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{{Infobox ET}}
{{Infobox ET}}
 
{{ED intro}}
'''27 equal divisions of the octave''' ('''27edo'''), or '''27(-tone) equal temperament''' ('''27tet''', '''27et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 27 [[equal]]ly large steps. Each step represents a frequency ratio of the 27th root of 2, or 44.4 [[cent]]s.


== Theory ==
== Theory ==
Assuming pure octaves, 27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. Its fifth and harmonic seventh are both sharp by 9{{c}}, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7{{c}}. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from [[superpyth|1/3-septimal-comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3-comma meantone|1/3-syntonic-comma meantone]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].


If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] sharply. The optimal step size for octave-shrinking in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614-edo, which corresponds to a step size of 44.3023 cents.
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]] [[tonality 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-11's, no-17's 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|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 triads, 3:5:7 and 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.


Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13+2/3 cents. The result is that [[6/5]], [[7/5]] and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from 1/3 [[septimal comma]] superpyth in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], resulting in three of them reaching a near perfect minor third/major sixth in both.
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.


27edo, with its 400 cent major third, tempers out the [[diesis]] of [[128/125]], and also the septimal comma, [[64/63]] (and hence [[126/125]] also). These it shares with 12edo, making some relationships familiar, and as a consequence they both support 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 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 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.


Though the [[7-limit]] tuning of 27edo 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 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; the major seventh 25\27 is less than a cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making it 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.
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.


Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant, 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.
=== Odd harmonics ===
{{Harmonics in equal|27}}


The 27-note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 edosteps, just one edostep 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 this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary 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 Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.
=== 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]], [[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.


=== Odd harmonics ===
=== Subsets and supersets ===
{{Harmonics in equal|27}}
Since 27 factors into primes as 3<sup>3</sup>, 27edo contains [[3edo]] and [[9edo]] as subsets.  Multiplying it by 3 gives [[81edo]], which is a good [[meantone]] tuning.


