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{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}


{{EDO intro|27}}
== 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]].


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]], since harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all tuned significantly sharp of just, and a step size of between 44.2 and 44.35 cents would be better in theory. 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.


However, assuming 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]], which is sharp by 13.7 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 and major sixth in both, with 19edo reaching a near-perfect [[6/5]] and 27edo reaching a near-perfect [[7/6]].
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 [[lesser diesis]] of [[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.
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 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, since a major third from C is C–D♯ rather than C–E and a minor third from C is now C–F♭ instead of C–E♭. The composer can decide for themselves 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 24: 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|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)
![[Walker Brightness Notation]]
! [[Interval region]]s
! colspan="2" | [[6L 1s]] Notation
! colspan="2" | [[Solfege]]s
! colspan="2" |[[Solfege|Solfeges]]
|-
|-
| 0
| 0
| 0.00
| 0.0
| [[1/1]]
| [[1/1]]
| P1
| P1
| perfect unison
| perfect unison
| D
| D
|unison
| unison
| perfect unison
| da
| 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
|diesis
| diesis
| aug 1sn, double-dim 2nd
| fra
| 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
|minor second
| minor second
| double-aug 1sn, dim 2nd
| fru
| 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
|neutral second
| neutral second
| minor 2nd
| ri
| 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
|small major second
| small major second
| major 2nd
| ro
| 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
|large major second
| large major second
| aug 2nd, double-dim 3rd
| ra
| D#, Ebbb
|ra
| re
| re
|-
|-
| 6
| 6
| 266.67
| 266.7
| [[7/6]]
| [[7/6]]
| m3
| m3
| minor 3rd
| minor 3rd
| F
| F
|subminor third
| subminor third
| double-aug 2nd, dim 3rd
| na
| 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 third
| minor third
| minor 3rd
| nu
| Eb
|nu
| me
| me
|-
|-
| 8
| 8
| 355.56
| 355.6
| [[16/13]]
| [[16/13]]
| ~3
| ~3
| mid 3rd
| mid 3rd
|^Gb
| ^Gb
|neutral third
| neutral third
| major 3rd
| mi
| 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#
|major third
| major third
| aug 3rd, double-dim 4th
| mo
| 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#
|supermajor third
| supermajor third
| double-aug 3rd, dim 4th
| ma
| Ex, Fbb
|ma
| mo
| mo
|-
|-
| 11
| 11
| 488.89
| 488.9
| [[4/3]]
| [[4/3]]
| P4
| P4
| perfect 4th
| perfect 4th
| G
| G
|fourth
| fourth
| minor 4th
| fa
| 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
|superfourth
| superfourth
| major 4th
| fu/sha
| 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
|small tritone
| small tritone
| aug 4th, double-dim 5th
| fi/shu
| 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
|large tritone
| large tritone
| double-aug 4th, dim 5th
| po/si
| 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#
|subfifth
| subfifth
| minor 5th
| pa/so
| Fx#, Gb
|pa/so
| sih
| sih
|-
|-
| 16
| 16
| 711.11
| 711.1
| [[3/2]]
| [[3/2]]
| P5
| P5
| perfect 5th
| perfect 5th
| A
| A
|fifth
| fifth
| major 5th
| sa
| 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
|subminor sixth
| subminor sixth
| aug 5th, double-dim 6th
| fla
| 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
|minor sixth
| minor sixth
| double-aug 5th, dim 6th
| flu
| Gx, Abb
|flu
| le
| le
|-
|-
| 19
| 19
| 844.44
| 844.4
| [[13/8]]
| [[13/8]]
| ~6
| ~6
| mid 6th
| mid 6th
| vA#
| vA#
|neutral sixth
| neutral sixth
| minor 6th
| li
| 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 sixth
| major sixth
| major 6th
| lo
| A
|lo
| la
| la
|-
|-
| 21
| 21
| 933.33
| 933.3
| [[12/7]]
| [[12/7]]
| M6
| M6
| major 6th
| major 6th
| B
| B
|supermajor sixth
| supermajor sixth
| aug 6th, double-dim 7th
| la
| 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
|harmonic seventh
| harmonic seventh
| double-aug 6th, dim 7th
| tha
| Ax, Bbb
|tha
| ta
| ta
|-
|-
| 23
| 23
| 1022.22
| 1022.2
| [[9/5]]
| [[9/5]]
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| Db
| Db
|large minor seventh
| large minor seventh
| minor 7th
| thu
| 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
|neutral seventh
| neutral seventh
| major 7th
| ti
| 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#
|major seventh
| major seventh
| aug 7th, double-dim 8ve
| to
| 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#
|supermajor seventh
| supermajor seventh
| double-aug 7th, dim 8ve
| ta
| Bx, Cb
|ta
| da
| da
|-
|-
| 27
| 27
| 1200.00
| 1200.0
| 2/1
| [[2/1]]
| P8
| P8
| 8ve
| 8ve
| D
| D
|octave
| octave
| 8ve
| da
| 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 ===
=== Interval quality and chord names in color notation ===
Line 374: Line 320:
|-
|-
! Quality
! Quality
! [[Color name|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:


