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{{interwiki
| de = 27edo
| en = 27edo
| es =
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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]]: Three fourths (C-Eb) in 19edo reach a near-perfect [[6/5]] and the same distance in 27edo reaches a near-perfect [[7/6]].


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, if a highly sharp-tending one. 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 ([0 20 33]) and 5:7:9 ([0 13 23]), via the [[BPS]] scale in [[43edt]], although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.


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]].
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.


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.
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.


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.
=== Odd harmonics ===
 
{{Harmonics in equal|27}}
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.


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.
=== As a tuning of other temperaments ===
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. The sharp 9/7 befits a [[generator]] for [[sensi]], which 27edo also supports, but a much better tuning is found in [[46edo]]. Another notable temperament 27edo supports is [[myna]], which divides the category of thirds into five different intervals: subminor, minor, neutral, major, and supermajor, representing 7/6, 6/5, 11/9~16/13, 5/4, and 9/7, 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 ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-1 right-2"
|-
|-
! #
! #
! Cents
! Cents
! Approximate Ratios*
! [[Interval region]]s
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
! Approximate ratios<ref group="note">As a 2.3.5.7.13.19-[[subgroup]] temperament, inconsistent intervals in ''italic''. </ref>
![[Walker Brightness Notation]]
! [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="2" | [[6L 1s]] Notation
! colspan="2" |[[Solfege|Solfeges]]
|-
|-
| 0
| 0
| 0.00
| 0.0
| Unison
| [[1/1]]
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 44.4
| Diesis
| [[28/27]], [[36/35]], [[39/38]], [[49/48]], [[50/49]], ''[[81/80]]''
| {{UDnote|step=1}}
|-
| 2
| 88.9
| Minor second
| ''[[16/15]]'', [[21/20]], [[25/24]], [[19/18]], [[20/19]]
| {{UDnote|step=2}}
|-
| 3
| 133.3
| Neutral second
| [[15/14]], [[14/13]], [[13/12]]
| {{UDnote|step=3}}
|-
| 4
| 177.8
| Small major second
| [[10/9]]
| {{UDnote|step=4}}
|-
| 5
| 222.2
| Large major second
| [[8/7]], [[9/8]]
| {{UDnote|step=5}}
|-
| 6
| 266.7
| Subminor third
| [[7/6]]
| {{UDnote|step=6}}
|-
| 7
| 311.1
| Minor third
| [[6/5]], [[19/16]]
| {{UDnote|step=7}}
|-
| 8
| 355.6
| Neutral third
| [[16/13]]
| {{UDnote|step=8}}
|-
| 9
| 400.0
| Major third
| [[5/4]], [[24/19]]
| {{UDnote|step=9}}
|-
| 10
| 444.4
| Supermajor third
| [[9/7]], [[13/10]]
| {{UDnote|step=10}}
|-
| 11
| 488.9
| Perfect fourth
| [[4/3]]
| {{UDnote|step=11}}
|-
| 12
| 533.3
| Superfourth
| [[19/14]], [[26/19]], [[27/20]], [[48/35]]
| {{UDnote|step=12}}
|-
| 13
| 577.8
| Small tritone
| [[7/5]], [[18/13]]
| {{UDnote|step=13}}
|-
| 14
| 622.2
| Large tritone
| [[10/7]], [[13/9]]
| {{UDnote|step=14}}
