32edo: Difference between revisions

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== Theory ==

''~~See~~ [[~~regular temperament~~]] for ~~more about what all this means~~ and ~~how~~ to ~~use it~~.''

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.

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. 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.

=== ~~Harmonics~~ ===

=== Odd harmonics ===

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.

=== 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 }}.

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{ED intro}}
 == Theory ==
-''See [[regular temperament]] for more about what all this means and how to use it.''
+edo 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.
-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. 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.
-=== Harmonics ===
+=== Odd harmonics ===
-edo 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.
 {{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 }}.
+See also [[32nd-octave temperaments]].
 == Intervals ==
@@ Line 19: / Line 25: @@
 ! Degree
 ! Cents
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 ! 13-limit Ratios
 ! Other
@@ Line 286: / Line 292: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo 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).
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
-{{Sharpness-sharp5a}}
+{{Sharpness-sharp5-szg}}
-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]]:
+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.
-{{Sharpness-sharp5}}
+=== Kite's ups and downs notation ===
+edo 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).
-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.
+{{Ups and downs sharpness}}
 === 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]].
+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 ====
@@ Line 322: / Line 328: @@
 {{Q-odd-limit intervals|32}}
-=== Zeta properties ===
+== Regular temperament 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.
+{| 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
+|}
-[[File:plot32.png|alt=plot32.png|plot32.png]]
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
-==== Zeta peak index ====
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-{{ZPI
+|-
-| zpi = 133
+! Periods<br>per 8ve
-| steps = 32.0701500181780
+! Generator*
-| step size = 37.4179727665700
+! Cents*
-| tempered height = 4.471728
+! Associated<br>ratio*
-| pure height = 2.687487
+! Temperaments
-| integral = 0.689412
+|-
-| gap = 12.537826
+| 1
-| octave = 1197.37512853024
+| 1\32
-| consistent = 4
+| 37.5
-| distinct = 4
+| 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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Delta-rational harmony ==
@@ Line 1,155: / Line 1,257: @@
 | 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]]
 == Instruments ==
-[[Lumatone mapping for 32edo]]
+* [[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 ===
 ; [[Brody Bigwood]]
 * [https://www.youtube.com/watch?v=yMokW3-0vIs ''Beyond the Grid''] (2024)
+; [[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]]
-* [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]]
-* [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]]
-* [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'']
 ; [[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)
 [[Category:Listen]]
 [[Category:Sixix]]
-{{todo|add rank 2 temperaments table}}
+{{Todo|add scales list}}