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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|27}}
{{ED intro}}
 
== Theory ==
== Theory ==
27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, 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 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55 cents, respectively.
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]].


However, assuming just octaves, 27edo's fifth and harmonic seventh are both sharp by nine cents, and the major third is the same 400 cent major third as [[12edo]], 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 [[superpyth|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 [[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.


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


Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|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.
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.


Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''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 [[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.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|27}}
{{Harmonics in equal|27}}
=== 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.
=== Subsets and supersets ===
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-all right-2 left-3"
|-
|-
! &#35;
! #
! Cents
! Cents
! Approximate Ratios<ref group="note">{{sg|2.3.5.7.13.19&nbsp;[[subgroup]]}}</ref>
! 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" | [[Solfege|Solfeges]]
! colspan="2" | [[Solfege]]s
|-
|-
| 0
| 0
| 0.00
| 0.0
| [[1/1]]
| [[1/1]]
| P1
| P1
Line 36: Line 43:
|-
|-
| 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
Line 46: Line 53:
|-
|-
| 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
Line 56: Line 63:
|-
|-
| 3
| 3
| 133.33
| 133.3
| [[15/14]], [[14/13]], [[13/12]]
| [[15/14]], [[14/13]], [[13/12]]
| vA1, ~2
| vA1, ~2
Line 66: Line 73:
|-
|-
| 4
| 4
| 177.78
| 177.8
| [[10/9]]
| [[10/9]]
| A1, vM2
| A1, vM2
Line 76: Line 83:
|-
|-
| 5
| 5
| 222.22
| 222.2
| [[8/7]], [[9/8]]
| [[8/7]], [[9/8]]
| M2
| M2
Line 86: Line 93:
|-
|-
| 6
| 6
| 266.67
| 266.7
| [[7/6]]
| [[7/6]]
| m3
| m3
Line 96: Line 103:
|-
|-
| 7
| 7
| 311.11
| 311.1
| [[6/5]], [[19/16]]
| [[6/5]], [[19/16]]
| ^m3
| ^m3
Line 106: Line 113:
|-
|-
| 8
| 8
| 355.56
| 355.6
| [[16/13]]
| [[16/13]]
| ~3
| ~3
Line 116: Line 123:
|-
|-
| 9
| 9
| 400.00
| 400.0
| [[5/4]], [[24/19]]
| [[5/4]], [[24/19]]
| vM3
| vM3
Line 126: Line 133:
|-
|-
| 10
| 10
| 444.44
| 444.4
| [[9/7]], [[13/10]]
| [[9/7]], [[13/10]]
| M3
| M3
Line 136: Line 143:
|-
|-
| 11
| 11
| 488.89
| 488.9
| [[4/3]]
| [[4/3]]
| P4
| P4
Line 146: Line 153:
|-
|-
| 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
Line 156: Line 163:
|-
|-
| 13
| 13
| 577.78
| 577.8
| [[7/5]], [[18/13]]
| [[7/5]], [[18/13]]
| ~4, ^d5
| ~4, ^d5
Line 166: Line 173:
|-
|-
| 14
| 14
| 622.22
| 622.2
| [[10/7]], [[13/9]]
| [[10/7]], [[13/9]]
| vA4, ~5
| vA4, ~5
Line 176: Line 183:
|-
|-
| 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
Line 186: Line 193:
|-
|-
| 16
| 16
| 711.11
| 711.1
| [[3/2]]
| [[3/2]]
| P5
| P5
Line 196: Line 203:
|-
|-
| 17
| 17
| 755.56
| 755.6
| [[14/9]], [[20/13]]
| [[14/9]], [[20/13]]
| m6
| m6
Line 206: Line 213:
|-
|-
| 18
| 18
| 800.00
| 800.0
| [[8/5]], [[19/12]]
| [[8/5]], [[19/12]]
| ^m6
| ^m6
Line 216: Line 223:
|-
|-
| 19
| 19
| 844.44
| 844.4
| [[13/8]]
| [[13/8]]
| ~6
| ~6
Line 226: Line 233:
|-
|-
| 20
| 20
| 888.89
| 888.9
| [[5/3]], [[32/19]]
| [[5/3]], [[32/19]]
| vM6
| vM6
Line 236: Line 243:
|-
|-
| 21
| 21
| 933.33
| 933.3
| [[12/7]]
| [[12/7]]
| M6
| M6
Line 246: Line 253:
|-
|-
| 22
| 22
| 977.78
| 977.8
| [[7/4]], [[16/9]]
| [[7/4]], [[16/9]]
| m7
| m7
Line 256: Line 263:
|-
|-
| 23
| 23
| 1022.22
| 1022.2
| [[9/5]]
| [[9/5]]
| ^m7
| ^m7
Line 266: Line 273:
|-
|-
| 24
| 24
| 1066.67
| 1066.7
| [[28/15]], [[13/7]], [[24/13]]
| [[13/7]], [[24/13]], [[28/15]]
| ~7
| ~7
| mid 7th
| mid 7th
Line 276: Line 283:
|-
|-
| 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
Line 286: Line 293:
|-
|-
| 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
Line 296: Line 303:
|-
|-
| 27
| 27
| 1200.00
| 1200.0
| 2/1
| [[2/1]]
| P8
| P8
| 8ve
| 8ve
Line 305: Line 312:
| do
| do
|}
|}
<references group="note" />


