27edo: Difference between revisions

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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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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]]

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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 17: / 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 312: / Line 312: @@
 | do
 |}
+<references group="note" />
 === Interval quality and chord names in color notation ===
@@ Line 325: / 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 592: / 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 778: / 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 1,145: / 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 1,199: / 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 1,210: / 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 1,259: / 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 1,300: / 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]]