|-
| 15
| 666.7
| Subfifth
| [[19/13]], [[28/19]], [[35/24]], [[40/27]]
| {{UDnote|step=15}}
|-
| 16
| 711.1
| Perfect fifth
| [[3/2]]
| {{UDnote|step=16}}
|-
| 17
| 755.6
| Subminor sixth
| [[14/9]], [[20/13]]
| {{UDnote|step=17}}
|-
| 18
| 800.0
| Minor sixth
| [[8/5]], [[19/12]]
| {{UDnote|step=18}}
|-
| 19
| 844.4
| Neutral sixth
| [[13/8]]
| {{UDnote|step=19}}
|-
| 20
| 888.9
| Major sixth
| [[5/3]], [[32/19]]
| {{UDnote|step=20}}
|-
| 21
| 933.3
| Supermajor sixth
| [[12/7]]
| {{UDnote|step=21}}
|-
| 22
| 977.8
| Harmonic seventh
| [[7/4]], [[16/9]]
| {{UDnote|step=22}}
|-
| 23
| 1022.2
| Large minor seventh
| [[9/5]]
| {{UDnote|step=23}}
|-
| 24
| 1066.7
| Neutral seventh
| [[13/7]], [[24/13]], [[28/15]]
| {{UDnote|step=24}}
|-
| 25
| 1111.1
| Major seventh
| ''[[15/8]]'', [[19/10]], [[36/19]], [[40/21]], [[48/25]]
| {{UDnote|step=25}}
|-
| 26
| 1155.6
| Supermajor seventh
| [[27/14]], [[35/18]], [[49/25]], [[96/49]], ''[[160/81]]''
| {{UDnote|step=26}}
|-
| 27
| 1200.0
| Octave
| [[2/1]]
| {{UDnote|step=27}}
|}
<references group="note" />
=== Proposed interval names and solfèges ===
{| class="wikitable center-all right-2 left-4 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and vm2)
! colspan="2" | [[Solfège]]s
|-
| 0
| 0.0
| P1
| P1
| perfect unison
| perfect unison
| D
| D
|unison
| da
| perfect unison
| C
|da
| do
| do
|-
|-
| 1
| 1
| 44.44
| 44.4
| [[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
| fra
| aug 1sn, double-dim 2nd
| C#, Dbbb
|fra
| di
| di
|-
|-
| 2
| 2
| 88.89
| 88.9
| [[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
| fru
| double-aug 1sn, dim 2nd
| Cx, Dbb
|fru
| ra
| ra
|-
|-
| 3
| 3
| 133.33
| 133.3
| [[15/14]], [[14/13]], [[13/12]]
| vA1, ~2
| vA1, ~2
| downaug 1sn, mid 2nd
| downaug 1sn, mid 2nd
| vD#, vvE
| vD#, vvE
|neutral second
| ri
| minor 2nd
| Db
|ri
| ru
| ru
|-
|-
| 4
| 4
| 177.78
| 177.8
| [[10/9]]
| A1, vM2
| A1, vM2
| aug 1sn, downmajor 2nd
| aug 1sn, downmajor 2nd
| D#, vE
| D#, vE
|small major second
| ro
| major 2nd
| D
|ro
| reh
| reh
|-
|-
| 5
| 5
| 222.22
| 222.2
| [[8/7]], [[9/8]]
| M2
| M2
| major 2nd
| major 2nd
| E
| E
|large major second
| ra
| aug 2nd, double-dim 3rd
| D#, Ebbb
|ra
| re
| re
|-
|-
| 6
| 6
| 266.67
| 266.7
| [[7/6]]
| m3
| m3
| minor 3rd
| minor 3rd
| F
| F
|subminor third
| na
| double-aug 2nd, dim 3rd
| Dx, Ebb
|na
| ma
| ma
|-
|-
| 7
| 7
| 311.11
| 311.1
| [[6/5]], [[19/16]]
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| Gb
| Gb
|minor third
| nu
| minor 3rd
| Eb
|nu
| me
| me
|-
|-
| 8
| 8
| 355.56
| 355.6
| [[16/13]]
| ~3
| ~3
| mid 3rd
| mid 3rd
|^Gb
| ^Gb
|neutral third
| mi
| major 3rd
| E
|mi
| mu
| mu
|-
|-
| 9
| 9
| 400.00
| 400.0
| [[5/4]], [[24/19]]
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
|major third
| mo
| aug 3rd, double-dim 4th
| E#, Fbbb
|mo
| mi
| mi
|-
|-
| 10
| 10
| 444.44
| 444.4
| [[9/7]], [[13/10]]
| M3
| M3
| major 3rd
| major 3rd
| F#
| F#
|supermajor third
| ma
| double-aug 3rd, dim 4th
| Ex, Fbb
|ma
| mo
| mo
|-
|-
| 11
| 11
| 488.89
| 488.9
| [[4/3]]
| P4
| P4
| perfect 4th
| perfect 4th
| G
| G
|fourth
| fa
| minor 4th
| Ex#, Fb
|fa
| fa
| fa
|-
|-
| 12
| 12
| 533.33
| 533.3
| [[27/20]], [[48/35]], [[19/14]], [[26/19]]