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 312: 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; &minus;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 358: 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
Line 399: Line 408:
| 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 ==
== Notation ==
Line 405: Line 414:
|+ style="font-size: 105%;" | Circle of fifths in 27edo
|+ style="font-size: 105%;" | Circle of fifths in 27edo
|- style="white-space: nowrap;"
|- style="white-space: nowrap;"
! rowspan="2" | [[Cent]]s !! colspan="6" | Note from C
!Cents
|- style="white-space: nowrap;"
! colspan="2" | Extended<br />Pythagorean<br />notation
! colspan="2" | Standard<br />notation !! colspan="2" | Quarter tone<br />notation
! colspan="2" | Quartertone<br />notation
|-
|-
| 0
| 0.0
| colspan="2" | C
| colspan="2" | C
| colspan="2" | A{{sesquisharp2}}
| colspan="2" | A{{sesquisharp2}}
|-
|-
| 711.11
| 711.1
| colspan="2" | G
| colspan="2" | G
| colspan="2" | E{{sesquisharp2}}
| colspan="2" | E{{sesquisharp2}}
|-
|-
| 222.22
| 222.2
| colspan="2" | D
| colspan="2" | D
| B{{sesquisharp2}} || F{{sesquiflat2}}
| B{{sesquisharp2}}
| F{{sesquiflat2}}
|-
|-
| 933.33
| 933.3
| colspan="2" | A
| colspan="2" | A
| colspan="2" | C{{sesquiflat2}}
| colspan="2" | C{{sesquiflat2}}
|-
|-
| 444.44
| 444.4
| colspan="2" | E
| colspan="2" | E
| colspan="2" | G{{sesquiflat2}}
| colspan="2" | G{{sesquiflat2}}
|-
|-
| 1155.55
| 1155.6
| colspan="2" | B
| colspan="2" | B
| colspan="2" | D{{sesquiflat2}}
| colspan="2" | D{{sesquiflat2}}
|-
|-
| 666.66
| 666.7
| colspan="2" | F&#x266F;
| colspan="2" | F♯
| colspan="2" | A{{sesquiflat2}}
| colspan="2" | A{{sesquiflat2}}
|-
|-
| 177.77
| 177.8
| colspan="2" | C&#x266F;
| colspan="2" | C♯
| colspan="2" | E{{sesquiflat2}}
| colspan="2" | E{{sesquiflat2}}
|-
|-
| 888.88
| 888.9
| colspan="2" | G&#x266F;
| colspan="2" | G♯
| colspan="2" | B{{sesquiflat2}}
| colspan="2" | B{{sesquiflat2}}
|-
|-
| 400
| 400.0
| colspan="2" | D&#x266F;
| colspan="2" | D♯
| colspan="2" | F{{demiflat2}}
| colspan="2" | F{{demiflat2}}
|-
|-
| 1111.11
| 1111.1
| colspan="2" | A&#x266F;
| colspan="2" | A♯
| colspan="2" | C{{demiflat2}}