Line 420: Line 367:
|-
|-
! [[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 435: 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 442: 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 449: 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 456: 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}}
|}


=== 15-odd-limit interval mappings ===
=== Extended Pythagorean notation ===
The following table shows how [[15-odd-limit intervals]] are represented in 27edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
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.
{| class="wikitable center-all mw-collapsible mw-collapsed"
 
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
=== 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 ♭.  
 
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
|-
! &#35;
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0.0
| C
| perfect unison
| [[1/1]]
|-
|-
! Interval, complement
| 1
! Error (abs, ¢)
| 44.4
! Error (rel, %)
| C#, Dbbb
| aug 1sn, triple-dim 2nd
| [[40/39]], [[45/44]], [[55/54]], [[81/80]]
|-
|-
| [[7/6]], [[12/7]]
| 2
| 0.204
| 88.9
| 0.5
| Cx, Dbb
| double-aug 1sn, double-dim 2nd
| [[16/15]], [[25/24]]
|-
|-
| ''[[15/11]], [[22/15]]''
| 3
| ''3.617''
| 133.3
| ''8.1''
| Db
| dim 2nd
| [[12/11]], [[13/12]]
|-
|-
| '''[[13/8]], [[16/13]]'''
| 4
| '''3.917'''
| 177.8
| '''8.8'''
| D
| perfect 2nd
| [[10/9]], [[11/10]]
|-
|-
| [[5/3]], [[6/5]]
| 5
| 4.530
| 222.2
| 10.2
| D#, Ebbb
| aug 2nd, double-dim 3rd
| [[9/8]]
|-
|-
| [[9/5]], [[10/9]]
| 6
| 4.626
| 266.7
| 10.4
| Dx, Ebb
| double-aug 2nd, dim 3rd
| [[15/13]]
|-
|-
| [[7/5]], [[10/7]]
| 7
| 4.734
| 311.1
| 10.7
| Eb
| minor 3rd
| [[6/5]]
|-
|-
| [[13/7]], [[14/13]]
| 8
| 5.035
| 355.6
| 11.3
| E
| major 3rd
| [[11/9]], [[16/13]]
|-
|-
| [[13/12]], [[24/13]]
| 9
| 5.239
| 400.0
| 11.8
| E#, Fbbb
| aug 3rd, double-dim 4th
| [[5/4]]
|-
|-
| ''[[11/9]], [[18/11]]''
| 10
| ''8.148''
| 444.4
| ''18.3''
| Ex, Fbb
| double-aug 3rd, dim 4th
| [[13/10]]
|-
|-
| '''[[7/4]], [[8/7]]'''
| 11
| '''8.952'''
| 488.9
| '''20.1'''
| Ex#, Fb
| minor 4th
| [[4/3]]
|-
|-
| '''[[3/2]], [[4/3]]'''
| 12
| '''9.156'''
| 533.3
| '''20.6'''
| F
| major 4th
| [[15/11]], [[27/20]]
|-
|-
| [[9/7]], [[14/9]]
| 13
| 9.360
| 577.8
| 21.1
| F#, Gbbb
| aug 4th, double-dim 5th
| [[11/8]], [[18/13]]
|-
|-
| [[13/10]], [[20/13]]
| 14
| 9.770
| 622.2
| 22.0
| Fx, Gbb
| double-aug 4th, dim 5th
| [[13/9]], [[16/11]]
|-
|-
| ''[[11/10]], [[20/11]]''
| 15
| ''12.774''
| 666.7
| ''28.7''
| Fx#, Gb
| minor 5th
| [[22/15]], [[40/27]]
|-
|-
| '''[[5/4]], [[8/5]]'''
| 16
| '''13.686'''
| 711.1
| '''30.8'''
| G
| major 5th
| [[3/2]]
|-
|-
| [[15/14]], [[28/15]]
| 17
| 13.891
| 755.6
| 31.3
| G#, Abbb
| aug 5th, double-dim 6th
| [[20/13]]
|-
|-
| [[13/9]], [[18/13]]
| 18
| 14.395
| 800.0
| 32.4
| Gx, Abb
| double-aug 5th, dim 6th
| [[8/5]]
|-
|-
| ''[[11/6]], [[12/11]]''
| 19
| ''17.304''
| 844.4
| ''38.9''
| Ab
| minor 6th
| [[13/8]], [[18/11]]
|-
| 20
| 888.9
| A
| major 6th
| [[5/3]]
|-
| 21
| 933.3
| A#, Bbbb
| aug 6th, double-dim 7th
| [[26/15]]
|-
|-
