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**Imported revision 219942206 - Original comment: **
 
- about contorsion (redundant, already stated it comes from 16edo)
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-13 13:08:14 UTC</tt>.<br>
: The original revision id was <tt>219942206</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //32 equal division// divides the octave into 32 equal parts of precisely 37.5 cents each. While even advocates of less-common edos can struggle to find something about it worth noting, it does provide an excellent tuning for [[Petr Parízek]]'s sixix temperament, which tempers out the 5-limit sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents.


== Theory ==
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]]. 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.


=Music=
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.
[[http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg|Sixix] by Petr Parízek</pre></div>
 
<h4>Original HTML content:</h4>
=== Odd harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;32edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;32 equal division&lt;/em&gt; divides the octave into 32 equal parts of precisely 37.5 cents each. While even advocates of less-common edos can struggle to find something about it worth noting, it does provide an excellent tuning for &lt;a class="wiki_link" href="/Petr%20Par%C3%ADzek"&gt;Petr Parízek&lt;/a&gt;'s sixix temperament, which tempers out the 5-limit sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents.&lt;br /&gt;
{{Harmonics in equal|32}}
&lt;br /&gt;
 
&lt;br /&gt;
=== As a tuning of other temperaments ===
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Music&lt;/h1&gt;
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.
[[&lt;!-- ws:start:WikiTextUrlRule:6:http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg --&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg" rel="nofollow"&gt;http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:6 --&gt;|Sixix] by Petr Parízek&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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 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.  
 
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]]. 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 2<sup>5</sup>, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.
 
See also [[32nd-octave temperaments]].
 
== Intervals ==
{| class="wikitable"
|-
! Degree
! Cents
! colspan="3" | [[Ups and downs notation]]
! 13-limit Ratios
! Other
|-
| 0
| 0.0
| P1
| perfect unison
| D
| 1/1
|-
| 1
| 37.5
| ^1, m2
| up unison, minor 2nd
| ^D, Eb
| 49/48, 50/49, 45/44
| 46/45, 52/51, 51/50
|-
| 2
| 75.0
| ^m2
| upminor 2nd
| ^Eb
| 22/21, 25/24
| 24/23, 23/22
|-
| 3
| 112.5
| ^^m2
| dupminor 2nd
| ^^Eb
| 16/15
| 49/46
|-
| 4
| 150.0
| vvM2
| dudmajor 2nd
| vvE
| 12/11, 49/45
| 25/23
|-
| 5
| 187.5
| A1, vM2
| aug 1sn, downmajor 2nd
| D#, vE
| 10/9, 39/35
| 19/17
|-
| 6
| 225.0
| M2
| major 2nd
| E
| 8/7, 25/22
| 57/50
|-
| 7
| 262.5
| m3
| minor 3rd
| F
| 7/6, 64/55
| 57/49
|-
| 8
| 300.0
| ^m3
| upminor 3rd
| ^F
| 6/5, 32/27
| 19/16
|-
| 9
| 337.5
| ^^m3
| dupminor 3rd
| ^^F
| 11/9, 39/32, 63/52
| 17/14, 28/23
|-
| 10
| 375.0
| vvM3
| dudmajor 3rd
| vvF#
| 5/4, 26/21, 56/45, 96/77
| 36/29
|-
| 11
| 412.5
| vM3
| downmajor 3rd
| vF#
| 14/11, 33/26, 80/63
| 19/15
|-
| 12
| 450.0
| M3
| major 3rd
| F#
| 13/10, 35/27, 64/49
| 22/17, 57/44
|-
| 13
| 487.5
| P4
| perfect 4th
| G
| 4/3, 33/25, 160/121
| 45/34, 85/64
|-
| 14
| 525.0
| ^4
| up 4th
| ^G
| 27/20, 110/81
| 19/14, 23/17
|-
| 15
| 562.5
| ^^4, ^d5
| dup 4th, updim 5th
| ^^G, ^Ab
| 18/13, 11/8
|-
| 16
| 600.0
| vvA4, ^^d5
| dudaug 4th, dupdim 5th
| vvG#, ^^Ab
| 7/5, 10/7, 99/70, 140/99
| 17/12, 12/17
|-
| 17
| 637.5
| vA4, vv5
| downaug 4th, dud 5th
| vG#, vvA
| 13/9, 16/11
|-
| 18
| 675.0
| v5
| down 5th
| vA
| 40/27, 81/55
| 28/19, 34/23
|-
| 19
| 712.5
| P5
| perfect 5th
| A
| 3/2, 50/33, 121/80
| 68/45, 128/85
|-
| 20
| 750.0
| m6
| minor 6th
| Bb
| 20/13, 54/35, 49/32
| 17/11, 88/57
|-
| 21
| 787.5
| ^m6
| upminor 6th
| ^Bb
| 11/7, 52/33, 63/40
| 30/19
|-
| 22
| 825.0
| ^^m6
| dupminor 6th
| ^^Bb
| 8/5, 21/13, 45/28, 77/48
| 29/18
|-
| 23
| 862.5
| vvM6
| dudmajor 6th
| vvB
| 18/11, 64/39, 104/63
| 28/17, 23/14
|-
| 24
| 900.0
| vM6
| downmajor 6th
| vB
| 5/3, 27/16
| 32/19
|-
| 25
| 937.5
| M6
| major 6th
| B
| 12/7, 55/32
| 98/57
|-
| 26
| 975.0
| m7
| minor 7th
| C
| 7/4, 44/25
| 100/57
|-
| 27
| 1012.5
| ^m7
| upminor 7th
| ^C
| 9/5, 70/39
| 34/19
|-
| 28
| 1050.0
| ^^m7
| dupminor 7th
| ^^C
| 11/6, 90/49
| 46/25
|-
| 29
| 1087.5
| vvM7
| dudmajor 7th
| vvC#
| 15/8
| 92/49
|-
| 30
| 1125.0
| vM7
| downmajor 7th
| vC#
| 21/11, 48/25
| 23/12, 44/23
|-
| 31
| 1162.5
| M7, v8
| major 7th, down 8ve
| C#, vD
| 96/49, 49/25, 88/45
| 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:
{{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]]
 
== 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
|}
 
== 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 ==
=== Modern renderings ===
; {{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]]
* ''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]]
* [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]]
* [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]]
* [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''] (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)
 
[[Category:Listen]]
[[Category:Sixix]]
{{Todo|add scales list}}
Retrieved from "https://en.xen.wiki/w/32edo"