27edo: Difference between revisions

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== Theory ==

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27edo, with its 400 cent major third, 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 allegedly Bohlen-Pierce 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.

Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes ~~quite~~ rough if extended to the 9 and above, unlike a true tritave based system.

Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|s=hair|d=med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, and a passable approximation of 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.

Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of ''a'' = 2 and ''s'' = 1%. This property is shared with all edos between around 24 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.

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=== ~~Notation~~ ===

== Notation ==

{| class="wikitable center-all floatright"

|+ style="white-space: nowrap;" | Circle of fifths in 27edo

! rowspan="2" | [[Cent]]s !! colspan="6" | Note

|-

! colspan="2" | Standard<br />notation !! colspan="2" | Quarter tone<br />notation

|-

| 0

| colspan="2" | C

| colspan="2" | A{{sesquisharp2}}

|-

| 711.11

| colspan="2" | G

| colspan="2" | E{{sesquisharp2}}

|-

| 222.22

| colspan="2" | D

| B{{sesquisharp2}} || F{{sesquiflat2}}

|-

| 933.33

| colspan="2" | A

| colspan="2" | C{{sesquiflat2}}

|-

| 444.44

| colspan="2" | E

| colspan="2" | G{{sesquiflat2}}

|-

| 1155.55

| colspan="2" | B

| colspan="2" | D{{sesquiflat2}}

|-

| 666.66

| colspan="2" | F♯

| colspan="2" | A{{sesquiflat2}}

|-

| 177.77

| colspan="2" | C♯

| colspan="2" | E{{sesquiflat2}}

|-

| 888.88

| colspan="2" | G♯

| colspan="2" | B{{sesquiflat2}}

|-

| 400

| colspan="2" | D♯

| colspan="2" | F{{demiflat2}}

|-

| 1111.11

| colspan="2" | A♯

| colspan="2" | C{{demiflat2}}

|-

| 622.22

| colspan="2" | E♯

| colspan="2" | G{{demiflat2}}

|-

| 133.33

| B♯

| F𝄫

| colspan="2" | D{{demiflat2}}

|-

| 844.44

| F𝄪

| C𝄫

| colspan="2" | A{{demiflat2}}

|-

| 355.56

| C𝄪

| G𝄫

| colspan="2" | E{{demiflat2}}

|-

| 1066.67

| G𝄪

| D𝄫

| colspan="2" | B{{demiflat2}}

|-

| 577.78

| D𝄪

| A𝄫

| colspan="2" | F{{demisharp2}}

|-

| 88.89

| A𝄪

| E𝄫

| colspan="2" | C{{demisharp2}}

|-

| 800

| E𝄪

| B𝄫

| colspan="2" | G{{demisharp2}}

|-

| 311.11

| B𝄪

| F♭

| colspan="2" | D{{demisharp2}}

|-

| 1022.22

| colspan="2" | C♭

| colspan="2" | A{{demisharp2}}

|-

| 533.33

| colspan="2" | G♭

| colspan="2" | E{{demisharp2}}

|-

| 44.44

| colspan="2" | D♭

| colspan="2" | B{{demisharp2}}

|-

| 755.56

| colspan="2" | A♭

| colspan="2" | F{{sesquisharp2}}

|-

| 266.67

| colspan="2" | E♭

| colspan="2" | C{{sesquisharp2}}

|-

| 977.78

| colspan="2" | B♭

| colspan="2" | G{{sesquisharp2}}

|-

| 488.89

| colspan="2" | F

| colspan="2" | D{{sesquisharp2}}

|-

| 0

| colspan="2" | C

| colspan="2" | A{{sesquisharp2}}

|-

|}

The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. With standard circle-of-fifths notation, a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats.The notes from C to D are C, D♭, C{{demisharp2}}, D{{demiflat2}}, C♯, and D, with some ascending intervals appearing to be descending on the staff.