== Intervals ==
== Intervals ==
Line 25: Line 27:
! #
! #
! Cents
! Cents
! Approximate Ratios*
! Approximate ratios<ref group="note">{{sg|27et|limit=2.3.5.7.13.19-[[subgroup]]}}</ref>
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and vm2)
! colspan="2" | [[6L 1s]] notation
! [[Interval region]]s
! Solfege
! colspan="2" | [[Solfege]]s
|-
|-
| 0
| 0
| 0.00
| 0.0
| [[1/1]]
| [[1/1]]
| P1
| P1
| perfect unison
| perfect unison
| D
| D
| perfect unison
| unison
| C
| da
| do
| do
|-
|-
| 1
| 1
| 44.44
| 44.4
| [[28/27]], [[36/35]], [[39/38]], [[49/48]], [[50/49]], [[81/80]]
| [[28/27]], [[36/35]], [[39/38]], [[49/48]], [[50/49]], ''[[81/80]]''
| ^1, m2
| ^1, m2
| up unison, minor 2nd
| up unison, minor 2nd
| ^D, Eb
| ^D, Eb
| aug 1sn, double-dim 2nd
| diesis
| C#, Dbbb
| fra
| di
| di
|-
|-
| 2
| 2
| 88.89
| 88.9
| [[16/15]], [[21/20]], [[25/24]], [[19/18]], [[20/19]]
| ''[[16/15]]'', [[21/20]], [[25/24]], [[19/18]], [[20/19]]
| ^^1, ^m2
| ^^1, ^m2
| dup unison, upminor 2nd
| dup unison, upminor 2nd
| ^^D, ^Eb
| ^^D, ^Eb
| double-aug 1sn, dim 2nd
| minor second
| Cx, Dbb
| fru
| ra
| ra
|-
|-
| 3
| 3
| 133.33
| 133.3
| [[15/14]], [[14/13]], [[13/12]]
| [[15/14]], [[14/13]], [[13/12]]
| vA1, ~2
| vA1, ~2
| downaug 1sn, mid 2nd
| downaug 1sn, mid 2nd
| vD#, vvE
| vD#, vvE
| minor 2nd
| neutral second
| Db
| ri
| ru
| ru
|-
|-
| 4
| 4
| 177.78
| 177.8
| [[10/9]]
| [[10/9]]
| A1, vM2
| A1, vM2
| aug 1sn, downmajor 2nd
| aug 1sn, downmajor 2nd
| D#, vE
| D#, vE
| major 2nd
| small major second
| D
| ro
| reh
| reh
|-
|-
| 5
| 5
| 222.22
| 222.2
| [[8/7]], [[9/8]]
| [[8/7]], [[9/8]]
| M2
| M2
| major 2nd
| major 2nd
| E
| E
| aug 2nd, double-dim 3rd
| large major second
| D#, Ebbb
| ra
| re
| re
|-
|-
| 6
| 6
| 266.67
| 266.7
| [[7/6]]
| [[7/6]]
| m3
| m3
| minor 3rd
| minor 3rd
| F
| F
| double-aug 2nd, dim 3rd
| subminor third
| Dx, Ebb
| na
| ma
| ma
|-
|-
| 7
| 7
| 311.11
| 311.1
| [[6/5]], [[19/16]]
| [[6/5]], [[19/16]]
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| Gb
| Gb
| minor 3rd
| minor third
| Eb
| nu
| me
| me
|-
|-
| 8
| 8
| 355.56
| 355.6
| [[16/13]]
| [[16/13]]
| ~3
| ~3
| mid 3rd
| mid 3rd
|^Gb
| ^Gb
| major 3rd
| neutral third
| E
| mi
| mu
| mu
|-
|-
| 9
| 9
| 400.00
| 400.0
| [[5/4]], [[24/19]]
| [[5/4]], [[24/19]]
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| aug 3rd, double-dim 4th
| major third
| E#, Fbbb
| mo
| mi
| mi
|-
|-
| 10
| 10
| 444.44
| 444.4
| [[9/7]], [[13/10]]
| [[9/7]], [[13/10]]
| M3
| M3
| major 3rd
| major 3rd
| F#
| F#
| double-aug 3rd, dim 4th
| supermajor third
| Ex, Fbb
| ma
| mo
| mo
|-
|-
| 11
| 11
| 488.89
| 488.9
| [[4/3]]
| [[4/3]]
| P4
| P4
| perfect 4th
| perfect 4th
| G
| G
| minor 4th
| fourth
| Ex#, Fb
| fa
| fa
| fa
|-
|-
| 12
| 12
| 533.33
| 533.3
| [[27/20]], [[48/35]], [[19/14]], [[26/19]]
| [[19/14]], [[26/19]], [[27/20]], [[48/35]]
| ^4
| ^4
| up 4th
| up 4th
| Ab
| Ab
| major 4th
| superfourth
| F
| fu/sha
| fih
| fih
|-
|-
| 13
| 13
| 577.78
| 577.8
| [[7/5]], [[18/13]]
| [[7/5]], [[18/13]]
| ~4, ^d5
| ~4, ^d5
| mid 4th, updim 5th
| mid 4th, updim 5th
| ^^G, ^Ab
| ^^G, ^Ab
| aug 4th, double-dim 5th
| small tritone
| F#, Gbbb
| fi/shu
| fi
| fi
|-
|-
| 14
| 14
| 622.22
| 622.2
| [[10/7]], [[13/9]]
| [[10/7]], [[13/9]]
| vA4, ~5
| vA4, ~5
| downaug 4th, mid 5th
| downaug 4th, mid 5th
| vG#, vvA
| vG#, vvA
| double-aug 4th, dim 5th
| large tritone
| Fx, Gbb
| po/si
| se
| se
|-
|-
| 15
| 15
| 666.67
| 666.7
| [[40/27]], [[35/24]], [[19/13]], [[28/19]]
| [[19/13]], [[28/19]], [[35/24]], [[40/27]]
| v5
| v5
| down fifth
| down fifth
| G#
| G#
| minor 5th
| subfifth
| Fx#, Gb
| pa/so
| sih
| sih
|-
|-
| 16
| 16
| 711.11
| 711.1
| [[3/2]]
| [[3/2]]
| P5
| P5
| perfect 5th
| perfect 5th
| A
| A
| major 5th
| fifth
| G
| sa
| so/sol
| so/sol
|-
|-
| 17
| 17
| 755.56
| 755.6
| [[14/9]], [[20/13]]
| [[14/9]], [[20/13]]
| m6
| m6
| minor 6th
| minor 6th
| Bb
| Bb
| aug 5th, double-dim 6th
| subminor sixth
| G#, Abbb
| fla
| lo
| lo
|-
|-
| 18
| 18
| 800.00
| 800.0
| [[8/5]], [[19/12]]
| [[8/5]], [[19/12]]
| ^m6
| ^m6
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| double-aug 5th, dim 6th
| minor sixth
| Gx, Abb
| flu
| le
| le
|-
|-
| 19
| 19
| 844.44
| 844.4
| [[13/8]]
| [[13/8]]
| ~6
| ~6
| mid 6th
| mid 6th
| vA#
| vA#
| minor 6th
| neutral sixth
| Ab
| li
| lu
| lu
|-
|-
| 20
| 20
| 888.89
| 888.9
| [[5/3]], [[32/19]]
| [[5/3]], [[32/19]]
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| A#
| A#
| major 6th
| major sixth
| A
| lo
| la
| la
|-
|-
| 21
| 21
| 933.33
| 933.3
| [[12/7]]
| [[12/7]]
| M6
| M6
| major 6th
| major 6th
| B
| B
| aug 6th, double-dim 7th
| supermajor sixth
| A#, Bbbb
| la
| li
| li
|-
|-
| 22
| 22
| 977.78
| 977.8
| [[7/4]], [[16/9]]
| [[7/4]], [[16/9]]
| m7
| m7
| minor 7th
| minor 7th
| C
| C
| double-aug 6th, dim 7th
| harmonic seventh
| Ax, Bbb
| tha
| ta
| ta
|-
|-
| 23
| 23
| 1022.22
| 1022.2
| [[9/5]]
| [[9/5]]
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| Db
| Db
| minor 7th
| large minor seventh
| Bb
| thu
| te
| te
|-
|-
| 24
| 24
| 1066.67
| 1066.7
| [[28/15]], [[13/7]], [[24/13]]
| [[13/7]], [[24/13]], [[28/15]]
| ~7
| ~7
| mid 7th
| mid 7th
| ^Db
| ^Db
| major 7th
| neutral seventh
| B
| ti
| tu
| tu
|-
|-
| 25
| 25
| 1111.11
| 1111.1
| [[15/8]], [[40/21]], [[48/25]], [[19/10]], [[36/19]]
| ''[[15/8]]'', [[19/10]], [[36/19]], [[40/21]], [[48/25]]
| vM7
| vM7
| downmajor 7th
| downmajor 7th
| vC#
| vC#
| aug 7th, double-dim 8ve
| major seventh
| B#, Cbb
| to
| ti
| ti
|-
|-
| 26
| 26
| 1155.56
| 1155.6
| [[27/14]], [[35/18]], [[96/49]], [[49/25]], [[160/81]]
| [[27/14]], [[35/18]], [[49/25]], [[96/49]], ''[[160/81]]''
| M7
| M7
| major 7th
| major 7th
| C#
| C#
| double-aug 7th, dim 8ve
| supermajor seventh
| Bx, Cb
| ta
| da
| da
|-
|-
| 27
| 27
| 1200.00
| 1200.0
| 2/1
| [[2/1]]
| P8
| P8
| 8ve
| 8ve
| D
| D
| 8ve
| octave
| C
| da
| do
| do
|}
|}
<nowiki/>* based on treating 27edo as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.
<references group="note" />