| ^4
| ^4
| up 4th
| up 4th
| Ab
| Ab
|superfourth
| fu/sha
| major 4th
| F
|fu/sha
| fih
| fih
|-
|-
| 13
| 13
| 577.78
| 577.8
| [[7/5]], [[18/13]]
| ~4, ^d5
| ~4, ^d5
| mid 4th, updim 5th
| mid 4th, updim 5th
| ^^G, ^Ab
| ^^G, ^Ab
|small tritone
| fi/shu
| aug 4th, double-dim 5th
| F#, Gbbb
|fi/shu
| fi
| fi
|-
|-
| 14
| 14
| 622.22
| 622.2
| [[10/7]], [[13/9]]
| vA4, ~5
| vA4, ~5
| downaug 4th, mid 5th
| downaug 4th, mid 5th
| vG#, vvA
| vG#, vvA
|large tritone
| po/si
| double-aug 4th, dim 5th
| Fx, Gbb
|po/si
| se
| se
|-
|-
| 15
| 15
| 666.67
| 666.7
| [[40/27]], [[35/24]], [[19/13]], [[28/19]]
| v5
| v5
| down fifth
| down fifth
| G#
| G#
|subfifth
| pa/so
| minor 5th
| Fx#, Gb
|pa/so
| sih
| sih
|-
|-
| 16
| 16
| 711.11
| 711.1
| [[3/2]]
| P5
| P5
| perfect 5th
| perfect 5th
| A
| A
|fifth
| sa
| major 5th
| G
|sa
| so/sol
| so/sol
|-
|-
| 17
| 17
| 755.56
| 755.6
| [[14/9]], [[20/13]]
| m6
| m6
| minor 6th
| minor 6th
| Bb
| Bb
|subminor sixth
| fla
| aug 5th, double-dim 6th
| G#, Abbb
|fla
| lo
| lo
|-
|-
| 18
| 18
| 800.00
| 800.0
| [[8/5]], [[19/12]]
| ^m6
| ^m6
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
|minor sixth
| flu
| double-aug 5th, dim 6th
| Gx, Abb
|flu
| le
| le
|-
|-
| 19
| 19
| 844.44
| 844.4
| [[13/8]]
| ~6
| ~6
| mid 6th
| mid 6th
| vA#
| vA#
|neutral sixth
| li
| minor 6th
| Ab
|li
| lu
| lu
|-
|-
| 20
| 20
| 888.89
| 888.9
| [[5/3]], [[32/19]]
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| A#
| A#
|major sixth
| lo
| major 6th
| A
|lo
| la
| la
|-
|-
| 21
| 21
| 933.33
| 933.3
| [[12/7]]
| M6
| M6
| major 6th
| major 6th
| B
| B
|supermajor sixth
| la
| aug 6th, double-dim 7th
| A#, Bbbb
|la
| li
| li
|-
|-
| 22
| 22
| 977.78
| 977.8
| [[7/4]], [[16/9]]
| m7
| m7
| minor 7th
| minor 7th
| C
| C
|harmonic seventh
| tha
| double-aug 6th, dim 7th
| Ax, Bbb
|tha
| ta
| ta
|-
|-
| 23
| 23
| 1022.22
| 1022.2
| [[9/5]]
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| Db
| Db
|large minor seventh
| thu
| minor 7th
| Bb
|thu
| te
| te
|-
|-
| 24
| 24
| 1066.67
| 1066.7
| [[28/15]], [[13/7]], [[24/13]]
| ~7
| ~7
| mid 7th
| mid 7th
| ^Db
| ^Db
|neutral seventh
| ti
| major 7th
| B
|ti
| tu
| tu
|-
|-
| 25
| 25
| 1111.11
| 1111.1
| [[15/8]], [[40/21]], [[48/25]], [[19/10]], [[36/19]]
| vM7
| vM7
| downmajor 7th
| downmajor 7th
| vC#
| vC#
|major seventh
| to
| aug 7th, double-dim 8ve
| B#, Cbb
|to
| ti
| ti
|-
|-
| 26
| 26
| 1155.56
| 1155.6
| [[27/14]], [[35/18]], [[96/49]], [[49/25]], [[160/81]]
| M7
| M7
| major 7th
| major 7th
| C#
| C#
|supermajor seventh
| ta
| double-aug 7th, dim 8ve
| Bx, Cb
|ta
| da
| da
|-
|-
| 27
| 27
| 1200.00
| 1200.0
| 2/1
| P8
| P8
| 8ve
| 8ve
| D
| D
|octave
| da
| 8ve
| C
|da
| do
| do
|}
|}
<nowiki/>* based on treating 27edo as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 374: Line 445:
|-
|-
! 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 492:
|-
|-
! [[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 507:
| 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 514:
| 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 521:
| 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 528:
| 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.
 
