27edo: Difference between revisions

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27edo, with its 400{{c}} major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.

Its step of 44.4{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having very [[harmonic entropy]] ~~and thus~~ is~~, in theory,~~ the ~~third-~~most dissonant ~~(third-closest to the peak at 46.4{{c}}, after neighboring [[26edo]] and its neighbor [[25edo]]), assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}~~. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

Its step of 44.4{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having very high [[harmonic entropy]]. In other words, there is a general perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

The [[chromatic semitone]] of 27edo, at 178{{c}}, is equal to a submajor second in size, meaning 27edo is a candidate for [[extraclassical tonality]] due to its sharp major third of 444 cents.

=== Odd harmonics ===

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=== Octave stretch ===

Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023{{c}}. More generally, narrowing the steps to between 44.2 and 44.35{{c}} would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55{{c}}, respectively.

Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023{{c}}. More generally, narrowing the steps to between 44.2 and 44.35{{c}} would be better in theory; [[43edt]], [[70ed6]], [[90ed10]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, 4.11, and 2.55{{c}}, respectively.

=== Subsets and supersets ===

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| do

|}

=== Interval quality and chord names in color notation ===

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| rowspan="2" | minor

| zo

| [a, b, 0, 1>

| {{monzo| a, b, 0, 1 }}

| 7/6, 7/4

|-

| fourthward wa

| [a, b>, b < −1

| {{monzo| a, b }}, {{nowrap|b < −1}}

| 32/27, 16/9

|-

| upminor

| gu

| [a, b, ~~-1>~~

| {{monzo| a, b, −1 }}

| 6/5, 9/5

|-

| rowspan="2" | mid

| tho

| [a, b, 0, 0, 0, 1>

| {{monzo| a, b, 0, 0, 0, 1 }}

| 13/12, 13/8

|-

| thu

| [a, b, 0, 0, 0, ~~-1>~~

| {{monzo| a, b, 0, 0, 0, −1 }}

| 16/13, 24/13

|-

| downmajor

| yo

| [a, b, 1>

| {{monzo| a, b, 1 }}

| 5/4, 5/3

|-

| rowspan="2" | major

| fifthward wa

| [a, b>, b > 1

| {{monzo| a, b }}, {{nowrap|b > 1}}

| 9/8, 27/16

|-

| ru

| [a, b, 0, ~~-1>~~

| {{monzo| a, b, 0, −1 }}

| 9/7, 12/7

|}

All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:

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=== 6L 1s (archeotonic) notation ===

The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 edostep in this instance, is denoted by ♯ and ♭.

The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 1 edostep in this instance, is denoted by ♯ and ♭.

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"

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{{Q-odd-limit intervals|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}}

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

~~{{ZPI~~

~~| zpi = 106~~

~~| steps = 27.0866140827635~~

~~| step size = 44.3023257293579~~

~~| tempered height = 6.069233~~

~~| pure height = 3.605183~~

~~| integral = 1.185939~~

~~| gap = 16.215619~~

~~| octave = 1196.16279469266~~

~~| consistent = 10~~

~~| distinct = 8~~

}}

== Regular temperament properties ==

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| 19th-partial chroma

|}

== Scales ==

=== MOS scales ===

* Superpyth ~~pentatonic~~ – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6

* Superpyth pentic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6

* Superpyth diatonic – Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1

* Superpyth chromatic – Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1

* Superpyth ~~hyperchromatic~~ – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1

* Superpyth enharmonic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1

* Augene[6] [[3L 3s]] (period = 9\27, gen = 2\27): 7 2 7 2 7 2

* Augene[9] [[3L 6s]] (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2

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* [[SNS (2/1, 3/2, 5/4)-7|5-limit / pental double harmonic major]]: 2 7 2 5 2 7 2

* enharmonic tetrachord octave species: 9 1 1 5 9 1 1, 1 9 1 5 1 9 1 (also Superpyth double harmonic major), 1 1 9 5 1 1 9

* [[The Pinetone System#The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4

* [[The Pinetone System #The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4

* [[The Pinetone System#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4

* [[The Pinetone System #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4

* [[The Pinetone System#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3

* [[The Pinetone System #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3

* [[The Pinetone System#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4

* [[The Pinetone System #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4

* [[The Pinetone System#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4

* [[The Pinetone System #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4

* [[The Pinetone System#Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3

* [[The Pinetone System #Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3

* 5-limit / pental double harmonic nonatonic (subset of Augene[12]): 2 5 2 2 5 2 5 2 2, 2 2 5 2 5 2 2 5 2 (Augene[9] [[4M]])