| ''[[11/7]], [[14/11]]''
| 22
| ''17.508''
| 977.8
| ''39.4''
| Ax, Bbb
| double-aug 6th, dim 7th
| [[16/9]]
|-
|-
| '''[[11/8]], [[16/11]]'''
| 23
| '''17.985'''
| 1022.2
| '''40.5'''
| Bb
| perfect 7th
| [[9/5]], [[20/11]]
|-
|-
| [[9/8]], [[16/9]]
| 24
| 18.312
| 1066.7
| 41.2
| B
| aug 7th
| [[11/6]], [[24/13]]
|-
|-
| [[15/13]], [[26/15]]
| 25
| 18.926
| 1111.1
| 42.6
| B#, Cbb
| double-aug 7th, double-dim 8ve
| [[15/8]], [[48/25]]
|-
|-
| ''[[15/8]], [[16/15]]''
| 26
| ''21.602''
| 1155.6
| ''48.6''
| Bx, Cb
| triple-aug 7th, dim 8ve
| [[39/20]], [[88/45]], [[108/55]], [[160/81]]
|-
|-
| [[13/11]], [[22/13]]
| 27
| 21.901
| 1200.0
| 49.3
| 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 585: 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 592: 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 599: 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 606: 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 613: 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 632: 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
| [[Ennealimmal]]<br>/niner (very inaccurate)
| [[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]]
! [[Color name]]
! Name
! Name
|-
|-
Line 717: Line 928:
| 41.06
| 41.06
| Trigu
| Trigu
| Augmented comma
| Augmented comma, lesser diesis
|-
|-
| 5
| 5
Line 724: Line 935:
| 27.66
| 27.66
| Saquadyo
| Saquadyo
| Tetracot comma
| Tetracot comma, minimal diesis
|-
|-
| 5
| 5
Line 788: Line 999:
| Triru-agu
| Triru-agu
| Orwellisma
| Orwellisma
|-
| 7
| <abbr title="40353607/40310784">(16 digits)</abbr>
| {{monzo| -11 -9 0 9 }}
| 1.84
| Tritrizo
| [[Septimal ennealimma]]
|-
|-
| 7
| 7
Line 818: Line 1,036:
|-
|-
| 11
| 11
| 77/75
| [[55/54]]
| {{monzo| 0 -1 -2 1 1 }}
| {{monzo| -1 -3 1 0 1 }}
| 45.56
| 31.77
|  
| Loyo
|  
| Telepathma
|-
|-
| 11
| 11
Line 844: Line 1,062:
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|-
| 13
| [[66/65]]
| {{monzo| 1 1 -1 0 1 -1 }}
| 26.43
| Thulogu
| Winmeanma
|-
|-
| 13
| 13
Line 850: Line 1,075:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap comma
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 858: Line 1,083:
| Thuthu
| Thuthu
| Tridecimal neutral thirds comma
| Tridecimal neutral thirds comma
|-
| 13
| [[325/324]]
| {{monzo| -2 -4 2 0 0 1 }}
| 5.34
| Thoyoyo
| Marveltwin comma
|-
|-
| 13
| 13
Line 865: Line 1,097:
| Thorugugu
| Thorugugu
| Ratwolfsma
| 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 ==
== 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)


Direct sunlight (this is its original/default tuning; subset of [[Sensi]][19])
== Instruments ==
* 44.444
[[File:27edo_Guitar.jpg|200px|thumb|right|Brendan Byrnes, guitarist]]
* 133.333
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.
* 488.889
* 711.111
* 755.555
* 1155.555
* 1200.000


Hypersakura (this is its original/default tuning; subset of Sensi[19])
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.
* 44.444
 
* 488.889
* [[Skip fretting system 27 2 9]]
* 711.111
 
* 755.555
27edo can also be played on the Lumatone, with various layouts discussed here.
* 1200.000
 