=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! quality
! Quality
! [[color name]]
! [[Color name]]
! monzo format
! Monzo format
! examples
! Examples
|-
|-
| rowspan="2" | minor
| rowspan="2" | minor
| zo
| zo
| {a, b, 0, 1}
| {{monzo| a, b, 0, 1 }}
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| fourthward wa
| fourthward wa
| {a, b}, b &lt; -1
| {{monzo| a, b }}, {{nowrap|b &lt; −1}}
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
| gu
| gu
| {a, b, -1}
| {{monzo| a, b, −1 }}
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| rowspan="2" | mid
| rowspan="2" | mid
| tho
| tho
| {a, b, 0, 0, 0, 1}
| {{monzo| a, b, 0, 0, 0, 1 }}
| 13/12, 13/8
| 13/12, 13/8
|-
|-
| thu
| thu
| {a, b, 0, 0, 0, -1}
| {{monzo| a, b, 0, 0, 0, −1 }}
| 16/13, 24/13
| 16/13, 24/13
|-
|-
| downmajor
| downmajor
| yo
| yo
| {a, b, 1}
| {{monzo| a, b, 1 }}
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| rowspan="2" | major
| rowspan="2" | major
| fifthward wa
| fifthward wa
| {a, b}, b &gt; 1
| {{monzo| a, b }}, {{nowrap|b &gt; 1}}
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| ru
| ru
| {a, b, 0, -1}
| {{monzo| a, b, 0, −1 }}
| 9/7, 12/7
| 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:
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:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! [[Color notation|color of the 3rd]]
! [[Color notation|Color of the 3rd]]
! JI chord
! JI chord
! notes as edosteps
! Notes as edosteps
! notes of C chord
! Notes of C chord
! written name
! Written name
! spoken name
! Spoken name
|-
|-
| zo
| zo
| 6:7:9
| 6:7:9
| 0-6-16
| 0–6–16
| C Eb G
| C–E♭–G
| Cm
| Cm
| C minor
| C minor
Line 378: Line 382:
| gu
| gu
| 10:12:15
| 10:12:15
| 0-7-16
| 0–7–16
| C ^Eb G
| C–F♭–G, C–E{{flatup}}–G
| C^m
| C^m
| C upminor
| C upminor
Line 385: Line 389:
| ilo
| ilo
| 18:22:27
| 18:22:27
| 0-8-16
| 0–8–16
| C vvE G
| C–E{{demiflat2}}–G
| C~
| C~
| C mid
| C mid
Line 392: Line 396:
| yo
| yo
| 4:5:6
| 4:5:6
| 0-9-16
| 0–9–16
| C vE G
| C–D♯–G, C–E{{naturaldown}}–G
| Cv
| Cv
| C downmajor or C down
| C downmajor or C down
Line 399: Line 403:
| ru
| ru
| 14:18:21
| 14:18:21
| 0-10-16
| 0–10–16
| C E G
| C–E–G
| C
| C
| C major or C
| C major or C
|}
|}
For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. See also the [[22edo]] page.
For a more complete list, see [[Ups and downs notation #Chords and chord progressions]]. See also the [[22edo]] page.


== JI approximation ==
== Notation ==
{| class="wikitable center-all floatright"
|+ style="font-size: 105%;" | Circle of fifths in 27edo
|- style="white-space: nowrap;"
!Cents
! colspan="2" | Extended<br />Pythagorean<br />notation
! colspan="2" | Quartertone<br />notation
|-
| 0.0
| colspan="2" | C
| colspan="2" | A{{sesquisharp2}}
|-
| 711.1
| colspan="2" | G
| colspan="2" | E{{sesquisharp2}}
|-
| 222.2
| colspan="2" | D
| B{{sesquisharp2}}
| F{{sesquiflat2}}
|-
| 933.3
| colspan="2" | A
| colspan="2" | C{{sesquiflat2}}
|-
| 444.4
| colspan="2" | E
| colspan="2" | G{{sesquiflat2}}
|-
| 1155.6
| colspan="2" | B
| colspan="2" | D{{sesquiflat2}}
|-
| 666.7
| colspan="2" | F♯
| colspan="2" | A{{sesquiflat2}}
|-
| 177.8
| colspan="2" | C♯
| colspan="2" | E{{sesquiflat2}}
|-
| 888.9
| colspan="2" | G♯
| colspan="2" | B{{sesquiflat2}}
|-
| 400.0
| colspan="2" | D♯
| colspan="2" | F{{demiflat2}}
|-
| 1111.1
| colspan="2" | A♯
| colspan="2" | C{{demiflat2}}
|-
| 622.2
| colspan="2" | E♯
| colspan="2" | G{{demiflat2}}
|-
| 133.3
| B♯
| F𝄫
| colspan="2" | D{{demiflat2}}
|-
| 844.4
| F𝄪
| C𝄫
| colspan="2" | A{{demiflat2}}
|-
| 355.6
| C𝄪
| G𝄫
| colspan="2" | E{{demiflat2}}
|-
| 1066.7
| G𝄪
| D𝄫
| colspan="2" | B{{demiflat2}}
|-
| 577.8
| D𝄪
| A𝄫
| colspan="2" | F{{demisharp2}}
|-
| 88.9
| A𝄪
| E𝄫
| colspan="2" | C{{demisharp2}}
|-
| 800.0
| E𝄪
| B𝄫
| colspan="2" | G{{demisharp2}}
|-
| 311.1
| B𝄪
| F♭
| colspan="2" | D{{demisharp2}}
|-
| 1022.2
| colspan="2" | C♭
| colspan="2" | A{{demisharp2}}
|-
| 533.3
| colspan="2" | G♭
| colspan="2" | E{{demisharp2}}
|-
| 44.4
| colspan="2" | D♭
| colspan="2" | B{{demisharp2}}
|-
| 755.6
| colspan="2" | A♭
| colspan="2" | F{{sesquisharp2}}
|-
| 266.7
| colspan="2" | E♭
| colspan="2" | C{{sesquisharp2}}
|-
| 977.8
| colspan="2" | B♭
| colspan="2" | G{{sesquisharp2}}
|-
| 488.9
| colspan="2" | F
| colspan="2" | D{{sesquisharp2}}
|-
| 0.0
| colspan="2" | C
| colspan="2" | A{{sesquisharp2}}
|}
 