== 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}}
|}


== JI approximation ==
=== 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.


=== 15-odd-limit interval mappings ===
=== Quartertone notation ===
The following table shows how [[15-odd-limit intervals]] are represented in 27edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
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.
{| class="wikitable center-all mw-collapsible mw-collapsed"
 
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp4-szg}}
 
=== Kite's ups and downs notation ===
27edo can also be notated with [[Kite's ups and downs notation|Kite's 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.
{{Ups and downs sharpness}}
 
=== 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"
|-
|-
! 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]]
|-
| 16
| 711.1
| G
| major 5th
| [[3/2]]
|-
|-
| [[13/9]], [[18/13]]
| 17
| 14.395
| 755.6
| 32.4
| 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]]
|-
|-
| ''[[11/6]], [[12/11]]''
| 21
| ''17.304''
| 933.3
| ''38.9''
| 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}}


=== Selected 17-limit intervals ===
== Approximation to JI ==
[[File:27ed2.svg|alt=alt : Your browser has no SVG support.]]
[[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 588: Line 922:
| 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 595: Line 929:
| 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 602: Line 936:
| 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 609: Line 943:
| 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 616: Line 950:
| 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 635: Line 970:
! 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)
| [[9L_9s]]
| [[9L 9s]]
|}
|}
In addition, 27edo can be used as a detempering target for [[ennealimmal]].


=== 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 720: Line 1,057:
| 41.06
| 41.06
| Trigu
| Trigu
| Augmented comma
| Augmented comma, lesser diesis
|-
|-
| 5
| 5
Line 727: Line 1,064:
| 27.66
| 27.66
| Saquadyo
| Saquadyo
| Tetracot comma
| Tetracot comma, minimal diesis
|-
|-
| 5
| 5
Line 828: Line 1,165:
|-
|-
| 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 854: Line 1,191:
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|-
| 13
| [[66/65]]
| {{monzo| 1 1 -1 0 1 -1 }}
| 26.43
| Thulogu
| Winmeanma
|-
|-
| 13
| 13
Line 860: Line 1,204:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap comma
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 868: Line 1,212:
| 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 875: Line 1,226:
| 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" />
 
== Octave stretch or compression ==
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. [[ZPI|106zpi]] is another possible choice.


== 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
* 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
* 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
* 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
* 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
* 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
=== JI chords ===
These are those [[overtone scale]]s 27edo approximates with neat-looking [[horogram]]s, which preserves their [[mapping]] well when rotated:
* [[5afdo]] (rotated): 6 5 5 4 7
* [[6afdo]]: 6 5 5 4 4 3
* [[7afdo]] (rotated): 3 3 5 5 4 4 3
* [[9afdo]] (rotated): 3 3 3 2 3 2 4 4 3
* [[15afdo]] (rotated): 2 2 2 2 2 1 2 1 2 1 2 1 3 2 2
* [[18afdo]]: 2 2 2 2 2 1 2 1 2 1 2 1 1 1 2 1 1 1
* [[21afdo]]: 2 2 1 2 1 2 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1
These are other [[JI]] chords that 27edo approximates well:
; [[12afdo]] without 17/12
* (11 tones)
* JI - 12:13:14:15:16:18:19:20:21:22:23:24
* Included edosteps - 0, 3, 6, 9, 11, 16, 18, 20, 22, 24, 25, 27
; an over-13 chord
* (9 tones)
* JI - 13:14:16:18:19:20:21:23:24:26
* Included edosteps - 0, 3, 8, 13, 15, 17, 19, 22, 24, 27
; an over-14 chord
* (9 tones)
* JI - 14:16:18:19:20:21:23:24:26:28
* Included edosteps - 0, 5, 10, 12, 14, 16, 19, 21, 24, 27
=== Other scales ===
; [[Pinetone]]
* 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
* [[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
; [[Superpyth]]
* 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
; [[Tonality diamond]]s
* 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


