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==Theory==
== Theory ==


If octaves are kept pure, 27edo divides the [[Octave|octave]] in 27 equal parts each exactly 44.444... [[cent|cent]]s in size. However, 27 is a prime candidate for [[octave_shrinking|octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5/4|third]], [[3/2|fifth]] and [[7/4|7/4]] sharply.
If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444... [[cent|cents]] in size. However, 27 is a prime candidate for [[Octave shrinking|octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5/4|third]], [[3/2|fifth]] and [[7/4]] sharply.


Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo|12edo]], sharp 13 2/3 cents. The result is that [[6/5|6/5]], [[7/5|7/5]] and especially [[7/6|7/6]] are all tuned more accurately than this.
Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13 2/3 cents. The result is that [[6/5]], [[7/5]] and especially [[7/6]] are all tuned more accurately than this.


27edo, with its 400 cent major third, tempers out the [[diesis|diesis]] of 128/125, and also the [[Septimal_comma|septimal comma]], 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo|22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.
27edo, with its 400 cent major third, tempers out the [[diesis]] of 128/125, and also the [[septimal comma]], 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.


Though the [[7-limit|7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent|consistent]]ly and distinctly--that is, everything in the [[7-limit diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament.
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|consistently]] and distinctly – that is, everything in the [[7-limit diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament.


Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[Harmonic_Entropy|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.
Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[Harmonic Entropy|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 very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 EDOsteps, just one EDOstep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it's 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.
The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 EDOsteps, just one EDOstep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it's 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.