~~There are eight enharmonic equivalences that do not involve microtonal accidentals:~~

* B♯ = F𝄫

The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. With standard [[circle-of-fifths notation]], a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats.The notes from C to D are C, D♭, C{{demisharp2}}, D{{demiflat2}}, C♯, and D, with some ascending intervals appearing to be descending on the staff.

* F𝄪 = C𝄫

* C𝄪 = G𝄫

* G𝄪 = D𝄫

* D𝄪 = A𝄫

* A𝄪 = E𝄫

* E𝄪 = B𝄫

* B𝄪 = F♭

Another notational implication is that, being a Superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) ~~C–E~~ is a 9/7 supermajor third and so the note located one major third above C must be notated as D♯ or E{{naturaldown}}. Conversely, the 6/5 minor third of a 10:12:15 chord is ~~technically~~ a diminished fourth, since (for example) ~~D–F~~ is a 7/6 subminor third and so the note located one minor third above D must be notated as either G♭ or F{{naturalup}}. The composer can decide for themselves which additional accidental pair is appropriate if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B♭ is not only familiar, though here very exaggerated, to those working with the Pythagorean scale (see [[53edo]]), but also to many classically trained violinists.

Another notational implication is that, being a Superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C–E is a 9/7 supermajor third and so the note located one major third above C must be notated as D♯ or E{{naturaldown}}. Conversely, the 6/5 minor third of a 10:12:15 chord is actually reached by a diminished fourth, since (for example) D–F is a 7/6 subminor third and so the note located one minor third above D must be notated as either G♭ or F{{naturalup}}. The composer can decide for themselves which additional accidental pair is appropriate if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B♭ is not only familiar, though here very exaggerated, to those working with the Pythagorean scale (see [[53edo]]), but also to many classically trained violinists.