* [[Lumatone mapping for 27edo]]


== Music ==
== Music ==
{{Catrel| 27edo tracks }}
{{Catrel| 27edo tracks }}
; [[Abnormality]]
* [https://www.youtube.com/watch?v=gfGNKd8SWWc ''Boiling''] (2024)
; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=Pr5E5brBGuw ''What Happens Next''] (2021)


; [[Beheld]]
; [[Beheld]]
Line 900: Line 1,234:
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=hDP8cfJqWOI ''microtonal improvisation in 27edo''] (2023)
* [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'']] (2012)
* [[: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)
Line 912: Line 1,249:
* [https://www.youtube.com/watch?v=nR8orkai8tQ ''Chorale in 27edo for Organ''] (2019)
* [https://www.youtube.com/watch?v=nR8orkai8tQ ''Chorale in 27edo for Organ''] (2019)


; [[Nae Ayy]]
; [[Herman Miller]]
* [https://www.youtube.com/watch?v=Pr5E5brBGuw ''What Happens Next''] (2021)
* ''[https://soundcloud.com/morphosyntax-1/nusu-laj-stille-nacht Stille Nacht (cover)]'' (2019)


; [[NullPointerException Music]]
; [[NullPointerException Music]]
Line 928: Line 1,265:


; [[Joel Taylor]]
; [[Joel Taylor]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina'']{{dead link}}
* [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


; [[User:Tristanbay|Tristan Bay]]
; [[Tristan Bay]]
*[https://youtu.be/R30aRbNtoIY ''Pitchblende''] (2023)
* [https://youtu.be/R30aRbNtoIY ''Pitchblende''] (2023)


;[[Chris Vaisvil]]
; [[Uncreative Name]]
*[http://micro.soonlabel.com/27edo/daily20111202-deep-chasm-zeta-cp-1.mp3 ''Chicago Pile-1''] (2011)
* [https://www.youtube.com/watch?v=dcQe6ebpGFU ''Autumn''] (2024) – in Blackdye, 27edo tuning


;[[Xotla]]
; [[Chris Vaisvil]]
*"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]
* [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]]

Latest revision as of 00:00, 16 August 2025

← 26edo 27edo 28edo →
Prime factorization 33
Step size 44.4444 ¢ 
Fifth 16\27 (711.111 ¢)
Semitones (A1:m2) 4:1 (177.8 ¢ : 44.44 ¢)
Consistency limit 9
Distinct consistency limit 7

27 equal divisions of the octave (abbreviated 27edo or 27ed2), also called 27-tone equal temperament (27tet) or 27 equal temperament (27et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 27 equal parts of about 44.4 ¢ each. Each step represents a frequency ratio of 21/27, or the 27th root of 2.

Theory

Assuming pure octaves, 27edo divides the octave in 27 equal parts each exactly 444⁄9 cents in size. Its fifth and harmonic seventh are both sharp by 9 ¢, and the major third is the same 400-cent major third as 12edo, sharp by 13.7 ¢. 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, 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.

Though 27edo's 7-limit tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both consistently 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 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 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.

27edo, with its 400 ¢ 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 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 ¢, 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 ¢, 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

Approximation of odd harmonics in 27edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +9.2 +13.7 +9.0 +18.3 -18.0 +3.9 -21.6 -16.1 +13.6 +18.1 -6.1
Relative (%) +20.6 +30.8 +20.1 +41.2 -40.5 +8.8 -48.6 -36.1 +30.6 +40.7 -13.6
Steps
(reduced) 		43
(16) 		63
(9) 		76
(22) 		86
(5) 		93
(12) 		100
(19) 		105
(24) 		110
(2) 		115
(7) 		119
(11) 		122
(14)

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 octave compression. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 ¢. More generally, narrowing the steps to between 44.2 and 44.35 ¢ 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 ¢, respectively.

Subsets and supersets

Since 27 factors into primes as 33, 27edo contains 3edo and 9edo as subsets. Multiplying it by 3 gives 81edo, which is a good meantone tuning.