==Intervals==
== Intervals ==


{| class="wikitable"
{| class="wikitable center-all right-2 left-3"
|-
|-
! style="text-align:center;" | Degree
! #
! style="text-align:center;" | Cents
! Cents
! colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|Ups and Downs Notation]]
! Approximate Ratios*
! style="text-align:center;" | Solfege
! colspan="3" | [[Ups and Downs Notation]]
! style="text-align:center;" | Approximate
! Solfege
Ratios*
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0.00
| 0.00
| style="text-align:center;" | P1
| 1/1
| style="text-align:center;" | perfect unison
| P1
| style="text-align:center;" | D
| perfect unison
| style="text-align:center;" | do
| D
| style="text-align:center;" | 1/1
| do
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 44.44
| 44.44
| style="text-align:center;" | ^1, m2
| 36/35, 49/48, 50/49
| style="text-align:center;" | up-unison, minor 2nd
| ^1, m2
| style="text-align:center;" | ^D, Eb
| up-unison, minor 2nd
| style="text-align:center;" | di
| ^D, Eb
| style="text-align:center;" | 36/35, 49/48, 50/49
| di
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 88.89
| 88.89
| style="text-align:center;" | ^m2
| 16/15, 21/20, 25/24
| style="text-align:center;" | upminor 2nd
| ^m2
| style="text-align:center;" | ^Eb
| upminor 2nd
| style="text-align:center;" | ra
| ^Eb
| style="text-align:center;" | 16/15, 21/20, 25/24
| ra
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 133.33
| 133.33
| style="text-align:center;" | ~2
| 14/13, 13/12
| style="text-align:center;" | mid 2nd
| ~2
| style="text-align:center;" | vvE
| mid 2nd
| style="text-align:center;" | ru
| vvE
| style="text-align:center;" | 14/13, 13/12
| ru
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 177.78
| 177.78
| style="text-align:center;" | vM2
| 10/9
| style="text-align:center;" | downmajor 2nd
| vM2
| style="text-align:center;" | vE
| downmajor 2nd
| style="text-align:center;" | reh
| vE
| style="text-align:center;" | 10/9
| reh
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 222.22
| 222.22
| style="text-align:center;" | M2
| 8/7, 9/8
| style="text-align:center;" | major 2nd
| M2
| style="text-align:center;" | E
| major 2nd
| style="text-align:center;" | re
| E
| style="text-align:center;" | 8/7, 9/8
| re
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 266.67
| 266.67
| style="text-align:center;" | m3
| 7/6
| style="text-align:center;" | minor 3rd
| m3
| style="text-align:center;" | F
| minor 3rd
| style="text-align:center;" | ma
| F
| style="text-align:center;" | 7/6
| ma
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 311.11
| 311.11
| style="text-align:center;" | ^m3
| 6/5
| style="text-align:center;" | upminor 3rd
| ^m3
| style="text-align:center;" | ^F
| upminor 3rd
| style="text-align:center;" | me
| ^F
| style="text-align:center;" | 6/5
| me
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 355.56
| 355.56
| style="text-align:center;" | ~3
| 16/13
| style="text-align:center;" | mid 3rd
| ~3
| style="text-align:center;" | ^^F
| mid 3rd
| style="text-align:center;" | mu
| ^^F
| style="text-align:center;" | 16/13
| mu
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 400
| 400.00
| style="text-align:center;" | vM3
| 5/4
| style="text-align:center;" | downmajor 3rd
| vM3
| style="text-align:center;" | vF#
| downmajor 3rd
| style="text-align:center;" | mi
| vF#
| style="text-align:center;" | 5/4
| mi
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 444.44
| 444.44
| style="text-align:center;" | M3
| 9/7, 13/10
| style="text-align:center;" | major 3rd
| M3
| style="text-align:center;" | F#
| major 3rd
| style="text-align:center;" | mo
| F#
| style="text-align:center;" | 9/7, 13/10
| mo
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 488.89
| 488.89
| style="text-align:center;" | P4
| 4/3
| style="text-align:center;" | perfect 4th
| P4
| style="text-align:center;" | G
| perfect 4th
| style="text-align:center;" | fa
| G
| style="text-align:center;" | 4/3
| fa
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 533.33
| 533.33
| style="text-align:center;" | ^4
| 49/36, 48/35
| style="text-align:center;" | up 4th
| ^4