=== Extended Pythagorean notation ===
27edo 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 5/4 above C must be notated as D♯. 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 6/5 above D must be notated as G♭. The diminished 2nd is a descending interval, thus A♯ is higher than B♭. Though here very exaggerated, this should be familiar to those working with the Pythagorean scale (see [[53edo]]), and also to many classically trained violinists.
 
=== Quartertone notation ===
Using standard [[chain-of-fifths notation]], a sharp (an augmented unison) raises a note by 4 edosteps, just one edostep beneath the following nominal, and the flat conversely lowers. The sharp is quite wide at about 178¢, sounding like a narrow major 2nd. C to C♯ describes the approximate 10/9 and 11/10 interval. 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 half-sharp is notated as a quartertone, but at about 89¢ it sounds more like a narrow semitone. The gamut from C to D is C, D♭, C{{demisharp2}}, D{{demiflat2}}, C♯, and D, with many ascending intervals appearing to be descending on the staff.
 
===Ups and downs notation===
27edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}
[[Alternative symbols for ups and downs notation|Alternatively,]] sharps and flats with arrows can be used, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}
 
=== Sagittal notation ===
This notation is a subset of the notation for [[54edo #Sagittal notation|54edo]].
 
==== Evo and Revo flavors ====
<imagemap>
File:27-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 487 0 647 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 270 106 [[8505/8192]]
rect 270 80 380 106 [[27/26]]
default [[File:27-EDO_Sagittal.svg]]
</imagemap>
 
==== Alternative Evo flavor ====
<imagemap>
File:27-EDO_Alternative_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 511 0 671 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 270 106 [[8505/8192]]
rect 270 80 380 106 [[27/26]]
default [[File:27-EDO_Alternative_Evo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:27-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 487 0 647 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 270 106 [[8505/8192]]
rect 270 80 380 106 [[27/26]]
default [[File:27-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.
 
=== 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 ♭.