| colspan="2" | C{{demiflat2}}
|-
|-
| 622.22
| 622.2
| colspan="2" | E&#x266F;
| colspan="2" | E♯
| colspan="2" | G{{demiflat2}}
| colspan="2" | G{{demiflat2}}
|-
|-
| 133.33
| 133.3
| B&#x266F;
| B♯
| F&#x1D12B;
| F𝄫
| colspan="2" | D{{demiflat2}}
| colspan="2" | D{{demiflat2}}
|-
|-
| 844.44
| 844.4
| F&#x1D12A;
| F𝄪
| C&#x1D12B;
| C𝄫
| colspan="2" | A{{demiflat2}}
| colspan="2" | A{{demiflat2}}
|-
|-
| 355.56
| 355.6
| C&#x1D12A;
| C𝄪
| G&#x1D12B;
| G𝄫
| colspan="2" | E{{demiflat2}}
| colspan="2" | E{{demiflat2}}
|-
|-
| 1066.67
| 1066.7
| G&#x1D12A;
| G𝄪
| D&#x1D12B;
| D𝄫
| colspan="2" | B{{demiflat2}}
| colspan="2" | B{{demiflat2}}
|-
|-
| 577.78
| 577.8
| D&#x1D12A;
| D𝄪
| A&#x1D12B;
| A𝄫
| colspan="2" | F{{demisharp2}}
| colspan="2" | F{{demisharp2}}
|-
|-
| 88.89
| 88.9
| A&#x1D12A;
| A𝄪
| E&#x1D12B;
| E𝄫
| colspan="2" | C{{demisharp2}}
| colspan="2" | C{{demisharp2}}
|-
|-
| 800
| 800.0
| E&#x1D12A;
| E𝄪
| B&#x1D12B;
| B𝄫
| colspan="2" | G{{demisharp2}}
| colspan="2" | G{{demisharp2}}
|-
|-
| 311.11
| 311.1
| B&#x1D12A;
| B𝄪
| F&#x266D;
| F♭
| colspan="2" | D{{demisharp2}}
| colspan="2" | D{{demisharp2}}
|-
|-
| 1022.22
| 1022.2
| colspan="2" | C&#x266D;
| colspan="2" | C♭
| colspan="2" | A{{demisharp2}}
| colspan="2" | A{{demisharp2}}
|-
|-
| 533.33
| 533.3
| colspan="2" | G&#x266D;
| colspan="2" | G♭
| colspan="2" | E{{demisharp2}}
| colspan="2" | E{{demisharp2}}
|-
|-
| 44.44
| 44.4
| colspan="2" | D&#x266D;
| colspan="2" | D♭
| colspan="2" | B{{demisharp2}}
| colspan="2" | B{{demisharp2}}
|-
|-
| 755.56
| 755.6
| colspan="2" | A&#x266D;
| colspan="2" | A♭
| colspan="2" | F{{sesquisharp2}}
| colspan="2" | F{{sesquisharp2}}
|-
|-
| 266.67
| 266.7
| colspan="2" | E&#x266D;
| colspan="2" | E♭
| colspan="2" | C{{sesquisharp2}}
| colspan="2" | C{{sesquisharp2}}
|-
|-
| 977.78
| 977.8
| colspan="2" | B&#x266D;
| colspan="2" | B♭
| colspan="2" | G{{sesquisharp2}}
| colspan="2" | G{{sesquisharp2}}
|-
|-
| 488.89
| 488.9
| colspan="2" | F
| colspan="2" | F
| colspan="2" | D{{sesquisharp2}}
| colspan="2" | D{{sesquisharp2}}
|-
|-
| 0
| 0.0
| colspan="2" | C
| colspan="2" | C
| colspan="2" | A{{sesquisharp2}}
| colspan="2" | A{{sesquisharp2}}
|-
|}
|}