| style="text-align:center;" | ^G
| up 4th
| style="text-align:center;" | fih
| ^G
| style="text-align:center;" | 49/36, 48/35
| fih
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 577.78
| 577.78
| style="text-align:center;" | ~4, vd5
| 7/5, 18/13
| style="text-align:center;" | mid 4th, updim 5th
| ~4, vd5
| style="text-align:center;" | ^^G, ^Ab
| mid 4th, updim 5th
| style="text-align:center;" | fi
| ^^G, ^Ab
| style="text-align:center;" | 7/5, 18/13
| fi
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 622.22
| 622.22
| style="text-align:center;" | vA4, ~5
| 10/7, 13/9
| style="text-align:center;" | downaug 4th, mid 5th
| vA4, ~5
| style="text-align:center;" | vG#, vvA
| downaug 4th, mid 5th
| style="text-align:center;" | se
| vG#, vvA
| style="text-align:center;" | 10/7, 13/9
| se
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 666.67
| 666.67
| style="text-align:center;" | v5
| 72/49, 35/24
| style="text-align:center;" | down fifth
| v5
| style="text-align:center;" | vA
| down fifth
| style="text-align:center;" | sih
| vA
| style="text-align:center;" | 72/49, 35/24
| sih
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 711.11
| 711.11
| style="text-align:center;" | P5
| 3/2
| style="text-align:center;" | perfect 5th
| P5
| style="text-align:center;" | A
| perfect 5th
| style="text-align:center;" | so/sol
| A
| style="text-align:center;" | 3/2
| so/sol
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | 755.56
| 755.56
| style="text-align:center;" | m6
| 14/9, 20/13
| style="text-align:center;" | minor 6th
| m6
| style="text-align:center;" | Bb
| minor 6th
| style="text-align:center;" | lo
| Bb
| style="text-align:center;" | 14/9, 20/13
| lo
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 800
| 800.00
| style="text-align:center;" | ^m6
| 8/5
| style="text-align:center;" | upminor 6th
| ^m6
| style="text-align:center;" | ^Bb
| upminor 6th
| style="text-align:center;" | le
| ^Bb
| style="text-align:center;" | 8/5
| le
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | 844.44
| 844.44
| style="text-align:center;" | ~6
| 13/8
| style="text-align:center;" | mid 6th
| ~6
| style="text-align:center;" | vvB
| mid 6th
| style="text-align:center;" | lu
| vvB
| style="text-align:center;" | 13/8
| lu
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" | 888.89
| 888.89
| style="text-align:center;" | vM6
| 5/3
| style="text-align:center;" | downmajor 6th
| vM6
| style="text-align:center;" | vB
| downmajor 6th
| style="text-align:center;" | la
| vB
| style="text-align:center;" | 5/3
| la
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 933.33
| 933.33
| style="text-align:center;" | M6
| 12/7
| style="text-align:center;" | major 6th
| M6
| style="text-align:center;" | B
| major 6th
| style="text-align:center;" | li
| B
| style="text-align:center;" | 12/7
| li
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:center;" | 977.78
| 977.78
| style="text-align:center;" | m7
| 7/4, 16/9
| style="text-align:center;" | minor 7th
| m7
| style="text-align:center;" | C
| minor 7th
| style="text-align:center;" | ta
| C
| style="text-align:center;" | 7/4, 16/9
| ta
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:center;" | 1022.22
| 1022.22
| style="text-align:center;" | ^m7
| 9/5
| style="text-align:center;" | upminor 7th
| ^m7
| style="text-align:center;" | ^C
| upminor 7th
| style="text-align:center;" | te
| ^C
| style="text-align:center;" | 9/5
| te
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" | 1066.67
| 1066.67
| style="text-align:center;" | ~7
| 13/7, 24/13
| style="text-align:center;" | mid 7th
| ~7
| style="text-align:center;" | ^^C
| mid 7th
| style="text-align:center;" | tu
| ^^C
| style="text-align:center;" | 13/7, 24/13
| tu
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:center;" | 1111.11
| 1111.11
| style="text-align:center;" | vM7
| 40/21
| style="text-align:center;" | downmajor 7th
| vM7
| style="text-align:center;" | vC#
| downmajor 7th
| style="text-align:center;" | ti
| vC#
| style="text-align:center;" | 40/21
| ti
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:center;" | 1155.56
| 1155.56
| style="text-align:center;" | M7
| 35/18, 96/49, 49/25