=== 15-odd-limit interval mappings ===
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
The following table shows how [[15-odd-limit intervals]] are represented in 27edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
|-
! Interval, complement
! &#35;
! Error (abs, ¢)
! Cents
! Error (rel, %)
! Note
! Name
! Associated ratio
|-
|-
| [[7/6]], [[12/7]]
| 0
| 0.204
| 0.0
| 0.5
| C
| perfect unison
| [[1/1]]
|-
|-
| ''[[15/11]], [[22/15]]''
| 1
| ''3.617''
| 44.4
| ''8.1''
| C#, Dbbb
| aug 1sn, triple-dim 2nd
| [[40/39]], [[45/44]], [[55/54]], [[81/80]]
|-
|-
| '''[[13/8]], [[16/13]]'''
| 2
| '''3.917'''
| 88.9
| '''8.8'''
| Cx, Dbb
| double-aug 1sn, double-dim 2nd
| [[16/15]], [[25/24]]
|-
|-
| [[5/3]], [[6/5]]
| 3
| 4.530
| 133.3
| 10.2
| Db
| dim 2nd
| [[12/11]], [[13/12]]
|-
|-
| [[9/5]], [[10/9]]
| 4
| 4.626
| 177.8
| 10.4
| D
| perfect 2nd
| [[10/9]], [[11/10]]
|-
|-
| [[7/5]], [[10/7]]
| 5
| 4.734
| 222.2
| 10.7
| D#, Ebbb
| aug 2nd, double-dim 3rd
| [[9/8]]
|-
|-
| [[13/7]], [[14/13]]
| 6
| 5.035
| 266.7
| 11.3
| Dx, Ebb
| double-aug 2nd, dim 3rd
| [[15/13]]
|-
|-
| [[13/12]], [[24/13]]
| 7
| 5.239
| 311.1
| 11.8
| Eb
| minor 3rd
| [[6/5]]
|-
|-
| ''[[11/9]], [[18/11]]''
| 8
| ''8.148''
| 355.6
| ''18.3''
| E
| major 3rd
| [[11/9]], [[16/13]]
|-
|-
| '''[[7/4]], [[8/7]]'''
| 9
| '''8.952'''
| 400.0
| '''20.1'''
| E#, Fbbb
| aug 3rd, double-dim 4th
| [[5/4]]
|-
|-
| '''[[3/2]], [[4/3]]'''
| 10
| '''9.156'''
| 444.4
| '''20.6'''
| Ex, Fbb
| double-aug 3rd, dim 4th
| [[13/10]]
|-
|-
| [[9/7]], [[14/9]]
| 11
| 9.360
| 488.9
| 21.1
| Ex#, Fb
| minor 4th
| [[4/3]]
|-
|-
| [[13/10]], [[20/13]]
| 12
| 9.770
| 533.3
| 22.0
| F
| major 4th
| [[15/11]], [[27/20]]
|-
|-
| ''[[11/10]], [[20/11]]''
| 13
| ''12.774''
| 577.8
| ''28.7''
| F#, Gbbb
| aug 4th, double-dim 5th
| [[11/8]], [[18/13]]
|-
|-
| '''[[5/4]], [[8/5]]'''
| 14
| '''13.686'''
| 622.2
| '''30.8'''
| Fx, Gbb
| double-aug 4th, dim 5th
| [[13/9]], [[16/11]]
|-
|-
| [[15/14]], [[28/15]]
| 15
| 13.891
| 666.7
| 31.3
| Fx#, Gb
| minor 5th
| [[22/15]], [[40/27]]
|-
|-
| [[13/9]], [[18/13]]
| 16
| 14.395
| 711.1
| 32.4
| G
| major 5th
| [[3/2]]
|-
|-
| ''[[11/6]], [[12/11]]''
| 17
| ''17.304''
| 755.6
| ''38.9''
| G#, Abbb
| aug 5th, double-dim 6th
| [[20/13]]
|-
|-
| ''[[11/7]], [[14/11]]''
| 18
| ''17.508''
| 800.0
| ''39.4''
| Gx, Abb
| double-aug 5th, dim 6th
| [[8/5]]
|-
|-
| '''[[11/8]], [[16/11]]'''
| 19
| '''17.985'''
| 844.4
| '''40.5'''
| Ab
| minor 6th
| [[13/8]], [[18/11]]
|-
|-
| [[9/8]], [[16/9]]
| 20
| 18.312
| 888.9
| 41.2
| A
| major 6th
| [[5/3]]
|-
|-
| [[15/13]], [[26/15]]
| 21
| 18.926
| 933.3
| 42.6
| A#, Bbbb
| aug 6th, double-dim 7th
| [[26/15]]
|-
|-
| ''[[15/8]], [[16/15]]''
| 22
| ''21.602''
| 977.8
| ''48.6''
| Ax, Bbb
| double-aug 6th, dim 7th
| [[16/9]]
|-
| 23
| 1022.2
| Bb
| perfect 7th
| [[9/5]], [[20/11]]
|-
| 24
| 1066.7
| B
| aug 7th
| [[11/6]], [[24/13]]
|-
|-
| [[13/11]], [[22/13]]
| 25
| 21.901
| 1111.1
| 49.3
| B#, Cbb
| double-aug 7th, double-dim 8ve
| [[15/8]], [[48/25]]
|-
| 26
| 1155.6
| Bx, Cb
| triple-aug 7th, dim 8ve
| [[39/20]], [[88/45]], [[108/55]], [[160/81]]
|-
| 27
| 1200.0
| C
| 8ve
| 2/1
|}
|}
{{15-odd-limit|27}}
{{clear}}
 
== Approximation to JI ==
[[File:27ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 27edo]]
 
=== Interval mappings ===
{{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}}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 528: Line 795:
| 2.3
| 2.3
| {{monzo| 43 -27 }}
| {{monzo| 43 -27 }}
| [{{val| 27 43 }}]
| {{mapping| 27 43 }}
| -2.89
| −2.89
| 2.88
| 2.88
| 6.50
| 6.50
Line 535: Line 802:
| 2.3.5
| 2.3.5
| 128/125, 20000/19683
| 128/125, 20000/19683
| [{{val| 27 43 63 }}]
| {{mapping| 27 43 63 }}
| -3.88
| −3.88
| 2.74
| 2.74
| 6.19
| 6.19
Line 542: Line 809:
| 2.3.5.7
| 2.3.5.7
| 64/63, 126/125, 245/243
| 64/63, 126/125, 245/243
| [{{val| 27 43 63 76 }}]
| {{mapping| 27 43 63 76 }}
| -3.70
| −3.71
| 2.39
| 2.39
| 5.40
| 5.40
Line 549: Line 816:
| 2.3.5.7.13
| 2.3.5.7.13
| 64/63, 91/90, 126/125, 169/168
| 64/63, 91/90, 126/125, 169/168
| [{{val| 27 43 63 76 100 }}]
| {{mapping| 27 43 63 76 100 }}
| -3.18
| −3.18
| 2.39
| 2.39
| 5.39
| 5.39
Line 556: Line 823:
| 2.3.5.7.13.19
| 2.3.5.7.13.19
| 64/63, 76/75, 91/90, 126/125, 169/168
| 64/63, 76/75, 91/90, 126/125, 169/168
| [{{val| 27 43 63 76 100 115 }}]
| {{mapping| 27 43 63 76 100 115 }}
| -3.18
| −3.18
| 2.18
| 2.18
| 4.92
| 4.92
|}
|}
* 27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next equal temperaments doing better in those subgroups are [[31edo|31]], 31, and [[46edo|46]], respectively.
* 27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next equal temperament that does better in this subgroup is [[53edo|53]].


