32edo: Difference between revisions

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== 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.

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=== As a tuning of other temperaments ===

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)(1/11), 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. The same generator can also be used for the much more inaccurate [[mohavila]] temperament in the 32bde val, which treats the 675~~-cent~~ fifth from [[16edo]] as 3/2 (basically it is [[mavila]]'s response to neutral third 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)(1/11), 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 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 ~~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{{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]].

~~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 than the sharp fifth, so~~ 32edo ~~cannot truly be considered a [[dual-fifth]] system~~.

=== Subsets and supersets ===

Since 32 is a power of two and factors as 25, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}. The next power of two edo is [[64edo]], doubling 32edo.

~~=== Subsets and supersets ===~~

[[96edo]] is an important superset of both 32edo and [[12edo]], being known for approximating JI very well.

~~Since 32~~ is ~~a power~~ of ~~two~~ and ~~factors as 25, 32edo contains subset edos {{EDOs| 2~~, ~~4, 8, and 16 }}~~.

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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.
+edo 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.
@@ Line 11: / Line 11: @@
 === As a tuning of other temperaments ===
-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 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. The same generator can also be used for the much more inaccurate [[mohavila]] temperament in the 32bde val, which treats the 675-cent fifth from [[16edo]] as 3/2 (basically it is [[mavila]]'s response to neutral third 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 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 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{{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]].
-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 than the sharp fifth, so 32edo cannot truly be considered a [[dual-fifth]] system.
+=== 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.
-=== Subsets and supersets ===
+[[96edo]] is an important superset of both 32edo and [[12edo]], being known for approximating JI very well.
-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]].
@@ Line 30: / Line 32: @@
 ! Cents
 ! 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
 |-
@@ Line 39: / Line 41: @@
 | D
 | 1/1
+|
 |-
 | 1
@@ Line 45: / Line 48: @@
 | up unison, minor 2nd
 | ^D, Eb
-| 49/48, 50/49, 45/44
+| ''[[25/24]]'', [[45/44]], [[49/48]]
 | 46/45, 52/51, 51/50
 |-
@@ Line 53: / Line 56: @@
 | upminor 2nd
 | ^Eb
-| 22/21, 25/24
+| [[22/21]], [[26/25]], ''[[33/32]]'', ''[[36/35]]'', ''[[81/80]]''
 | 24/23, 23/22
 |-
@@ Line 61: / Line 64: @@
 | dupminor 2nd
 | ^^Eb
-| 16/15
+| ''[[13/12]]'', [[15/14]], [[16/15]]
 | 49/46
 |-
@@ Line 69: / Line 72: @@
 | dudmajor 2nd
 | vvE
-| 12/11, 49/45
+| ''[[10/9]]'', [[12/11]], ''[[14/13]]'', [[35/32]]
 | 25/23
 |-
@@ Line 77: / Line 80: @@
 | aug 1sn, downmajor 2nd
 | D#, vE
-| 10/9, 39/35