* 5-limit / pental double harmonic decatonic (subset of Augene[12]): 2 5 2 2 3 2 2 5 2 2

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* [[Blackdye]] / [[syntonic dipentatonic]] (superset of [[Zarlino]]): 1 4 2 4 1 4 2 4 1 4

* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of [[Zarlino]]): 3 2 4 2 3 2 4 2 3 2

* Direct sunlight (original/default tuning; subset of [[Sensi]][19]): 1 2 8 5 1 9 1 ((1, 3, 11, 16, 17, 26, 27)\27)

Direct sunlight (~~this is its~~ original/default tuning; subset of [[Sensi]][19])

* Hypersakura (original/default tuning; subset of Sensi[19]): 1 10 5 1 10 ((1 11 16 17 27)\27)

* ~~44.444~~

* 133.333

* 488.889

* 711.111

* 755.555

* 1155.555

* 1200.000

Hypersakura (~~this is its~~ original/default tuning; subset of Sensi[19])

* 44.444

* 488.889

* 711.111

* 755.555

* 1200.000

== Instruments ==

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; [[Bryan Deister]]

* [https://www.youtube.com/watch?v=hDP8cfJqWOI ''microtonal improvisation in 27edo''] (2023)

; [[Francium]]

* [https://www.youtube.com/watch?v=3Ty3FpmAdGA ''Happy Birthday in 27edo''] (2025)

; [[Igliashon Jones]]

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; [[Xotla]]

* "Funkrotonal" from ''Microtonal Allsorts'' (2023) – [https://open.spotify.com/track/1zjNkbm8kIkuCxtodyFCL0 Spotify] | [https://xotla.bandcamp.com/track/funkrotonal-27edo Bandcamp] | [https://www.youtube.com/watch?v=7gt1BBJuJsE YouTube]

~~== Notes ==~~

~~<references group="note" />~~

[[Category:Augene]]