27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next equal temperaments doing better in those subgroups are [[31edo|31]], 31, and [[46edo|46]], respectively.
=== Uniform maps ===
 
{{Uniform map|edo=27}}
27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next equal temperament that does better in this subgroup is [[53edo|53]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 575: Line 843:
! Generator
! Generator
! Temperaments
! Temperaments
! MOS Scales
! Mos scales
|-
|-
| 1
| 1
| 1\27
| 1\27
| [[Quartonic]]/quarto
| [[Quartonic]] / quarto (27e) / quartz (27)
|
|
|-
|-
| 1
| 1
| 2\27
| 2\27
| [[Octacot]]/octocat
| [[Octacot]] / octocat (27e)
| [[1L_12s]], [[13L_1s]]
| [[1L 12s]], [[13L 1s]]
|-
|-
| 1
| 1
| 4\27
| 4\27
| [[Tetracot]]/modus/wollemia
| [[Tetracot]] (27e) / modus (27e) / wollemia (27e)
| [[1L_5s]], [[6L_1s]], [[7L_6s]], [[7L_13s]]
| [[1L 5s]], [[6L 1s]], [[7L 6s]], [[7L 13s]]
|-
|-
| 1
| 1
| 5\27
| 5\27
| [[Machine]]/kumonga
| [[Machine]] (27)<br>[[Kumonga]] (27e)
| [[1L_4s]], [[5L_1s]], [[5L_6s]], [[11L_5s]]
| [[1L 4s]], [[5L 1s]], [[5L 6s]], [[11L 5s]]
|-
|-
| 1
| 1
| 7\27
| 7\27
| [[Myna]]/coleto/minah/[[oolong]]
| [[Myna]] (27e) / coleto (27e) / myno (27)<br>[[Oolong]] (27e)
| [[4L_3s]], [[4L_7s]], [[4L_11s]], [[4L_15s]], [[4L_19s]]
| [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]]
|-
|-
| 1
| 1
| 8\27
| 8\27
| [[Beatles]]/ringo
| [[Beatles]] (27e) / ringo (27e) / beetle (27)
| [[3L_4s]], [[7L_3s]], [[10L_7s]]
| [[3L 4s]], [[7L 3s]], [[10L 7s]]
|-
|-
| 1
| 1
| 10\27
| 10\27
| [[Sensi]]/sensis
| [[Sensi]]
| [[3L_2s]], [[3L_5s]], [[8L_3s]], [[8L_11s]]
| [[3L 2s]], [[3L 5s]], [[8L 3s]], [[8L 11s]]
|-
|-
| 1
| 1
| 11\27
| 11\27
| [[Superpyth]]
| [[Superpyth]] (27e)
| [[5L_2s]], [[5L_7s]], [[5L_12s]], [[5L_17s]]
| [[5L 2s]], [[5L 7s]], [[5L 12s]], [[5L 17s]]
|-
|-
| 1
| 1
| 13\27
| 13\27
| Fervor
| [[Fervor]] (27e)
| [[2L_3s]], [[2L_5s]], [[2L_7s]], [[2L_9s]], [[2L_11s]], etc ... [[2L_23s]]
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], etc. … [[2L 23s]]
|-
|-
| 3
| 3
| 1\27
| 1\27
| [[Semiaug]]/hemiaug
| [[Hemiaug]] (27e)
|
|
|-
|-
| 3
| 3
| 2\27
| 2\27
| [[Augmented]]/[[Augene]]/ogene
| [[Augene]] (27e) / Eugene (27)
| [[3L_3s]], [[3L_6s]], [[3L_9s]], [[12L_3s]]
| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]
|-
|-
| 3
| 3
| 4\27
| 4\27
| [[Oodako]]/[[terrain]]
| [[Oodako]] (27e)<br>[[Terrain]]
| [[3L_3s]], [[6L_3s]], [[6L_9s]], [[6L_15s]]
| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15s]]
|-
|-
| 9
| 9
| 1\27
| 1\27
| Terrible version of [[Ennealimmal]]<br>/niner
| [[Niner]] (27e)<br>[[Ennealimmal]] (out of tune)
| [[9L_9s]]
| [[9L 9s]]
|}
|}


=== Commas ===
=== Commas ===
27edo [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 27 43 63 76 93 100 }}.)
27et [[tempering out|tempers out]] the following [[commas]]. (Note: This assumes the patent [[val]], {{val| 27 43 63 76 93 100 }}.)