Direct sunlight (this is its original/default tuning; subset of [[Sensi]][19])
; [[5-limit]] scales:
* 44.444
* 5-limit / pental double harmonic  hexatonic (Augmented[6] [[4M]]): 2 7 2 7 7 2, 7 7 2 2 7 2
* 133.333
* 5-limit / pental tetrachordal major: 5 4 2 5 5 4 2
* 488.889
* 5-limit / pental tetrachordal minor: 5 2 4 5 5 2 4
* 711.111
* 5-limit / pental melodic minor: 5 2 4 5 4 5 2
* 755.555
* 5-limit / pental harmonic minor: 5 2 4 5 2 7 2
* 1155.555
* 5-limit / pental harmonic major: 5 4 2 5 2 7 2
* 1200.000
* [[SNS (2/1, 3/2, 5/4)-7|5-limit / pental double harmonic major]]: 2 7 2 5 2 7 2
* 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]])


Hypersakura (this is its original/default tuning; subset of Sensi[19])
; Miscellaneous
* 44.444
* [[Blackdye]] / [[syntonic dipentatonic]] (superset of [[Zarlino]]): 1 4 2 4 1 4 2 4 1 4
* 488.889
* [[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
* 711.111
* enharmonic trichord octave species: 9 2 5 9 2, 2 9 5 2 9
* 755.555
* 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
* 1200.000
* [[Zarlino]] / Ptolemy diatonic, "just" major: 5 4 2 5 4 5 2
* "Just" minor (inverse of "just" major): 5 2 4 5 2 5 49
* Direct sunlight{{idio}} (original/default tuning; subset of [[Sensi]][19]): 1 2 8 5 1 9 1
* Hypersakura{{idio}} (original/default tuning; subset of Sensi[19]): 1 10 5 1 10
* [[Maeve Gutierrez#Gutierrez wisp scale|Gutierrez wisp scale]]{{idio}} ''(scale's [[period]] is [[nonoctave]])''
* [[Maeve Gutierrez#Will-o-wisps' scale|Lambeth will-o-wisps' scale]]{{idio}} ''(scale's [[period]] is [[nonoctave]])''
* [[User:BudjarnLambeth/Augene18 subsets in 97ed12]]
 
== 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 ==
{{Catrel| 27edo tracks }}
{{Catrel| 27edo tracks }}
=== Modern renderings ===
; {{W|Scott Joplin}}
* [https://www.youtube.com/shorts/5vRudUCuyqc ''Maple Leaf Rag''] (1899) – arranged with syntonic chroma adjustment for harpsichord and rendered by Claudi Meneghin (2025)
=== 21st century===
; [[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 903: Line 1,404:


; [[Gregoire Blanc]]
; [[Gregoire Blanc]]
* [https://youtu.be/a4-JhcaZSUs?feature=shared ''A microtonal teatime jam''] (2023)
* [https://www.youtube.com/watch?v=a4-JhcaZSUs ''A microtonal teatime jam''] (2023)


; [[Brendan Byrnes]]
; [[Brendan Byrnes]]
* [https://youtu.be/sWaqlAgSWcc ''Sunspots''] (2022)
* [https://www.youtube.com/watch?v=sWaqlAgSWcc ''Sunspots''] (2022)
* ''27 EDO Etude'' (2022)
** [https://brendanbyrnes.bandcamp.com/track/27-edo-etude on Bandcamp]
** [https://m.youtube.com/watch?v=Lml2cfJW9QI on YouTube] (with sheet music)
* [https://www.youtube.com/watch?v=lywpWPBYQi0 ''Istril Bloom''] (2025)
 
; [[Flora Canou]]
* [https://soundcloud.com/floracanou/prelude-the-triad-challenge?in=floracanou/sets/totmc-suite "Prelude: the Triad Challenge"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in superpyth, 70ed6 tuning