Intervals

# Cents Approximate ratios[note 1] Ups and downs notation (EUs: v4A1 and vm2) Interval regions Solfeges
0 0.0 1/1 P1 perfect unison D unison da do
1 44.4 28/27, 36/35, 39/38, 49/48, 50/49, 81/80 ^1, m2 up unison, minor 2nd ^D, Eb diesis fra di
2 88.9 16/15, 21/20, 25/24, 19/18, 20/19 ^^1, ^m2 dup unison, upminor 2nd ^^D, ^Eb minor second fru ra
3 133.3 15/14, 14/13, 13/12 vA1, ~2 downaug 1sn, mid 2nd vD#, vvE neutral second ri ru
4 177.8 10/9 A1, vM2 aug 1sn, downmajor 2nd D#, vE small major second ro reh
5 222.2 8/7, 9/8 M2 major 2nd E large major second ra re
6 266.7 7/6 m3 minor 3rd F subminor third na ma
7 311.1 6/5, 19/16 ^m3 upminor 3rd Gb minor third nu me
8 355.6 16/13 ~3 mid 3rd ^Gb neutral third mi mu
9 400.0 5/4, 24/19 vM3 downmajor 3rd vF# major third mo mi
10 444.4 9/7, 13/10 M3 major 3rd F# supermajor third ma mo
11 488.9 4/3 P4 perfect 4th G fourth fa fa
12 533.3 19/14, 26/19, 27/20, 48/35 ^4 up 4th Ab superfourth fu/sha fih
13 577.8 7/5, 18/13 ~4, ^d5 mid 4th, updim 5th ^^G, ^Ab small tritone fi/shu fi
14 622.2 10/7, 13/9 vA4, ~5 downaug 4th, mid 5th vG#, vvA large tritone po/si se
15 666.7 19/13, 28/19, 35/24, 40/27 v5 down fifth G# subfifth pa/so sih
16 711.1 3/2 P5 perfect 5th A fifth sa so/sol
17 755.6 14/9, 20/13 m6 minor 6th Bb subminor sixth fla lo
18 800.0 8/5, 19/12 ^m6 upminor 6th ^Bb minor sixth flu le
19 844.4 13/8 ~6 mid 6th vA# neutral sixth li lu
20 888.9 5/3, 32/19 vM6 downmajor 6th A# major sixth lo la
21 933.3 12/7 M6 major 6th B supermajor sixth la li
22 977.8 7/4, 16/9 m7 minor 7th C harmonic seventh tha ta
23 1022.2 9/5 ^m7 upminor 7th Db large minor seventh thu te
24 1066.7 13/7, 24/13, 28/15 ~7 mid 7th ^Db neutral seventh ti tu
25 1111.1 15/8, 19/10, 36/19, 40/21, 48/25 vM7 downmajor 7th vC# major seventh to ti
26 1155.6 27/14, 35/18, 49/25, 96/49, 160/81 M7 major 7th C# supermajor seventh ta da
27 1200.0 2/1 P8 8ve D octave da do
  1. ↑ Based on treating 27et as a 2.3.5.7.13.19-subgroup temperament; other approaches are also possible.

Interval quality and chord names in color notation

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

Quality Color name Monzo format Examples
minor zo [a, b, 0, 1⟩ 7/6, 7/4
fourthward wa [a, b⟩, b < −1 32/27, 16/9
upminor gu [a, b, −1⟩ 6/5, 9/5
mid tho [a, b, 0, 0, 0, 1⟩ 13/12, 13/8
thu [a, b, 0, 0, 0, −1⟩ 16/13, 24/13
downmajor yo [a, b, 1⟩ 5/4, 5/3
major fifthward wa [a, b⟩, b > 1 9/8, 27/16
ru [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:

Color of the 3rd JI chord Notes as edosteps Notes of C chord Written name Spoken name
zo 6:7:9 0–6–16 C–E♭–G Cm C minor
gu 10:12:15 0–7–16 C–F♭–G, C–E⁠ ⁠–G C^m C upminor
ilo 18:22:27 0–8–16 C–E⁠ ⁠–G C~ C mid
yo 4:5:6 0–9–16 C–D♯–G, C–E⁠ ⁠–G Cv C downmajor or C down
ru 14:18:21 0–10–16 C–E–G C C major or C

For a more complete list, see Ups and downs notation #Chords and chord progressions. See also the 22edo page.

