
| +3+?+2
| 0.00051
| 0.00051
|-
|-
| 0,17,21,29
| 0,17,21,29
| +3 +? +2
| +3+?+2
| 0.00090
| 0.00090
|-
|-
| 0,17,22,28
| 0,17,22,28
| +2 +? +1
| +2+?+1
| 0.00062
| 0.00062
|-
|-
| 0,17,23,27
| 0,17,23,27
| +3 +? +1
| +3+?+1
| 0.00039
| 0.00039
|-
|-
| 0,18,25,31
| 0,18,25,31
| +2 +? +1
| +2+?+1
| 0.00007
| 0.00007
|-
|-
| 0,18,26,30
| 0,18,26,30
| +3 +? +1
| +3+?+1
| 0.00001
| 0.00001
|-
| 0,19,20,27
| +2 +? +1
| 0.00139
|-
|-
| 0,19,21,30
| 0,19,21,30
| +3 +? +2
| +3+?+2
| 0.00014
| 0.00014
|-
|-
| 0,20,21,26
| 0,20,21,26
| +3 +? +1
| +3+?+1
| 0.00032
| 0.00032
|-
| 0,20,23,30
| +2 +? +1
| 0.00110
|-
|-
| 0,21,24,29
| 0,21,24,29
| +3 +? +1
| +3+?+1
| 0.00026
| 0.00026
|}
|}
== Octave stretch or compression ==
Whether [[octave stretch]], shrink or neither is advised for 32edo depends on which [[val]]s one wishes to use.
For 32, pure-octaves, or slight compression (~0.5{{c}}), works well.
For 32f, moderate compression (~2{{c}}) works well. This is close to [[zpi|133zpi]] (32.07edo).
For 32c or 32cf, substantial compression (3-4{{c}}) is well suited.
For 32be, substantial ''stretch'' works (~5{{c}}). This is close to [[zpi|132zpi]] (31.86edo).
The graph shows [[zeta]] near 32edo.
[[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 ==
* [[Lumatone mapping for 32edo]]


== 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]]
* [https://www.youtube.com/shorts/nTQfjPjeee8 ''32edo improv''] (2025)
* ''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]]
* [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 rank 2 temperaments table|add lumatone mapping}}
{{Todo|add scales list}}
Retrieved from "https://en.xen.wiki/w/32edo"