; [[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)
* [https://www.youtube.com/shorts/FSPUebavRCQ ''27edo waltz''] (2025)
* [https://www.youtube.com/shorts/izpEen38Sps ''27edo improv''] (2025)
* ''Flies Control My Pain - 27edo'' (2026)
** [https://www.youtube.com/shorts/sKnjDPEOQtc <nowiki>[short 1]</nowiki>] (using [[tetracot]] Lumatone mapping)
** [https://www.youtube.com/shorts/QEebNJkcIlE <nowiki>[short 2]</nowiki>] (using [[Starling_temperaments#Kumonga|kumonga]] Lumatone mapping)
; [[Francium]]
* [https://www.youtube.com/watch?v=3Ty3FpmAdGA ''Happy Birthday in 27edo''] (2025)
* "Router-Pseudoscientist" from ''TOTMC 2025'' (2025) – [https://open.spotify.com/track/5qrXYuhz3XOEaUyFvP4ldp Spotify] | [https://francium223.bandcamp.com/track/router-pseudoscientist Bandcamp] | [https://www.youtube.com/watch?v=Wfg2gWW9qZg YouTube]
* [https://www.youtube.com/watch?v=hY0zo6MqQtU ''Waltz No. 11 in A flat major''] (2026)
* [https://www.youtube.com/watch?v=wY43YLa17s4 ''Plane Sonatina No. 4''] (2026)
; [[groundfault]]
* From ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/album/a-new-dusk Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo YouTube]
** "Back Stalk"
** "Superior Intermedial" – in part, the rest being in 31edo
** "Revelation of Your Forever"
* "Sakura Blade Minivan", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/sakura-blade-minivan-27-35edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1436 YouTube (23:56–27:58)] – in part, the rest being in 35edo


; [[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 918: Line 1,444:
; [[Peter Kosmorsky]]
; [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=7QcwKlK6z4c ''miniature prelude and fugue''] (2011)
* [https://www.youtube.com/watch?v=7QcwKlK6z4c ''miniature prelude and fugue''] (2011)
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=JrpcIkElKQc ''Will-O-Wisps''] (2025) – uses his "will-o-wisps' scale"{{idio}} tuned to 27edo


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [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)
* [https://www.youtube.com/watch?v=ntnFso-3T_I ''Chaconne in 27edo, for Baroque Quartet''] (2025)


; [[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 929: Line 1,459:


; [[Dustin Schallert]]
; [[Dustin Schallert]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 ''Tetracot Perc-Sitar'']{{dead link}} (on [https://soundcloud.com/dustin-schallert/tetracot-perc-sitar SoundCloud]){{dead link}}
* [https://web.archive.org/web/20201127015111/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 ''Tetracot Perc-Sitar'']
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 ''Tetracot Jam'']{{dead link}} (on [https://soundcloud.com/dustin-schallert/tetracot-jam SoundCloud]){{dead link}}
* [https://web.archive.org/web/20201129105050/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 ''Tetracot Jam'']
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3 ''Tetracot Pump'']{{dead link}} (on [https://soundcloud.com/dustin-schallert/tetracot-pump SoundCloud]){{dead link}}
* [https://web.archive.org/web/20201127012230/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3 ''Tetracot Pump''] – all in modus, 27edo tuning
* [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}}


Line 938: Line 1,468:


; [[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
 
; [[The Evil Doings Of An Intergalactic Skeleton]]
* [https://soundcloud.com/unfaced-bones/the-taste-of-pure-saccharin-27edo ''the taste of pure saccharine''] (2025)


; [[User:Tristanbay|Tristan Bay]]
; [[Tristan Bay]]
*[https://youtu.be/R30aRbNtoIY ''Pitchblende''] (2023)
* [https://www.youtube.com/watch?v=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]
* [https://web.archive.org/web/20231121072342/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:Listen]]
[[Category:Listen]]
[[Category:Augmented]]
[[Category:Sensi]]
[[Category:Superpyth]]
[[Category:Tetracot]]
[[Category:Twentuning]]
[[Category:Twentuning]]
Retrieved from "https://en.xen.wiki/w/27edo"