Notation

Circle of fifths in 27edo
Cents Extended
Pythagorean
notation 		Quartertone
notation
0.0 C A⁠ ⁠
711.1 G E⁠ ⁠
222.2 D B⁠ ⁠ F⁠ ⁠
933.3 A C⁠ ⁠
444.4 E G⁠ ⁠
1155.6 B D⁠ ⁠
666.7 F♯ A⁠ ⁠
177.8 C♯ E⁠ ⁠
888.9 G♯ B⁠ ⁠
400.0 D♯ F⁠ ⁠
1111.1 A♯ C⁠ ⁠
622.2 E♯ G⁠ ⁠
133.3 B♯ F𝄫 D⁠ ⁠
844.4 F𝄪 C𝄫 A⁠ ⁠
355.6 C𝄪 G𝄫 E⁠ ⁠
1066.7 G𝄪 D𝄫 B⁠ ⁠
577.8 D𝄪 A𝄫 F⁠ ⁠
88.9 A𝄪 E𝄫 C⁠ ⁠
800.0 E𝄪 B𝄫 G⁠ ⁠
311.1 B𝄪 F♭ D⁠ ⁠
1022.2 C♭ A⁠ ⁠
533.3 G♭ E⁠ ⁠
44.4 D♭ B⁠ ⁠
755.6 A♭ F⁠ ⁠
266.7 E♭ C⁠ ⁠
977.8 B♭ G⁠ ⁠
488.9 F D⁠ ⁠
0.0 C A⁠ ⁠

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 ⁠, D⁠ ⁠, 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.

Step offset 0 1 2 3 4 5 6 7 8 9
Sharp symbol               
Flat symbol
                

Alternatively, sharps and flats with arrows can be used, borrowed from extended Helmholtz–Ellis notation:

Step offset 0 1 2 3 4 5 6 7 8 9
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 
 
 
 		 

Sagittal notation

This notation is a subset of the notation for 54edo.

Evo and Revo flavors

Sagittal notationPeriodic table of EDOs with sagittal notation81/808505/819227/26

Alternative Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/808505/819227/26

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/808505/819227/26

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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 LLLLLLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 edostep in this instance, is denoted by ♯ and ♭.

# Cents Note Name Associated ratio
0 0.0 C perfect unison 1/1
1 44.4 C#, Dbbb aug 1sn, triple-dim 2nd 40/39, 45/44, 55/54, 81/80
2 88.9 Cx, Dbb double-aug 1sn, double-dim 2nd 16/15, 25/24
3 133.3 Db dim 2nd 12/11, 13/12
4 177.8 D perfect 2nd 10/9, 11/10
5 222.2 D#, Ebbb aug 2nd, double-dim 3rd 9/8
6 266.7 Dx, Ebb double-aug 2nd, dim 3rd 15/13
7 311.1 Eb minor 3rd 6/5
8 355.6 E major 3rd 11/9, 16/13
9 400.0 E#, Fbbb aug 3rd, double-dim 4th 5/4
10 444.4 Ex, Fbb double-aug 3rd, dim 4th 13/10
11 488.9 Ex#, Fb minor 4th 4/3
12 533.3 F major 4th 15/11, 27/20
13 577.8 F#, Gbbb aug 4th, double-dim 5th 11/8, 18/13
14 622.2 Fx, Gbb double-aug 4th, dim 5th 13/9, 16/11
15 666.7 Fx#, Gb minor 5th 22/15, 40/27
16 711.1 G major 5th 3/2
17 755.6 G#, Abbb aug 5th, double-dim 6th 20/13
18 800.0 Gx, Abb double-aug 5th, dim 6th 8/5
19 844.4 Ab minor 6th 13/8, 18/11
20 888.9 A major 6th 5/3
21 933.3 A#, Bbbb aug 6th, double-dim 7th 26/15
22 977.8 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
25 1111.1 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

