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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 {{nowrap|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 [[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. This is, according to some metrics, one of the most optimal tunings of archy temperament, as it splits the error rather cleanly between 3 and 7 (accounting for the higher error tolerance on lower harmonics). 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.


=== Odd harmonics ===
=== Odd 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 ==
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! Degree
! Degree
! Cents
! Cents
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
! 13-limit Ratios
! 13-limit Ratios
! Other
! Other
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[[File:plot32.png|alt=plot32.png|plot32.png]]
[[File:plot32.png|alt=plot32.png|plot32.png]]


==== Zeta peak index ====
== Regular temperament properties ==
{{ZPI
{| class="wikitable center-4 center-5 center-6"
| zpi = 133
|-
| steps = 32.0701500181780
! rowspan="2" | [[Subgroup]]
| step size = 37.4179727665700
! rowspan="2" | [[Comma list]]
| tempered height = 4.471728
! rowspan="2" | [[Mapping]]
| pure height = 2.687487
! rowspan="2" | Optimal<br>8ve stretch (¢)
| integral = 0.689412
! colspan="2" | Tuning error
| gap = 12.537826
|-
| octave = 1197.37512853024
! [[TE error|Absolute]] (¢)
| consistent = 4
! [[TE simple badness|Relative]] (%)
| distinct = 4
|-
}}
| 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 ==
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; [[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]]
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[[Category:Listen]]
[[Category:Listen]]
[[Category:Sixix]]
[[Category:Sixix]]
{{todo|add rank 2 temperaments table}}
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