+| ''[[11/10]]''
 | 19/17
 |-
@@ Line 85: / Line 88: @@
 | major 2nd
 | E
-| 8/7, 25/22
+| [[8/7]], ''[[9/8]]''
 | 57/50
 |-
@@ Line 93: / Line 96: @@
 | minor 3rd
 | F
-| 7/6, 64/55
+| [[7/6]], ''[[13/11]]'', [[15/13]]
 | 57/49
 |-
@@ Line 101: / Line 104: @@
 | upminor 3rd
 | ^F
-| 6/5, 32/27
+| [[33/28]]
 | 19/16
 |-
@@ Line 109: / Line 112: @@
 | dupminor 3rd
 | ^^F
-| 11/9, 39/32, 63/52
+| ''[[6/5]]'', [[11/9]], [[39/32]]
 | 17/14, 28/23
 |-
@@ Line 117: / Line 120: @@
 | dudmajor 3rd
 | vvF#
-| 5/4, 26/21, 56/45, 96/77
+| [[5/4]], [[16/13]], [[27/22]]
 | 36/29
 |-
@@ Line 125: / Line 128: @@
 | downmajor 3rd
 | vF#
-| 14/11, 33/26, 80/63
+| [[14/11]]
 | 19/15
 |-
@@ Line 133: / Line 136: @@
 | major 3rd
 | F#
-| 13/10, 35/27, 64/49
+| [[9/7]], [[13/10]], ''[[32/25]]'', [[64/49]]
 | 22/17, 57/44
 |-
@@ Line 141: / Line 144: @@
 | perfect 4th
 | G
-| 4/3, 33/25, 160/121
+| [[4/3]], [[21/16]]
 | 45/34, 85/64
 |-
@@ Line 149: / Line 152: @@
 | up 4th
 | ^G
-| 27/20, 110/81
+| [[15/11]]
 | 19/14, 23/17
 |-
@@ Line 157: / Line 160: @@
 | dup 4th, updim 5th
 | ^^G, ^Ab
-| 18/13, 11/8
+| [[11/8]], [[48/35]], ''[[27/20]]''
+|
 |-
 | 16
@@ Line 164: / Line 168: @@
 | dudaug 4th, dupdim 5th
 | vvG#, ^^Ab
-| 7/5, 10/7, 99/70, 140/99
+| [[7/5]], [[10/7]], [[45/32]], [[64/45]]
 | 17/12, 12/17
 |-
@@ Line 172: / Line 176: @@
 | downaug 4th, dud 5th
 | vG#, vvA
-| 13/9, 16/11
+| [[16/11]], [[35/24]], ''[[40/27]]''
+|
 |-
 | 18
@@ Line 179: / Line 184: @@
 | down 5th
 | vA
-| 40/27, 81/55
+| [[22/15]]
 | 28/19, 34/23
 |-
@@ Line 187: / Line 192: @@
 | perfect 5th
 | A
-| 3/2, 50/33, 121/80
+| [[3/2]], [[32/21]]
 | 68/45, 128/85
 |-
@@ Line 195: / Line 200: @@
 | minor 6th
 | Bb
-| 20/13, 54/35, 49/32
+| [[14/9]], [[20/13]], ''[[25/16]]'', [[49/32]]
 | 17/11, 88/57
 |-
@@ Line 203: / Line 208: @@
 | upminor 6th
 | ^Bb
-| 11/7, 52/33, 63/40
+| [[11/7]]
 | 30/19
 |-
@@ Line 211: / Line 216: @@
 | dupminor 6th
 | ^^Bb
-| 8/5, 21/13, 45/28, 77/48
+| [[8/5]], [[13/8]], [[44/27]]
 | 29/18
 |-
@@ Line 219: / Line 224: @@
 | dudmajor 6th
 | vvB
-| 18/11, 64/39, 104/63
+| ''[[5/3]]'', [[18/11]], [[64/39]]
 | 28/17, 23/14
 |-
@@ Line 227: / Line 232: @@
 | downmajor 6th
 | vB
-| 5/3, 27/16
+| [[56/33]]
 | 32/19
 |-
@@ Line 235: / Line 240: @@
 | major 6th
 | B
-| 12/7, 55/32
+| [[12/7]], ''[[22/13]]'', [[26/15]]
 | 98/57
 |-
@@ Line 243: / Line 248: @@
 | minor 7th
 | C
-| 7/4, 44/25
+| [[7/4]], ''[[16/9]]''
 | 100/57
 |-
@@ Line 251: / Line 256: @@
 | upminor 7th
 | ^C
-| 9/5, 70/39
+| [[20/11]]
 | 34/19
 |-
@@ Line 259: / Line 264: @@
 | dupminor 7th
 | ^^C
-| 11/6, 90/49
+| ''[[9/5]]'', [[11/6]], ''[[13/7]]'', [[64/35]]
 | 46/25
 |-
@@ Line 267: / Line 272: @@
 | dudmajor 7th
 | vvC#
-| 15/8
+| [[15/8]], ''[[24/13]]'', [[28/15]]
 | 92/49
 |-
@@ Line 275: / Line 280: @@
 | downmajor 7th
 | vC#
-| 21/11, 48/25
+| [[21/11]], [[25/13]], ''[[64/33]]'', ''[[35/18]]'', ''[[160/81]]''
 | 23/12, 44/23
 |-
@@ Line 283: / Line 288: @@
 | major 7th, down 8ve
 | C#, vD
-| 96/49, 49/25, 88/45
+| ''[[48/25]]'', [[88/45]], [[96/49]]
 | 45/23, 51/26, 100/51
 |-
@@ Line 291: / Line 296: @@
 | 8ve
 | D
-| 2/1
+| [[2/1]]
 |
 |}
@@ Line 1,339: / Line 1,344: @@
 * [https://www.youtube.com/watch?v=LI-lMuAYws8 ''Revelation''] (2026)
+== Notes ==
+<references group="note"/>
 [[Category:Listen]]
 [[Category:Sixix]]
 {{Todo|add scales list}}

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