| style="text-align:center;" | major 7th
| M7
| style="text-align:center;" | C#
| major 7th
| style="text-align:center;" | da
| C#
| style="text-align:center;" | 35/18, 96/49, 49/25
| da
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:center;" | 1200
| 1200.00
| style="text-align:center;" | P8
| 2/1
| style="text-align:center;" | 8ve
| P8
| style="text-align:center;" | D
| 8ve
| style="text-align:center;" | do
| D
| style="text-align:center;" | 2/1
| do
|}
|}
*based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.
<nowiki/>* based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | quality
! quality
! | [[Kite's color notation|color]]
! [[Color notation|color]]
! | monzo format
! monzo format
! | examples
! examples
|-
|-
| style="text-align:center;" | minor
| minor
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | {a, b, 0, 1}
| {a, b, 0, 1}
| style="text-align:center;" | 7/6, 7/4
| 7/6, 7/4
|-
|-
| style="text-align:center;" | "
| "
| style="text-align:center;" | fourthward wa
| fourthward wa
| style="text-align:center;" | {a, b}, b &lt; -1
| {a, b}, b &lt; -1
| style="text-align:center;" | 32/27, 16/9
| 32/27, 16/9
|-
|-
| style="text-align:center;" | upminor
| upminor
| style="text-align:center;" | gu
| gu
| style="text-align:center;" | {a, b, -1}
| {a, b, -1}
| style="text-align:center;" | 6/5, 9/5
| 6/5, 9/5
|-
|-
| style="text-align:center;" | mid
| mid
| style="text-align:center;" | tho
| tho
| style="text-align:center;" | {a, b, 0, 0, 0, 1}
| {a, b, 0, 0, 0, 1}
| style="text-align:center;" | 13/12, 13/8
| 13/12, 13/8
|-
|-
| style="text-align:center;" | "
| "
| style="text-align:center;" | thu
| thu
| style="text-align:center;" | {a, b, 0, 0, 0, -1}
| {a, b, 0, 0, 0, -1}
| style="text-align:center;" | 16/13, 24/13
| 16/13, 24/13
|-
|-
| style="text-align:center;" | downmajor
| downmajor
| style="text-align:center;" | yo
| yo
| style="text-align:center;" | {a, b, 1}
| {a, b, 1}
| style="text-align:center;" | 5/4, 5/3
| 5/4, 5/3
|-
|-
| style="text-align:center;" | major
| major
| style="text-align:center;" | fifthward wa
| fifthward wa
| style="text-align:center;" | {a, b}, b &gt; 1
| {a, b}, b &gt; 1
| style="text-align:center;" | 9/8, 27/16
| 9/8, 27/16
|-
|-
| style="text-align:center;" | "
| "
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | {a, b, 0, -1}
| {a, b, 0, -1}
| style="text-align:center;" | 9/7, 12/7
| 9/7, 12/7
|}
|}
All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:
All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | [[Kite's 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
|-
|-
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | 6:7:9
| 6:7:9
| style="text-align:center;" | 0-6-16
| 0-6-16
| style="text-align:center;" | C Eb G
| C Eb G
| style="text-align:center;" | Cm
| Cm
| style="text-align:center;" | C minor
| C minor
|-
|-
| style="text-align:center;" | gu
| gu
| style="text-align:center;" | 10:12:15
| 10:12:15
| style="text-align:center;" | 0-7-16
| 0-7-16
| style="text-align:center;" | C ^Eb G
| C ^Eb G
| style="text-align:center;" | C^m
| C^m
| style="text-align:center;" | C upminor
| C upminor
|-
|-
| style="text-align:center;" | ilo
| ilo
| style="text-align:center;" | 18:22:27
| 18:22:27
| style="text-align:center;" | 0-8-16
| 0-8-16
| style="text-align:center;" | C vvE G
| C vvE G
| style="text-align:center;" | C~
| C~
| style="text-align:center;" | C mid
| C mid
|-
|-
| style="text-align:center;" | yo
| yo
| style="text-align:center;" | 4:5:6
| 4:5:6
| style="text-align:center;" | 0-9-16
| 0-9-16
| style="text-align:center;" | C vE G
| C vE G
| style="text-align:center;" | Cv
| Cv
| style="text-align:center;" | C downmajor or C down
| C downmajor or C down
|-
|-
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | 14:18:21
| 14:18:21
| style="text-align:center;" | 0-10-16
| 0-10-16
| style="text-align:center;" | C E G
| C E G
| style="text-align:center;" | C
| C
| style="text-align:center;" | C major or C
| C major or C
|}
|}
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. See also the [[22edo|22edo]] page.
For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. See also the [[22edo]] page.