Approximation to JI

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Selected 19-limit intervals approximated in 27edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 27edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 27edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 0.5
15/11, 22/15 3.617 8.1
13/8, 16/13 3.917 8.8
5/3, 6/5 4.530 10.2
9/5, 10/9 4.626 10.4
7/5, 10/7 4.734 10.7
13/7, 14/13 5.035 11.3
13/12, 24/13 5.239 11.8
11/9, 18/11 8.148 18.3
7/4, 8/7 8.952 20.1
3/2, 4/3 9.156 20.6
9/7, 14/9 9.360 21.1
13/10, 20/13 9.770 22.0
11/10, 20/11 12.774 28.7
5/4, 8/5 13.686 30.8
15/14, 28/15 13.891 31.3
13/9, 18/13 14.395 32.4
11/6, 12/11 17.304 38.9
11/7, 14/11 17.508 39.4
11/8, 16/11 17.985 40.5
9/8, 16/9 18.312 41.2
15/13, 26/15 18.926 42.6
15/8, 16/15 21.602 48.6
13/11, 22/13 21.901 49.3
15-odd-limit intervals in 27edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 0.5
13/8, 16/13 3.917 8.8
5/3, 6/5 4.530 10.2
9/5, 10/9 4.626 10.4
7/5, 10/7 4.734 10.7
13/7, 14/13 5.035 11.3
13/12, 24/13 5.239 11.8
7/4, 8/7 8.952 20.1
3/2, 4/3 9.156 20.6
9/7, 14/9 9.360 21.1
13/10, 20/13 9.770 22.0
5/4, 8/5 13.686 30.8
15/14, 28/15 13.891 31.3
13/9, 18/13 14.395 32.4
11/8, 16/11 17.985 40.5
9/8, 16/9 18.312 41.2
15/13, 26/15 18.926 42.6
13/11, 22/13 21.901 49.3
15/8, 16/15 22.842 51.4
11/7, 14/11 26.936 60.6
11/6, 12/11 27.141 61.1
11/10, 20/11 31.671 71.3
11/9, 18/11 36.297 81.7
15/11, 22/15 40.827 91.9
15-odd-limit intervals by 27e val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
7/6, 12/7 0.204 0.5
15/11, 22/15 3.617 8.1
13/8, 16/13 3.917 8.8
5/3, 6/5 4.530 10.2
9/5, 10/9 4.626 10.4
7/5, 10/7 4.734 10.7
13/7, 14/13 5.035 11.3
13/12, 24/13 5.239 11.8
11/9, 18/11 8.148 18.3
7/4, 8/7 8.952 20.1
3/2, 4/3 9.156 20.6
9/7, 14/9 9.360 21.1
13/10, 20/13 9.770 22.0
11/10, 20/11 12.774 28.7
5/4, 8/5 13.686 30.8
15/14, 28/15 13.891 31.3
13/9, 18/13 14.395 32.4
11/6, 12/11 17.304 38.9
11/7, 14/11 17.508 39.4
9/8, 16/9 18.312 41.2
15/13, 26/15 18.926 42.6
13/11, 22/13 22.543 50.7
15/8, 16/15 22.842 51.4
11/8, 16/11 26.460 59.5

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [43 -27⟩ [⟨27 43]] −2.89 2.88 6.50
2.3.5 128/125, 20000/19683 [⟨27 43 63]] −3.88 2.74 6.19
2.3.5.7 64/63, 126/125, 245/243 [⟨27 43 63 76]] −3.71 2.39 5.40
2.3.5.7.13 64/63, 91/90, 126/125, 169/168 [⟨27 43 63 76 100]] −3.18 2.39 5.39
2.3.5.7.13.19 64/63, 76/75, 91/90, 126/125, 169/168 [⟨27 43 63 76 100 115]] −3.18 2.18 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 31, 31, and 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 53.

Uniform maps

13-limit uniform maps between 26.8 and 27.2
Min. size Max. size Wart notation Map
26.7385 26.8145 27bcdf ⟨27 42 62 75 93 99]
26.8145 26.8887 27cdf ⟨27 43 62 75 93 99]
26.8887 26.8936 27cd ⟨27 43 62 75 93 100]
26.8936 26.9173 27c ⟨27 43 62 76 93 100]
26.9173 27.0276 27 ⟨27 43 63 76 93 100]
27.0276 27.1589 27e ⟨27 43 63 76 94 100]
27.1589 27.2498 27eff ⟨27 43 63 76 94 101]

Rank-2 temperaments

Periods
per 8ve 		Generator Temperaments Mos scales
1 1\27 Quartonic / quarto (27e) / quartz (27)
1 2\27 Octacot / octocat (27e) 1L 12s, 13L 1s
1 4\27 Tetracot (27e) / modus (27e) / wollemia (27e) 1L 5s, 6L 1s, 7L 6s, 7L 13s
1 5\27 Machine (27)
Kumonga (27e) 		1L 4s, 5L 1s, 5L 6s, 11L 5s
1 7\27 Myna (27e) / coleto (27e) / myno (27)
Oolong (27e) 		4L 3s, 4L 7s, 4L 11s, 4L 15s, 4L 19s
1 8\27 Beatles (27e) / ringo (27e) / beetle (27) 3L 4s, 7L 3s, 10L 7s
1 10\27 Sensi 3L 2s, 3L 5s, 8L 3s, 8L 11s
1 11\27 Superpyth (27e) 5L 2s, 5L 7s, 5L 12s, 5L 17s
1 13\27 Fervor (27e) 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s, etc. … 2L 23s
3 1\27 Hemiaug (27e)
3 2\27 Augene (27e) / Eugene (27) 3L 3s, 3L 6s, 3L 9s, 12L 3s
3 4\27 Oodako (27e)
Terrain 		3L 3s, 6L 3s, 6L 9s, 6L 15s
9 1\27 Niner (27e)
Ennealimmal (out of tune) 		9L 9s

Commas

27et tempers out the following commas. (Note: This assumes the patent val, ⟨27 43 63 76 93 100].)