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic Limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
! [[Color name]]
! [[Color name]]
! Name(s)
! Name
|-
|-
| 5
| 5
Line 660: Line 928:
| 41.06
| 41.06
| Trigu
| Trigu
| Diesis, augmented comma
| Augmented comma, lesser diesis
|-
|-
| 5
| 5
Line 667: Line 935:
| 27.66
| 27.66
| Saquadyo
| Saquadyo
| Minimal diesis, Tetracot comma
| Tetracot comma, minimal diesis
|-
|-
| 5
| 5
Line 674: Line 942:
| 13.40
| 13.40
| Sepgu
| Sepgu
| Medium semicomma, Sensipent comma
| Sensipent comma
|-
|-
| 5
| 5
Line 695: Line 963:
| 27.26
| 27.26
| Ru
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
| Septimal comma
|-
|-
| 7
| 7
Line 702: Line 970:
| 14.52
| 14.52
| Quinzogu
| Quinzogu
| Trimyna
| Trimyna comma
|-
|-
| 7
| 7
Line 709: Line 977:
| 14.19
| 14.19
| Zozoyo
| Zozoyo
| Sensamagic
| Sensamagic comma
|-
|-
| 7
| 7
Line 716: Line 984:
| 13.79
| 13.79
| Zotrigu
| Zotrigu
| Septimal semicomma, Starling comma
| Starling comma
|-
|-
| 7
| 7
Line 723: Line 991:
| 13.47
| 13.47
| Rurutriyo
| Rurutriyo
| Octagar
| Octagar comma
|-
|-
| 7
| 7
Line 730: Line 998:
| 13.07
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma, Orwell comma
| Orwellisma
|-
| 7
| <abbr title="40353607/40310784">(16 digits)</abbr>
| {{monzo| -11 -9 0 9 }}
| 1.84
| Tritrizo
| [[Septimal ennealimma]]
|-
|-
| 7
| 7
Line 759: Line 1,034:
| Trizogugu
| Trizogugu
| [[Landscape comma]]
| [[Landscape comma]]
|-
| 11
| [[55/54]]
| {{monzo| -1 -3 1 0 1 }}
| 31.77
| Loyo
| Telepathma
|-
|-
| 11
| 11
Line 772: Line 1,054:
| 9.69
| 9.69
| Saluzo
| Saluzo
| Pentacircle
| Pentacircle comma
|-
|-
| 11
| 11
Line 780: Line 1,062:
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|-
| 13
| [[66/65]]
| {{monzo| 1 1 -1 0 1 -1 }}
| 26.43
| Thulogu
| Winmeanma
|-
|-
| 13
| 13
Line 786: Line 1,075:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
| 13
| [[512/507]]
| {{monzo| 9 -1 0 0 0 -2 }}
| 16.99
| Thuthu
| Tridecimal neutral thirds comma
|-
| 13
| [[325/324]]
| {{monzo| -2 -4 2 0 0 1 }}
| 5.34
| Thoyoyo
| Marveltwin comma
|-
| 13
| [[351/350]]
| {{monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Thorugugu
| Ratwolfsma
|-
| 13
| [[31213/31104]]
| {{monzo| -7 -5 0 4 0 1 }}
| 6.06
| Thoquadzo
| Praveensma
|-
| 17
| [[85/84]]
| {{monzo| -2 -1 1 -1 0 0 1 }}
| 20.49
| Soruyo
| Monk comma
|-
| 17
| [[154/153]]
| {{monzo| 1 -2 0 1 1 0 -1 }}
| 11.28
| Sulozo
| Augustma
|-
| 19
| [[77/76]]
| {{monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.63
| Nulozo
| Small undevicesimal ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|}
|}
<references/>
<references group="note" />
 
== Scales ==
=== MOS scales ===
{{Main|List of MOS scales in 27edo}}
* 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 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
* Augene[12] [[3L 9s]] (period = 9\27, gen = 2\27): 3 2 2 2 3 2 2 2 3 2 2 2
* Augene[15] [[12L 3s]] (period = 9\27, gen = 2\27): 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2
* Beatles[7] [[3L 4s]] (gen = 8\27): 3 5 3 5 3 5 3
* Beatles[10] [[7L 3s]] (gen = 8\27): 3 3 2 3 3 2 3 3 2 3
* Beatles[17] [[10L 7s]] (gen = 8\27): 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 2 1
* Sensi[5] [[3L 2s]] (gen = 10\27): 7 3 7 3 7
* Sensi[8] [[3L 5s]] (gen = 10\27): 3 4 3 3 4 3 3 4
* Sensi[11] [[8L 3s]] (gen = 10\27): 3 3 1 3 3 3 1 3 3 3 1
* Machine[5] [[1L 4s]] (gen = 5\27): 5 5 5 5 7
* Machine[6] [[5L 1s]] (gen = 5\27): 5 5 5 5 5 2
* Machine[11] [[5L 6s]] (gen = 5\27): 2 3 2 3 2 3 2 3 2 3 2
* Machine[16] [[11L 5s]] (gen = 5\27): 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2
* Tetracot[6] [[1L 5s]] (gen = 4\27): 4 4 4 4 4 7
* Tetracot[7] [[6L 1s]] (gen = 4\27): 4 4 4 4 4 4 3
* Tetracot[13] [[7L 6s]] (gen = 4\27): 3 1 3 1 3 1 3 1 3 1 3 1 3
* Tetracot[20] [[7L 13s]] (gen = 4\27): 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1
* Octacot[13] [[1L 12s]] (gen = 2\27): 2 2 2 2 2 2 2 2 2 2 2 2 3
* Octacot[14] [[13L 1s]] (gen = 2\27): 2 2 2 2 2 2 2 2 2 2 2 2 2 1
* Myna[7] [[4L 3s]] (gen = 7\27): 6 1 6 1 6 1 6
* Myna[11] [[4L 7s]] (gen = 7\27): 5 1 1 5 1 1 5 1 1 5 1
* Myna[15] [[4L 11s]] (gen = 7\27): 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1
* Myna[19] [[4L 15s]] (gen = 7\27): 3 1 1 1 1 3 1 1 1 1 3 1 1 1 1 3 1 1 1
 
=== Other scales ===
* 5-limit / pental / [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]]: 5 4 7 4 7
* 5-limit / pental / [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]]: 7 4 5 7 4
* enharmonic trichord octave species: 9 2 5 9 2, 2 9 5 2 9
* 5-limit / pental double harmonic  hexatonic (Augmented[6] [[4M]]): 2 7 2 7 7 2, 7 7 2 2 7 2
* Superpyth melodic minor – Superpyth 2|4 #6 #7 or 5|1 b3: 5 1 5 5 5 5 1
* Superpyth harmonic minor – Superpyth 2|4 #7: 5 1 5 5 1 9 1
* Superpyth harmonic major – Superpyth 5|1 b6: 5 5 1 5 1 9 1
* Superpyth double harmonic major – Superpyth 5|1 b2 b6: 1 9 1 5 1 9 1
* [[Zarlino]] / Ptolemy diatonic, "just" major: 5 4 2 5 4 5 2
* "Just" minor (inverse of "just" major): 5 2 4 5 2 5 4
* 5-limit / pental tetrachordal major: 5 4 2 5 5 4 2
* 5-limit / pental tetrachordal minor: 5 2 4 5 5 2 4
* 5-limit / pental melodic minor: 5 2 4 5 4 5 2
* 5-limit / pental harmonic minor: 5 2 4 5 2 7 2
* 5-limit / pental harmonic major: 5 4 2 5 2 7 2
* 5-odd limit tonality diamond: 7 2 2 5 2 2 7
* 7-odd limit tonality diamond: 5 1 1 2 2 2 1 2 2 2 1 1 5
* 9-odd limit tonality diamond: 4 1 1 1 2 1 1 2 1 2 1 1 2 1 1 1 4
* [[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 #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 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 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
* 5-limit / pental double harmonic chromatic: 2 2 3 2 2 3 2 2 2 3 2 2, 2 2 3 2 2 2 3 2 2 3 2 2 (Augene[12] [[4M]])
* [[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)
* Hypersakura (original/default tuning; subset of Sensi[19]): 1 10 5 1 10 ((1 11 16 17 27)\27)
 