== Intervals ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|27}}
 == Theory ==
@@ Line 9: / Line 8: @@
 edo, with its 400 cent major third, 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 allegedly Bohlen-Pierce 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.
-Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.
+Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|s=hair|d=med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, and a passable approximation of 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.
 Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of ''a'' = 2 and ''s'' = 1%. This property is shared with all edos between around 24 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.
@@ Line 16: / Line 15: @@
 {{Harmonics in equal|27}}
-=== Notation ===
+== Notation ==
+{| class="wikitable center-all floatright"
+|+ style="white-space: nowrap;" | Circle of fifths in 27edo
+! rowspan="2" | [[Cent]]s !! colspan="6" | Note
+|-
+! colspan="2" | Standard<br />notation !! colspan="2" | Quarter tone<br />notation
+|-
+| 0
+| colspan="2" | C
+| colspan="2" | A{{sesquisharp2}}
+|-
+| 711.11
+| colspan="2" | G
+| colspan="2" | E{{sesquisharp2}}
+|-
+| 222.22
+| colspan="2" | D
+| B{{sesquisharp2}} || F{{sesquiflat2}}
+|-
+| 933.33
+| colspan="2" | A
+| colspan="2" | C{{sesquiflat2}}
+|-
+| 444.44
+| colspan="2" | E
+| colspan="2" | G{{sesquiflat2}}
+|-
+| 1155.55
+| colspan="2" | B
+| colspan="2" | D{{sesquiflat2}}
+|-
+| 666.66
+| colspan="2" | F&#x266F;
+| colspan="2" | A{{sesquiflat2}}
+|-
+| 177.77
+| colspan="2" | C&#x266F;
+| colspan="2" | E{{sesquiflat2}}
+|-
+| 888.88
+| colspan="2" | G&#x266F;
+| colspan="2" | B{{sesquiflat2}}
+|-
+| 400
+| colspan="2" | D&#x266F;
+| colspan="2" | F{{demiflat2}}
+|-
+| 1111.11
+| colspan="2" | A&#x266F;
+| colspan="2" | C{{demiflat2}}
+|-
+| 622.22
+| colspan="2" | E&#x266F;
+| colspan="2" | G{{demiflat2}}
+|-
+| 133.33
+| B&#x266F;
+| F&#x1D12B;
+| colspan="2" | D{{demiflat2}}
+|-
+| 844.44
+| F&#x1D12A;
+| C&#x1D12B;
+| colspan="2" | A{{demiflat2}}
+|-
+| 355.56
+| C&#x1D12A;
+| G&#x1D12B;
+| colspan="2" | E{{demiflat2}}
+|-
+| 1066.67
+| G&#x1D12A;
+| D&#x1D12B;
+| colspan="2" | B{{demiflat2}}
+|-
+| 577.78
+| D&#x1D12A;
+| A&#x1D12B;
+| colspan="2" | F{{demisharp2}}
+|-
+| 88.89
+| A&#x1D12A;
+| E&#x1D12B;
+| colspan="2" | C{{demisharp2}}
+|-
+| 800
+| E&#x1D12A;
+| B&#x1D12B;
+| colspan="2" | G{{demisharp2}}
+|-
+| 311.11
+| B&#x1D12A;
+| F&#x266D;
+| colspan="2" | D{{demisharp2}}
+|-
+| 1022.22
+| colspan="2" | C&#x266D;
+| colspan="2" | A{{demisharp2}}
+|-
+| 533.33
+| colspan="2" | G&#x266D;
+| colspan="2" | E{{demisharp2}}
+|-
+| 44.44
+| colspan="2" | D&#x266D;
+| colspan="2" | B{{demisharp2}}
+|-
+| 755.56
+| colspan="2" | A&#x266D;
+| colspan="2" | F{{sesquisharp2}}
+|-
+| 266.67
+| colspan="2" | E&#x266D;
+| colspan="2" | C{{sesquisharp2}}
+|-
+| 977.78
+| colspan="2" | B&#x266D;
+| colspan="2" | G{{sesquisharp2}}
+|-
+| 488.89
+| colspan="2" | F
+| colspan="2" | D{{sesquisharp2}}
+|-
+| 0
+| colspan="2" | C
+| colspan="2" | A{{sesquisharp2}}
+|-
+|}
 {{sharpness-sharp4}}
-The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. With standard circle-of-fifths notation, a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats.The notes from C to D are C, D♭, C{{demisharp2}}, D{{demiflat2}}, C♯, and D, with some ascending intervals appearing to be descending on the staff.
-There are eight enharmonic equivalences that do not involve microtonal accidentals:
-* B♯ = F𝄫
+The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. With standard [[circle-of-fifths notation]], a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats.The notes from C to D are C, D&#x266D;, C{{demisharp2}}, D{{demiflat2}}, C&#x266F;, and D, with some ascending intervals appearing to be descending on the staff.
-* F𝄪 = C𝄫
-* C𝄪 = G𝄫
-* G𝄪 = D𝄫
-* D𝄪 = A𝄫
-* A𝄪 = E𝄫
-* E𝄪 = B𝄫
-* B𝄪 = F♭
-Another notational implication is that, being a Superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C–E is a 9/7 supermajor third and so the note located one major third above C must be notated as D♯ or E{{naturaldown}}. Conversely, the 6/5 minor third of a 10:12:15 chord is technically a diminished fourth, since (for example) D–F is a 7/6 subminor third and so the note located one minor third above D must be notated as either G♭ or F{{naturalup}}. The composer can decide for themselves which additional accidental pair is appropriate if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B♭ is not only familiar, though here very exaggerated, to those working with the Pythagorean scale (see [[53edo]]), but also to many classically trained violinists.
+Another notational implication is that, being a Superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C&ndash;E is a 9/7 supermajor third and so the note located one major third above C must be notated as D♯ or E{{naturaldown}}. Conversely, the 6/5 minor third of a 10:12:15 chord is actually reached by a diminished fourth, since (for example) D&ndash;F is a 7/6 subminor third and so the note located one minor third above D must be notated as either G&#x266D; or F{{naturalup}}. The composer can decide for themselves which additional accidental pair is appropriate if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B&#x266D; is not only familiar, though here very exaggerated, to those working with the Pythagorean scale (see [[53edo]]), but also to many classically trained violinists.
+{{clear}}
 == Intervals ==