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name
5 128/125 [7 0 -3⟩ 41.06 Trigu Augmented comma, lesser diesis
5 20000/19683 [5 -9 4⟩ 27.66 Saquadyo Tetracot comma, minimal diesis
5 78732/78125 [2 9 -7⟩ 13.40 Sepgu Sensipent comma
5 (26 digits) [1 -27 18⟩ 0.86 Satritribiyo Ennealimma
7 686/675 [1 -3 -2 3⟩ 27.99 Trizo-agugu Senga
7 64/63 [6 -2 0 -1⟩ 27.26 Ru Septimal comma
7 50421/50000 [-4 1 -5 5⟩ 14.52 Quinzogu Trimyna comma
7 245/243 [0 -5 1 2⟩ 14.19 Zozoyo Sensamagic comma
7 126/125 [1 2 -3 1⟩ 13.79 Zotrigu Starling comma
7 4000/3969 [5 -4 3 -2⟩ 13.47 Rurutriyo Octagar comma
7 1728/1715 [6 3 -1 -3⟩ 13.07 Triru-agu Orwellisma
7 (16 digits) [-11 -9 0 9⟩ 1.84 Tritrizo Septimal ennealimma
7 (12 digits) [-6 -8 2 5⟩ 1.12 Quinzo-ayoyo Wizma
7 2401/2400 [-5 -1 -2 4⟩ 0.72 Bizozogu Breedsma
7 4375/4374 [-1 -7 4 1⟩ 0.40 Zoquadyo Ragisma
7 (12 digits) [-4 6 -6 3⟩ 0.33 Trizogugu Landscape comma
11 55/54 [-1 -3 1 0 1⟩ 31.77 Loyo Telepathma
11 99/98 [-1 2 0 -2 1⟩ 17.58 Loruru Mothwellsma
11 896/891 [7 -4 0 1 -1⟩ 9.69 Saluzo Pentacircle comma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
13 66/65 [1 1 -1 0 1 -1⟩ 26.43 Thulogu Winmeanma
13 91/90 [-1 -2 -1 1 0 1⟩ 19.13 Thozogu Superleap comma, biome comma
13 512/507 [9 -1 0 0 0 -2⟩ 16.99 Thuthu Tridecimal neutral thirds comma
13 325/324 [-2 -4 2 0 0 1⟩ 5.34 Thoyoyo Marveltwin comma
13 351/350 [-1 3 -2 -1 0 1⟩ 4.94 Thorugugu Ratwolfsma
13 31213/31104 [-7 -5 0 4 0 1⟩ 6.06 Thoquadzo Praveensma
17 85/84 [-2 -1 1 -1 0 0 1⟩ 20.49 Soruyo Monk comma
17 154/153 [1 -2 0 1 1 0 -1⟩ 11.28 Sulozo Augustma
19 77/76 [2 -1 -2 0 0 0 0 1⟩ 22.63 Nulozo Small undevicesimal ninth tone
19 96/95 [5 1 -1 0 0 0 0 -1⟩ 18.13 Nugu 19th-partial chroma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Scales

MOS scales

Main article: 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 / Pinetone major pentatonic: 5 4 7 4 7
  • 5-limit / pental / 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
  • 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
  • Pinetone diatonic: 4 3 4 5 4 3 4
  • Pinetone major-harmonic octatonic: 4 3 4 2 3 4 3 4
  • Pinetone minor-harmonic octatonic: 4 3 2 4 3 4 4 3
  • Pinetone diminished octatonic / Porcusmine: 3 4 2 4 3 4 3 4
  • Pinetone harmonic diminished: 3 4 2 5 2 4 3 4
  • 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 / 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

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.

27edo can also be played on the Lumatone, with various layouts discussed here.

Music

See also: Category:27edo tracks
Abnormality
Nae Ayy
Beheld
Gregoire Blanc
Brendan Byrnes
Bryan Deister
Francium
Igliashon Jones
Peter Kosmorsky
Claudi Meneghin
Herman Miller
NullPointerException Music
Dustin Schallert
Gene Ward Smith
Joel Taylor
Tristan Bay
Uncreative Name
  • Autumn (2024) – in Blackdye, 27edo tuning
Chris Vaisvil
Xotla
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