== Instruments ==
[[File:27edo_Guitar.jpg|200px|thumb|right|Brendan Byrnes, guitarist]]
While playing 27edo instruments requires significantly more frets or keys than 12edo, it is still possible to create physical instruments that can play all its notes. Probably the most notable of these is owned by Brendan Byrnes and played on some of his tracks listed in the music section.
 
However, the frets are very close together and playing high up the neck requires careful use of fingernails for fretting. A skip-fretted guitar would have notes only slightly closer together than 12edo and be easier to play, particularly when tuned in the configuration detailed below.
 
* [[Skip fretting system 27 2 9]]
 
27edo can also be played on the Lumatone, with various layouts discussed here.
 
* [[Lumatone mapping for 27edo]]


== Music ==
== Music ==
; [[Gene Ward Smith]]
{{Catrel| 27edo tracks }}
* [https://www.archive.org/details/MusicForYourEars ''Music For Your Ears''] [https://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] – the central portion is in 27edo, the rest in [[46edo]].
 
; [[Abnormality]]
* [https://www.youtube.com/watch?v=gfGNKd8SWWc ''Boiling''] (2024)
 
; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=Pr5E5brBGuw ''What Happens Next''] (2021)
 
; [[Beheld]]
* [https://www.youtube.com/watch?v=JH4zrwGqv6A ''Thick vibe''] (2023)
 
; [[Gregoire Blanc]]
* [https://youtu.be/a4-JhcaZSUs?feature=shared ''A microtonal teatime jam''] (2023)
 
; [[Brendan Byrnes]]
* [https://youtu.be/sWaqlAgSWcc ''Sunspots''] (2022)
 
; [[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]]
; [[Igliashon Jones]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 ''Sad Like Winter Leaves'']{{dead link}}
* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 ''Sad Like Winter Leaves''] – in Augene[12] tuned to 27edo
* [[:File:Superpythagorean_Waltz.mp3|''Superpythagorean Waltz'']]
* [[:File:Superpythagorean_Waltz.mp3|''Superpythagorean Waltz'']] (2012)
* [https://pixelarchipelago.bandcamp.com/track/stuttering-anticipation-27edo ''Stuttering Anticipation''] (2021)
* [https://pixelarchipelago.bandcamp.com/track/stuttering-anticipation-27edo ''Stuttering Anticipation''] (2021)


; [[Joel Taylor]]
; [[Peter Kosmorsky]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina'']{{dead link}}
* [https://www.youtube.com/watch?v=7QcwKlK6z4c ''miniature prelude and fugue''] (2011)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=nR8orkai8tQ ''Chorale in 27edo for Organ''] (2019)


; [[Peter Kosmorsky]]
; [[Herman Miller]]
* [https://www.youtube.com/watch?v=7QcwKlK6z4c ''miniature prelude and fugue'']
* ''[https://soundcloud.com/morphosyntax-1/nusu-laj-stille-nacht Stille Nacht (cover)]'' (2019)


; [[Chris Vaisvil]]
; [[NullPointerException Music]]
* [http://micro.soonlabel.com/27edo/daily20111202-deep-chasm-zeta-cp-1.mp3 ''Chicago Pile-1'']
* [https://www.youtube.com/watch?v=II817QeOHoQ ''Edolian - Adventure''] (2020)


; [[Dustin Schallert]]
; [[Dustin Schallert]]
Line 814: Line 1,261:
* [https://soundcloud.com/dustin-schallert/27-edo-guitar-1 ''27-EDO Guitar 1'']{{dead link}}
* [https://soundcloud.com/dustin-schallert/27-edo-guitar-1 ''27-EDO Guitar 1'']{{dead link}}


; [[Brendan Byrnes]]
; [[Gene Ward Smith]]
* [https://youtu.be/sWaqlAgSWcc ''Sunspots'']
* [https://www.archive.org/details/MusicForYourEars ''Music For Your Ears''] [https://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] – the central portion is in 27edo, the rest in [[46edo]].
 
; [[Joel Taylor]]
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] – in Augene[12] tuned to 27edo
 
; [[Tristan Bay]]
* [https://youtu.be/R30aRbNtoIY ''Pitchblende''] (2023)
 
; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=dcQe6ebpGFU ''Autumn''] (2024) – in Blackdye, 27edo tuning
 
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/27edo/daily20111202-deep-chasm-zeta-cp-1.mp3 ''Chicago Pile-1''] (2011)


== See also ==
; [[Xotla]]
* [[Lumatone mapping for 27edo]]
* "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]


[[Category:Augene]]
[[Category:Augene]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Sensi]]
[[Category:Superpyth]]
[[Category:Tetracot]]
[[Category:Twentuning]]
[[Category:Twentuning]]
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