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. 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)(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. 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 factors into primes as 25, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.

== Intervals ==

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! Degree

! Cents

! colspan="3" | [[Ups and ~~Downs Notation~~]]

! colspan="3" | [[Ups and downs notation]]

! 13-limit Ratios

! Other

Line 284:

Line 290:

== Notation ==

=== Ups and downs notation ===

32edo 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).

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.

=== 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]].

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default [[File:32-EDO_Revo_Sagittal.svg]]

</imagemap>

~~=== Ups and downs notation ===~~

Another way to notate the microtonal notes is to use [[Alternative symbols for ups and downs notation#Sharp-5|ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

~~{{Sharpness-sharp5}}~~

~~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.~~

== Approximation to JI ==

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[[File:plot32.png|alt=plot32.png|plot32.png]]

==== ~~Zeta peak index~~ ====

== Regular temperament properties ==

{| ~~class~~="~~wikitable center~~-~~all~~"

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 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

|-

~~! colspan="~~3" | ~~Tuning~~

| 2.3.5.7

~~! colspan~~=~~"3"~~ | ~~Strength~~

| 50/49, 64/63, 3125/2916

~~! colspan~~="2" | ~~Closest edo~~

| {{Mapping| 32 51 75 90 }} (32)

~~! colspan~~="2" | ~~Integer limit~~

| -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

|-

! ~~ZPI~~

! Periods per 8ve

~~! Steps~~ per ~~octave~~

! Generator*

~~! Step size (cents)~~

! Cents*

~~! Height~~

! Associated ratio*

~~! Integral~~

! Temperaments

~~! Gap~~

! ~~Edo~~

! ~~Octave (cents)~~

! ~~Consistent~~

! ~~Distinct~~

|-

| [[~~133zpi~~]]

| 1

| 32.~~0701500181780~~

| 1\32

| 37.~~4179727665700~~

| 37.5

| 4.~~471728~~

| 49/48

| 0.~~689412~~

| [[Slender]] (32)

| 12.~~537826~~

|-

| ~~32edo~~

| 1

| ~~1197~~.~~37512853024~~

| 9\32

| 4

| 262.5

| 4

| 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 (1\32)

| 487.5 (37.5)

| 4/3 (36/35)

| [[Octonion]] (32cf)

|-

| 16

| 14\33 (1\32)

| 487.5 (37.5)

| 4/3 (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 ==

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; [[Petr Pařízek]]

* [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']

; [[Billy Stiltner]]

* [https://billystiltner.bandcamp.com/album/1332 ''1332'']

; [[Chris Vaisvil]]

Line 1,190:

Line 1,286:

[[Category:Listen]]

[[Category:Sixix]]

~~{{todo|add rank 2 temperaments table}}~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|32}}
+{{ED intro}}
 == Theory ==
+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 factors into primes as 2<sup>5</sup>, 32edo contains subset edos {{EDOs| 2, 4, 8, and 16 }}.
 == Intervals ==
@@ Line 17: / Line 23: @@
 ! Degree
 ! Cents
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 ! 13-limit Ratios
 ! Other
@@ Line 284: / Line 290: @@
 == Notation ==
+=== Ups and downs 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).
+{{Sharpness-sharp5a}}
+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]]:
+{{Sharpness-sharp5}}
+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.
 === 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]].
@@ Line 306: / Line 322: @@
 default [[File:32-EDO_Revo_Sagittal.svg]]
 </imagemap>
-=== Ups and downs notation ===
-Another way to notate the microtonal notes is to use [[Alternative symbols for ups and downs notation#Sharp-5|ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
-{{Sharpness-sharp5}}
-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.
 == Approximation to JI ==
@@ Line 322: / Line 331: @@
 [[File:plot32.png|alt=plot32.png|plot32.png]]
-==== Zeta peak index ====
+== Regular temperament properties ==
-{| class="wikitable center-all"
+{| 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
 |-
-! colspan="3" | Tuning
+| 2.3.5.7
-! colspan="3" | Strength
+| 50/49, 64/63, 3125/2916
-! colspan="2" | Closest edo
+| {{Mapping| 32 51 75 90 }} (32)
-! colspan="2" | Integer limit
+| -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
 |-
-! ZPI
+! Periods<br>per 8ve
-! Steps per octave
+! Generator*
-! Step size (cents)
+! Cents*
-! Height
+! Associated<br>ratio*
-! Integral
+! Temperaments
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
 |-
-| [[133zpi]]
+| 1
-| 32.0701500181780
+| 1\32
-| 37.4179727665700
+| 37.5
-| 4.471728
+| 49/48
-| 0.689412
+| [[Slender]] (32)
-| 12.537826
+|-
-| 32edo
+| 1
-| 1197.37512853024
+| 9\32
-| 4
+| 262.5
-| 4
+| 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 ==
@@ Line 1,180: / Line 1,273: @@
 ; [[Petr Pařízek]]
 * [http://micro.soonlabel.com/petr_parizek/3125_2916_temp_q32.ogg ''Sixix'']
+; [[Billy Stiltner]]
+* [https://billystiltner.bandcamp.com/album/1332 ''1332'']
 ; [[Chris Vaisvil]]
@@ Line 1,190: / Line 1,286: @@
 [[Category:Listen]]
 [[Category:Sixix]]
-{{todo|add rank 2 temperaments table}}