==Rank two temperaments==
== Rank two temperaments ==
[[List_of_27edo_rank_two_temperaments_by_badness|List of 27edo rank two temperaments by badness]]


[[List_of_edo-distinct_27e_rank_two_temperaments|List of edo-distinct 27e rank two temperaments]]
[[List of 27edo rank two temperaments by badness]]
 
[[List of edo-distinct 27e rank two temperaments]]


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Periods
! Periods<br>per octave
 
! Generator
per octave
! Temperaments
! | Generator
! | Temperaments
|-
|-
| | 1
| 1
| | 1\27
| 1\27
| | [[Quartonic|Quartonic]]/Quarto
| [[Quartonic]]/Quarto
|-
|-
| | 1
| 1
| | 2\27
| 2\27
| | [[Octacot|Octacot]]/Octocat
| [[Octacot]]/Octocat
|-
|-
| | 1
| 1
| | 4\27
| 4\27
| | [[Tetracot|Tetracot]]/Modus/Wollemia
| [[Tetracot]]/Modus/Wollemia
|-
|-
| | 1
| 1
| | 5\27
| 5\27
| | [[Machine|Machine]]/Kumonga
| [[Machine]]/Kumonga
|-
|-
| | 1
| 1
| | 7\27
| 7\27
| | [[Myna|Myna]]/Coleto/Minah
| [[Myna]]/Coleto/Minah
|-
|-
| | 1
| 1
| | 8\27
| 8\27
| | [[Beatles|Beatles]]/Ringo
| [[Beatles]]/Ringo
|-
|-
| | 1
| 1
| | 10\27
| 10\27
| | [[Sensi|Sensi]]/Sensis
| [[Sensi]]/Sensis
|-
|-
| | 1
| 1
| | 11\27
| 11\27
| | [[Superpyth|Superpyth]]
| [[Superpyth]]
|-
|-
| | 1
| 1
| | 13\27
| 13\27
| | Fervor
| Fervor
|-
|-
| | 3
| 3
| | 1\27
| 1\27
| | [[Semiaug|Semiaug]]/Hemiaug
| [[Semiaug]]/Hemiaug
|-
|-
| | 3
| 3
| | 2\27
| 2\27
| | [[augmented|Augmented]]/[[Augene|Augene]]/Ogene
| [[Augmented]]/[[Augene]]/Ogene
|-
|-
| | 3
| 3
| | 4\27
| 4\27
| | Oodako
| Oodako
|-
|-
| | 9
| 9
| | 1\27
| 1\27
| | Terrible version of [[Ennealimmal|Ennealimmal]]
| Terrible version of [[Ennealimmal]]<br>/Niner
 
/ Niner
|}
|}


==Commas==
== Commas ==
27 EDO tempers out the following [[commas]]. (Note: This assumes the val &lt; 27 43 63 76 93 100 |.)
27 EDO tempers out the following [[commas]]. (Note: This assumes the val {{val|27 43 63 76 93 100}}.)


