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== Theory ==
== 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.  
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.  
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.  
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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 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]].
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]].


=== Subsets and supersets ===
=== Subsets and supersets ===
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! 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
|-
|-
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| D
| D
| 1/1
| 1/1
|
|-
|-
| 1
| 1
Line 47: Line 48:
| up unison, minor 2nd
| up unison, minor 2nd
| ^D, Eb
| ^D, Eb
| 49/48, 50/49, 45/44
| ''[[25/24]]'', [[45/44]], [[49/48]]
| 46/45, 52/51, 51/50
| 46/45, 52/51, 51/50
|-
|-
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| 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
|-
|-
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| dupminor 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
| 16/15
| ''[[13/12]]'', [[15/14]], [[16/15]]
| 49/46
| 49/46
|-
|-
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| dudmajor 2nd
| dudmajor 2nd
| vvE
| vvE
| 12/11, 49/45
| ''[[10/9]]'', [[12/11]], ''[[14/13]]'', [[35/32]]
| 25/23
| 25/23
|-
|-
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| aug 1sn, downmajor 2nd
| aug 1sn, downmajor 2nd
| D#, vE
| D#, vE
| 10/9, 39/35
| ''[[11/10]]''
| 19/17
| 19/17
|-
|-
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| major 2nd
| major 2nd
| E
| E
| 8/7, 25/22
| [[8/7]], ''[[9/8]]''
| 57/50
| 57/50
|-
|-
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| minor 3rd
| minor 3rd
| F
| F
| 7/6, 64/55
| [[7/6]], ''[[13/11]]'', [[15/13]]
| 57/49
| 57/49
|-
|-
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| upminor 3rd
| upminor 3rd
| ^F
| ^F
| 6/5, 32/27
| [[33/28]]
| 19/16
| 19/16
|-
|-
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| 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
|-
|-
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| dudmajor 3rd
| dudmajor 3rd
| vvF#
| vvF#
| 5/4, 26/21, 56/45, 96/77
| [[5/4]], [[16/13]], [[27/22]]
| 36/29
| 36/29
|-
|-
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| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| 14/11, 33/26, 80/63
| [[14/11]]
| 19/15
| 19/15
|-
|-
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| 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
|-
|-
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| perfect 4th
| perfect 4th
| G
| G
| 4/3, 33/25, 160/121
| [[4/3]], [[21/16]]
| 45/34, 85/64
| 45/34, 85/64
|-
|-
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| up 4th
| up 4th
| ^G
| ^G
| 27/20, 110/81
| [[15/11]]
| 19/14, 23/17
| 19/14, 23/17
|-
|-
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| dup 4th, updim 5th
| dup 4th, updim 5th
| ^^G, ^Ab
| ^^G, ^Ab
| 18/13, 11/8
| [[11/8]], [[48/35]], ''[[27/20]]''
|
|-
|-
| 16
| 16
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| 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
|-
|-
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| downaug 4th, dud 5th
| downaug 4th, dud 5th
| vG#, vvA
| vG#, vvA
| 13/9, 16/11
| [[16/11]], [[35/24]], ''[[40/27]]''
|
|-
|-
| 18
| 18
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| down 5th
| down 5th
| vA
| vA
| 40/27, 81/55
| [[22/15]]
| 28/19, 34/23
| 28/19, 34/23
|-
|-
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| perfect 5th
| perfect 5th
| A
| A
| 3/2, 50/33, 121/80
| [[3/2]], [[32/21]]
| 68/45, 128/85
| 68/45, 128/85
|-
|-
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| 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
|-
|-
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| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| 11/7, 52/33, 63/40
| [[11/7]]
| 30/19
| 30/19
|-
|-
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| dupminor 6th
| dupminor 6th
| ^^Bb
| ^^Bb
| 8/5, 21/13, 45/28, 77/48
| [[8/5]], [[13/8]], [[44/27]]
| 29/18
| 29/18
|-
|-
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| 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 229: Line 232:
| downmajor 6th
| downmajor 6th
| vB
| vB
| 5/3, 27/16
| [[56/33]]
| 32/19
| 32/19
|-
|-
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| major 6th
| major 6th
| B
| B
| 12/7, 55/32
| [[12/7]], ''[[22/13]]'', [[26/15]]
| 98/57
| 98/57
|-
|-
Line 245: Line 248:
| minor 7th
| minor 7th
| C
| C
| 7/4, 44/25
| [[7/4]], ''[[16/9]]''
| 100/57
| 100/57
|-
|-
Line 253: Line 256:
| upminor 7th
| upminor 7th
| ^C
| ^C
| 9/5, 70/39
| [[20/11]]
| 34/19
| 34/19
|-
|-
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| dupminor 7th
| dupminor 7th
| ^^C
| ^^C
| 11/6, 90/49
| ''[[9/5]]'', [[11/6]], ''[[13/7]]'', [[64/35]]
| 46/25
| 46/25
|-
|-
Line 269: Line 272:
| dudmajor 7th
| dudmajor 7th
| vvC#
| vvC#
| 15/8
| [[15/8]], ''[[24/13]]'', [[28/15]]
| 92/49
| 92/49
|-
|-
Line 277: 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 285: Line 288:
| major 7th, down 8ve
| major 7th, down 8ve
| C#, vD
| C#, vD
| 96/49, 49/25, 88/45
| ''[[48/25]]'', [[88/45]], [[96/49]]
| 45/23, 51/26, 100/51
| 45/23, 51/26, 100/51
|-
|-
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| 8ve
| 8ve
| D
| D
| 2/1
| [[2/1]]
|  
|  
|}
|}
Line 1,341: Line 1,344:
* [https://www.youtube.com/watch?v=LI-lMuAYws8 ''Revelation''] (2026)
* [https://www.youtube.com/watch?v=LI-lMuAYws8 ''Revelation''] (2026)


== Notes ==
<references group="note"/>
[[Category:Listen]]
[[Category:Listen]]
[[Category:Sixix]]
[[Category:Sixix]]
{{Todo|add scales list}}
{{Todo|add scales list}}
Retrieved from "https://en.xen.wiki/w/32edo"