@@ Line 9: / Line 9: @@
 edo, with its 400{{c}} major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.
-Its step of 44.4{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having very [[harmonic entropy]] and thus is, in theory, the third-most dissonant (third-closest to the peak at 46.4{{c}}, after neighboring [[26edo]] and its neighbor [[25edo]]), assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
+Its step of 44.4{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having very high [[harmonic entropy]]. In other words, there is a general perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
+The [[chromatic semitone]] of 27edo, at 178{{c}}, is equal to a submajor second in size, meaning 27edo is a candidate for [[extraclassical tonality]] due to its sharp major third of 444 cents.
 === Odd harmonics ===
@@ Line 15: / Line 17: @@
 === Octave stretch ===
-Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023{{c}}. More generally, narrowing the steps to between 44.2 and 44.35{{c}} would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55{{c}}, respectively.
+Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023{{c}}. More generally, narrowing the steps to between 44.2 and 44.35{{c}} would be better in theory; [[43edt]], [[70ed6]], [[90ed10]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, 4.11, and 2.55{{c}}, respectively.
 === Subsets and supersets ===
@@ Line 310: / Line 312: @@
 | do
 |}
+<references group="note" />
 === Interval quality and chord names in color notation ===
@@ Line 323: / Line 326: @@
 | rowspan="2" | minor
 | zo
-| [a, b, 0, 1>
+| {{monzo| a, b, 0, 1 }}
 | 7/6, 7/4
 |-
 | fourthward wa
-| [a, b>, b < −1
+| {{monzo| a, b }}, {{nowrap|b &lt; −1}}
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| [a, b, -1>
+| {{monzo| a, b, −1 }}
 | 6/5, 9/5
 |-
 | rowspan="2" | mid
 | tho
-| [a, b, 0, 0, 0, 1>
+| {{monzo| a, b, 0, 0, 0, 1 }}
 | 13/12, 13/8
 |-
 | thu
-| [a, b, 0, 0, 0, -1>
+| {{monzo| a, b, 0, 0, 0, −1 }}
 | 16/13, 24/13
 |-
 | downmajor
 | yo
-| [a, b, 1>
+| {{monzo| a, b, 1 }}
 | 5/4, 5/3
 |-
 | rowspan="2" | major
 | fifthward wa
-| [a, b>, b > 1
+| {{monzo| a, b }}, {{nowrap|b &gt; 1}}
 | 9/8, 27/16
 |-
 | ru
-| [a, b, 0, -1>
+| {{monzo| a, b, 0, −1 }}
 | 9/7, 12/7
 |}
 All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:
@@ Line 590: / Line 594: @@
 === 6L 1s (archeotonic) notation ===
-The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 edostep in this instance, is denoted by ♯ and ♭.
+The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 1 edostep in this instance, is denoted by ♯ and ♭.
 {| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
@@ Line 776: / Line 780: @@
 {{Q-odd-limit intervals|27}}
 {{Q-odd-limit intervals|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 106
-| steps = 27.0866140827635
-| step size = 44.3023257293579
-| tempered height = 6.069233
-| pure height = 3.605183
-| integral = 1.185939
-| gap = 16.215619
-| octave = 1196.16279469266
-| consistent = 10
-| distinct = 8
-}}
 == Regular temperament properties ==
@@ Line 1,143: / Line 1,133: @@
 | 19th-partial chroma
 |}
+<references group="note" />
 == Scales ==
 === MOS scales ===
 {{Main|List of MOS scales in 27edo}}
-* Superpyth pentatonic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6
+* Superpyth pentic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6
 * Superpyth diatonic – Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1
 * Superpyth chromatic – Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1
-* Superpyth hyperchromatic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1
+* Superpyth enharmonic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1
 * Augene[6] [[3L 3s]] (period = 9\27, gen = 2\27): 7 2 7 2 7 2
 * Augene[9] [[3L 6s]] (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2
@@ Line 1,197: / Line 1,188: @@
 * [[SNS (2/1, 3/2, 5/4)-7|5-limit / pental double harmonic major]]: 2 7 2 5 2 7 2
 * enharmonic tetrachord octave species: 9 1 1 5 9 1 1, 1 9 1 5 1 9 1 (also Superpyth double harmonic major), 1 1 9 5 1 1 9
-* [[The Pinetone System#The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4
+* [[The Pinetone System #The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4
-* [[The Pinetone System#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4
+* [[The Pinetone System #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4
-* [[The Pinetone System#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3
+* [[The Pinetone System #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3
-* [[The Pinetone System#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4
+* [[The Pinetone System #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4
-* [[The Pinetone System#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4
+* [[The Pinetone System #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4
-* [[The Pinetone System#Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3
+* [[The Pinetone System #Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3
 * 5-limit / pental double harmonic nonatonic (subset of Augene[12]): 2 5 2 2 5 2 5 2 2, 2 2 5 2 5 2 2 5 2 (Augene[9] [[4M]])
 * 5-limit / pental double harmonic decatonic (subset of Augene[12]): 2 5 2 2 3 2 2 5 2 2
@@ Line 1,208: / Line 1,199: @@
 * [[Blackdye]] / [[syntonic dipentatonic]] (superset of [[Zarlino]]): 1 4 2 4 1 4 2 4 1 4
 * [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of [[Zarlino]]): 3 2 4 2 3 2 4 2 3 2
+* Direct sunlight (original/default tuning; subset of [[Sensi]][19]): 1 2 8 5 1 9 1 ((1, 3, 11, 16, 17, 26, 27)\27)
-Direct sunlight (this is its original/default tuning; subset of [[Sensi]][19])
+* Hypersakura (original/default tuning; subset of Sensi[19]): 1 10 5 1 10 ((1 11 16 17 27)\27)
-* 44.444
-* 133.333
-* 488.889
-* 711.111
-* 755.555
-* 1155.555
-* 1200.000
-Hypersakura (this is its original/default tuning; subset of Sensi[19])
-* 44.444
-* 488.889
-* 711.111
-* 755.555
-* 1200.000
 == Instruments ==
@@ Line 1,257: / Line 1,234: @@
 ; [[Bryan Deister]]
 * [https://www.youtube.com/watch?v=hDP8cfJqWOI ''microtonal improvisation in 27edo''] (2023)
+; [[Francium]]
+* [https://www.youtube.com/watch?v=3Ty3FpmAdGA ''Happy Birthday in 27edo''] (2025)
 ; [[Igliashon Jones]]
@@ Line 1,298: / Line 1,278: @@
 ; [[Xotla]]
 * "Funkrotonal" from ''Microtonal Allsorts'' (2023) – [https://open.spotify.com/track/1zjNkbm8kIkuCxtodyFCL0 Spotify] | [https://xotla.bandcamp.com/track/funkrotonal-27edo Bandcamp] | [https://www.youtube.com/watch?v=7gt1BBJuJsE YouTube]
-== Notes ==
-<references group="note" />
 [[Category:Augene]]