{| class="wikitable"
{| class="wikitable center-all left-2 right-3"
|-
|-
! | [[Ratio]]
! | [[Ratio]]
! | [[Monzo]]
! | [[Monzo]]
! | Cents
! | Cents
![[Color notation/Temperament Names|Color Name]]
! [[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 1
! | Name 2
! | Name 2
! | Name 3
! | Name 3
|-
|-
| style="text-align:center;" | 128/125
| 128/125
| |<nowiki> | 7 0 -3 </nowiki>&gt;
| |<nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:right;" | 41.06
| 41.06
| style="text-align:center;" |Trigu
| Trigu
| style="text-align:center;" | Diesis
| Diesis
| style="text-align:center;" | Augmented Comma
| Augmented Comma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 20000/19683
| 20000/19683
| |<nowiki> | 5 -9 4 </nowiki>&gt;
| |<nowiki> | 5 -9 4 </nowiki>&gt;
| style="text-align:right;" | 27.66
| 27.66
| style="text-align:center;" |Saquadyo
| Saquadyo
| style="text-align:center;" | Minimal Diesis
| Minimal Diesis
| style="text-align:center;" | Tetracot Comma
| Tetracot Comma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 78732/78125
| 78732/78125
| |<nowiki> | 2 9 -7 </nowiki>&gt;
| |<nowiki> | 2 9 -7 </nowiki>&gt;
| style="text-align:right;" | 13.40
| 13.40
| style="text-align:center;" |Sepgu
| Sepgu
| style="text-align:center;" | Medium Semicomma
| Medium Semicomma
| style="text-align:center;" | Sensipent Comma
| Sensipent Comma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 4711802/4709457
| 4711802/4709457
| |<nowiki> | 1 -27 18 </nowiki>&gt;
| |<nowiki> | 1 -27 18 </nowiki>&gt;
| style="text-align:right;" | 0.86
| 0.86
| style="text-align:center;" |Satritribiyo
| Satritribiyo
| style="text-align:center;" | Ennealimma
| Ennealimma
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 686/675
| 686/675
| |<nowiki> | 1 -3 -2 3 </nowiki>&gt;
| |<nowiki> | 1 -3 -2 3 </nowiki>&gt;
| style="text-align:right;" | 27.99
| 27.99
| style="text-align:center;" |Trizo-agugu
| Trizo-agugu
| style="text-align:center;" | Senga
| Senga
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 64/63
| 64/63
| |<nowiki> | 6 -2 0 -1 </nowiki>&gt;
| |<nowiki> | 6 -2 0 -1 </nowiki>&gt;
| style="text-align:right;" | 27.26
| 27.26
| style="text-align:center;" |Ru
| Ru
| style="text-align:center;" | Septimal Comma
| Septimal Comma
| style="text-align:center;" | Archytas' Comma
| Archytas' Comma
| style="text-align:center;" | Leipziger Komma
| Leipziger Komma
|-
|-
| style="text-align:center;" | 50421/50000
| 50421/50000
| |<nowiki> | -4 1 -5 5 </nowiki>&gt;
| |<nowiki> | -4 1 -5 5 </nowiki>&gt;
| style="text-align:right;" | 14.52
| 14.52
| style="text-align:center;" |Quinzogu
| Quinzogu
| style="text-align:center;" | Trimyna
| Trimyna
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 245/243
| 245/243
| |<nowiki> | 0 -5 1 2 </nowiki>&gt;
| |<nowiki> | 0 -5 1 2 </nowiki>&gt;
| style="text-align:right;" | 14.19
| 14.19
| style="text-align:center;" |Zozoyo
| Zozoyo
| style="text-align:center;" | Sensamagic
| Sensamagic
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 126/125
| 126/125
| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| style="text-align:right;" | 13.79
| 13.79
| style="text-align:center;" |Zotrigu
| Zotrigu
| style="text-align:center;" | Septimal Semicomma
| Septimal Semicomma
| style="text-align:center;" | Starling Comma
| Starling Comma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 4000/3969
| 4000/3969
| |<nowiki> | 5 -4 3 -2 </nowiki>&gt;
| |<nowiki> | 5 -4 3 -2 </nowiki>&gt;
| style="text-align:right;" | 13.47
| 13.47
| style="text-align:center;" |Rurutriyo
| Rurutriyo
| style="text-align:center;" | Octagar
| Octagar
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 1728/1715
| 1728/1715
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| style="text-align:right;" | 13.07
| 13.07
| style="text-align:center;" |Triru-agu
| Triru-agu
| style="text-align:center;" | Orwellisma
| Orwellisma
| style="text-align:center;" | Orwell Comma
| Orwell Comma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 420175/419904
| 420175/419904
| |<nowiki> | -6 -8 2 5 </nowiki>&gt;
| |<nowiki> | -6 -8 2 5 </nowiki>&gt;
| style="text-align:right;" | 1.12
| 1.12
| style="text-align:center;" |Quinzo-ayoyo
| Quinzo-ayoyo
| style="text-align:center;" | Wizma
| Wizma
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 2401/2400
| 2401/2400
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| style="text-align:right;" | 0.72
| 0.72
| style="text-align:center;" |Bizozogu
| Bizozogu
| style="text-align:center;" | Breedsma
| Breedsma
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 4375/4374
| 4375/4374
| |<nowiki> | -1 -7 4 1 </nowiki>&gt;
| |<nowiki> | -1 -7 4 1 </nowiki>&gt;
| style="text-align:right;" | 0.40
| 0.40
| style="text-align:center;" |Zoquadyo
| Zoquadyo
| style="text-align:center;" | Ragisma
| Ragisma
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 250047/250000
| 250047/250000
| |<nowiki> | -4 6 -6 3 </nowiki>&gt;
| |<nowiki> | -4 6 -6 3 </nowiki>&gt;
| style="text-align:right;" | 0.33
| 0.33
| style="text-align:center;" |Trizogugu
| Trizogugu
| style="text-align:center;" | Landscape Comma
| Landscape Comma
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 99/98
| 99/98
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| style="text-align:right;" | 17.58
| 17.58
| style="text-align:center;" |Loruru
| Loruru
| style="text-align:center;" | Mothwellsma
| Mothwellsma
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 896/891
| 896/891
| |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
| |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
| style="text-align:right;" | 9.69
| 9.69
| style="text-align:center;" |Saluzo
| Saluzo
| style="text-align:center;" | Pentacircle
| Pentacircle
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 385/384
| 385/384
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| style="text-align:right;" | 4.50
| 4.50
| style="text-align:center;" |Lozoyo
| Lozoyo
| style="text-align:center;" | Keenanisma
| Keenanisma
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 91/90
| 91/90
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| style="text-align:right;" | 19.13
| 19.13
| style="text-align:center;" |Thozogu
| Thozogu
| style="text-align:center;" | Superleap
| Superleap
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|}
|}


==Music==
== Music ==


* [http://www.archive.org/details/MusicForYourEars Music For Your Ears] ([http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play]) by [[Gene Ward Smith]] The central portion is in 27edo, the rest in [[46edo]].
* [http://www.archive.org/details/MusicForYourEars Music For Your Ears] ([http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play]) by [[Gene Ward Smith]] The central portion is in 27edo, the rest in [[46edo]].

Revision as of 00:54, 4 July 2020

Theory

If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444... cents in size. However, 27 is a prime candidate for octave shrinking, and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the third, fifth and 7/4 sharply.

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as 12edo, sharp 13 2/3 cents. The result is that 6/5, 7/5 and especially 7/6 are all tuned more accurately than this.

27edo, with its 400 cent major third, tempers out the diesis of 128/125, and also the septimal comma, 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with 22edo tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.

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 consistently and distinctly – that is, everything in the 7-limit diamond is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament.

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 very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 EDOsteps, just one EDOstep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it's 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.

Intervals

# Cents Approximate Ratios* Ups and Downs Notation Solfege
0 0.00 1/1 P1 perfect unison D do
1 44.44 36/35, 49/48, 50/49 ^1, m2 up-unison, minor 2nd ^D, Eb di
2 88.89 16/15, 21/20, 25/24 ^m2 upminor 2nd ^Eb ra
3 133.33 14/13, 13/12 ~2 mid 2nd vvE ru
4 177.78 10/9 vM2 downmajor 2nd vE reh
5 222.22 8/7, 9/8 M2 major 2nd E re
6 266.67 7/6 m3 minor 3rd F ma
7 311.11 6/5 ^m3 upminor 3rd ^F me
8 355.56 16/13 ~3 mid 3rd ^^F mu
9 400.00 5/4 vM3 downmajor 3rd vF# mi
10 444.44 9/7, 13/10 M3 major 3rd F# mo
11 488.89 4/3 P4 perfect 4th G fa
12 533.33 49/36, 48/35 ^4 up 4th ^G fih
13 577.78 7/5, 18/13 ~4, vd5 mid 4th, updim 5th ^^G, ^Ab fi
14 622.22 10/7, 13/9 vA4, ~5 downaug 4th, mid 5th vG#, vvA se
15 666.67 72/49, 35/24 v5 down fifth vA sih
16 711.11 3/2 P5 perfect 5th A so/sol
17 755.56 14/9, 20/13 m6 minor 6th Bb lo
18 800.00 8/5 ^m6 upminor 6th ^Bb le
19 844.44 13/8 ~6 mid 6th vvB lu
20 888.89 5/3 vM6 downmajor 6th vB la
21 933.33 12/7 M6 major 6th B li
22 977.78 7/4, 16/9 m7 minor 7th C ta
23 1022.22 9/5 ^m7 upminor 7th ^C te
24 1066.67 13/7, 24/13 ~7 mid 7th ^^C tu
25 1111.11 40/21 vM7 downmajor 7th vC# ti
26 1155.56 35/18, 96/49, 49/25 M7 major 7th C# da
27 1200.00 2/1 P8 8ve D do

* based on treating 27-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.

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

quality color 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 Eb G Cm C minor
gu 10:12:15 0-7-16 C ^Eb G C^m C upminor
ilo 18:22:27 0-8-16 C vvE G C~ C mid
yo 4:5:6 0-9-16 C vE 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.

Rank two temperaments

List of 27edo rank two temperaments by badness

List of edo-distinct 27e rank two temperaments

Periods
per octave 		Generator Temperaments
1 1\27 Quartonic/Quarto
1 2\27 Octacot/Octocat
1 4\27 Tetracot/Modus/Wollemia
1 5\27 Machine/Kumonga
1 7\27 Myna/Coleto/Minah
1 8\27 Beatles/Ringo
1 10\27 Sensi/Sensis
1 11\27 Superpyth
1 13\27 Fervor
3 1\27 Semiaug/Hemiaug
3 2\27 Augmented/Augene/Ogene
3 4\27 Oodako
9 1\27 Terrible version of Ennealimmal
/Niner

Commas

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

Ratio Monzo Cents Color Name Name 1 Name 2 Name 3
128/125 | 7 0 -3 > 41.06 Trigu Diesis Augmented Comma
20000/19683 | 5 -9 4 > 27.66 Saquadyo Minimal Diesis Tetracot Comma
78732/78125 | 2 9 -7 > 13.40 Sepgu Medium Semicomma Sensipent Comma
4711802/4709457 | 1 -27 18 > 0.86 Satritribiyo Ennealimma
686/675 | 1 -3 -2 3 > 27.99 Trizo-agugu Senga
64/63 | 6 -2 0 -1 > 27.26 Ru Septimal Comma Archytas' Comma Leipziger Komma
50421/50000 | -4 1 -5 5 > 14.52 Quinzogu Trimyna
245/243 | 0 -5 1 2 > 14.19 Zozoyo Sensamagic
126/125 | 1 2 -3 1 > 13.79 Zotrigu Septimal Semicomma Starling Comma
4000/3969 | 5 -4 3 -2 > 13.47 Rurutriyo Octagar
1728/1715 | 6 3 -1 -3 > 13.07 Triru-agu Orwellisma Orwell Comma
420175/419904 | -6 -8 2 5 > 1.12 Quinzo-ayoyo Wizma
2401/2400 | -5 -1 -2 4 > 0.72 Bizozogu Breedsma
4375/4374 | -1 -7 4 1 > 0.40 Zoquadyo Ragisma
250047/250000 | -4 6 -6 3 > 0.33 Trizogugu Landscape Comma
99/98 | -1 2 0 -2 1 > 17.58 Loruru Mothwellsma
896/891 | 7 -4 0 1 -1 > 9.69 Saluzo Pentacircle
385/384 | -7 -1 1 1 1 > 4.50 Lozoyo Keenanisma
91/90 | -1 -2 -1 1 0 1 > 19.13 Thozogu Superleap

Music

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