The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Helmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. With standard [[circle-of-fifths notation]], a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats.The notes from C to D are C, D&#x266D;, C{{demisharp2}}, D{{demiflat2}}, C&#x266F;, and D, with some ascending intervals appearing to be descending on the staff.
=== 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.


Another notational implication is that, being a Superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C&ndash;E is a 9/7 supermajor third and so the note located one major third above C must be notated as D♯ or E{{naturaldown}}. Conversely, the 6/5 minor third of a 10:12:15 chord is actually reached by a diminished fourth, since (for example) D&ndash;F is a 7/6 subminor third and so the note located one minor third above D must be notated as either G&#x266D; or F{{naturalup}}. The composer can decide for themselves which additional accidental pair is appropriate if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B&#x266D; is not only familiar, though here very exaggerated, to those working with the Pythagorean scale (see [[53edo]]), but also to many classically trained violinists.
=== 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.


{{sharpness-sharp4}}
===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 ===
=== 6L 1s (archeotonic) notation ===
The notation of Tetracot[7]. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma {{nowrap|(L &minus; s)}}, 1 step in this instance, is denoted by &#x266F; and &#x266D;.  
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"
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
Line 557: Line 613:
| 44.4
| 44.4
| C#, Dbbb
| C#, Dbbb
| aug 1sn, double-dim 2nd
| aug 1sn, triple-dim 2nd
| [[40/39]], [[45/44]], [[55/54]], [[81/80]]
| [[40/39]], [[45/44]], [[55/54]], [[81/80]]
|-
|-
Line 563: Line 619:
| 88.9
| 88.9
| Cx, Dbb
| Cx, Dbb
| double-aug 1sn, dim 2nd
| double-aug 1sn, double-dim 2nd
| [[16/15]], [[25/24]]
| [[16/15]], [[25/24]]
|-
|-
Line 569: Line 625:
| 133.3
| 133.3
| Db
| Db
| minor 2nd
| dim 2nd
| [[12/11]], [[13/12]]
| [[12/11]], [[13/12]]
|-
|-
Line 575: Line 631:
| 177.8
| 177.8
| D
| D
| major 2nd
| perfect 2nd
| [[10/9]], [[11/10]]
| [[10/9]], [[11/10]]
|-
|-
Line 689: Line 745:
| 1022.2
| 1022.2
| Bb
| Bb
| minor 7th
| perfect 7th
| [[9/5]], [[20/11]]
| [[9/5]], [[20/11]]
|-
|-
Line 695: Line 751:
| 1066.7
| 1066.7
| B
| B
| major 7th
| aug 7th
| [[11/6]], [[24/13]]
| [[11/6]], [[24/13]]
|-
|-
Line 701: Line 757:
| 1111.1
| 1111.1
| B#, Cbb
| B#, Cbb
| aug 7th, double-dim 8ve
| double-aug 7th, double-dim 8ve
| [[15/8]], [[48/25]]
| [[15/8]], [[48/25]]
|-
|-
Line 707: Line 763:
| 1155.6
| 1155.6
| Bx, Cb
| Bx, Cb
| double-aug 7th, dim 8ve
| triple-aug 7th, dim 8ve
| [[39/20]], [[88/45]], [[108/55]], [[160/81]]
| [[39/20]], [[88/45]], [[108/55]], [[160/81]]
|-
|-
Line 720: Line 776:
== Approximation to JI ==
== Approximation to JI ==
[[File:27ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 27edo]]
[[File:27ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 27edo]]
=== Interval mappings ===
=== Interval mappings ===
{{15-odd-limit|27}}
{{Q-odd-limit intervals|27}}
{{15-odd-limit|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}}
{{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 ==
Line 739: Line 796:
| {{monzo| 43 -27 }}
| {{monzo| 43 -27 }}
| {{mapping| 27 43 }}
| {{mapping| 27 43 }}
| &minus;2.89
| −2.89
| 2.88
| 2.88
| 6.50
| 6.50
Line 746: Line 803:
| 128/125, 20000/19683
| 128/125, 20000/19683
| {{mapping| 27 43 63 }}
| {{mapping| 27 43 63 }}
| &minus;3.88
| −3.88
| 2.74
| 2.74
| 6.19
| 6.19
Line 753: Line 810:
| 64/63, 126/125, 245/243
| 64/63, 126/125, 245/243
| {{mapping| 27 43 63 76 }}
| {{mapping| 27 43 63 76 }}
| &minus;3.70
| −3.71
| 2.39
| 2.39
| 5.40
| 5.40
Line 760: Line 817:
| 64/63, 91/90, 126/125, 169/168
| 64/63, 91/90, 126/125, 169/168
| {{mapping| 27 43 63 76 100 }}
| {{mapping| 27 43 63 76 100 }}
| &minus;3.18
| −3.18
| 2.39
| 2.39
| 5.39
| 5.39
Line 767: Line 824:
| 64/63, 76/75, 91/90, 126/125, 169/168
| 64/63, 76/75, 91/90, 126/125, 169/168
| {{mapping| 27 43 63 76 100 115 }}
| {{mapping| 27 43 63 76 100 115 }}
| &minus;3.18
| −3.18
| 2.18
| 2.18
| 4.92
| 4.92
Line 773: Line 830:
* 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 (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 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]].
=== Uniform maps ===
{{Uniform map|edo=27}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 852: Line 912:


=== Commas ===
=== Commas ===
27edo [[tempers out]] the following [[commas]]. (Note: This assumes the patent [[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"
Line 1,073: Line 1,133:
| 19th-partial chroma
| 19th-partial chroma
|}
|}
<references group="note" />


== Scales ==
== Scales ==
=== MOS scales ===
=== MOS scales ===
{{Main|List of MOS scales in 27edo}}
{{Main|List of MOS scales in 27edo}}
* Superpyth pentatonic &ndash; Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6
* Superpyth pentic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6
* Superpyh diatonic &ndash; Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1
* Superpyth diatonic – Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1
* Superpyth chromatic &ndash; Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1
* Superpyth chromatic – Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1
* Superpyth hyperchromatic &ndash; Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 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[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[9] [[3L 6s]] (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2
Line 1,111: Line 1,172:
* enharmonic trichord octave species: 9 2 5 9 2, 2 9 5 2 9
* 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
* 5-limit / pental double harmonic  hexatonic (Augmented[6] [[4M]]): 2 7 2 7 7 2, 7 7 2 2 7 2
* Superpyth melodic minor &ndash; Superpyth 2|4 #6 #7 or 5|1 b3: 5 1 5 5 5 5 1
* Superpyth melodic minor – Superpyth 2|4 #6 #7 or 5|1 b3: 5 1 5 5 5 5 1
* Superpyth harmonic minor &ndash; Superpyth 2|4 #7: 5 1 5 5 1 9 1
* Superpyth harmonic minor – Superpyth 2|4 #7: 5 1 5 5 1 9 1
* Superpyth harmonic major &ndash; Superpyth 5|1 b6: 5 5 1 5 1 9 1
* Superpyth harmonic major – Superpyth 5|1 b6: 5 5 1 5 1 9 1
* Superpyth double harmonic major &ndash; Superpyth 5|1 b2 b6: 1 9 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
* [[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
* "Just" minor (inverse of "just" major): 5 2 4 5 2 5 4
Line 1,127: Line 1,188:
* [[SNS (2/1, 3/2, 5/4)-7|5-limit / pental double harmonic major]]: 2 7 2 5 2 7 2
* [[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
* 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 #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 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 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 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 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
* [[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 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 decatonic (subset of Augene[12]): 2 5 2 2 3 2 2 5 2 2
Line 1,138: Line 1,199:
* [[Blackdye]] / [[syntonic dipentatonic]] (superset of [[Zarlino]]): 1 4 2 4 1 4 2 4 1 4
* [[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
* [[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)
Direct sunlight (this is its original/default tuning; subset of [[Sensi]][19])
* Hypersakura (original/default tuning; subset of Sensi[19]): 1 10 5 1 10 ((1 11 16 17 27)\27)
* 44.444
* 133.333
* 488.889
* 711.111
* 755.555
* 1155.555
* 1200.000
 
Hypersakura (this is its original/default tuning; subset of Sensi[19])
* 44.444
* 488.889
* 711.111
* 755.555
* 1200.000


== Instruments ==
== Instruments ==
Line 1,187: 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]]
* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 ''Sad Like Winter Leaves''] &ndash; in Augene[12] tuned to 27edo
* [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 1,200: Line 1,250:


; [[Herman Miller]]
; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/nusu-laj-stille-nacht Stille Nacht (cover)]'' (2018)
* ''[https://soundcloud.com/morphosyntax-1/nusu-laj-stille-nacht Stille Nacht (cover)]'' (2019)


; [[NullPointerException Music]]
; [[NullPointerException Music]]
Line 1,212: Line 1,262:


; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* [https://www.archive.org/details/MusicForYourEars ''Music For Your Ears''] [https://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] &ndash; the central portion is in 27edo, the rest in [[46edo]].
* [https://www.archive.org/details/MusicForYourEars ''Music For Your Ears''] [https://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] – the central portion is in 27edo, the rest in [[46edo]].


; [[Joel Taylor]]
; [[Joel Taylor]]
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] &ndash; in Augene[12] tuned to 27edo
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] – in Augene[12] tuned to 27edo


; [[Tristan Bay]]
; [[Tristan Bay]]
* [https://youtu.be/R30aRbNtoIY ''Pitchblende''] (2023)
* [https://youtu.be/R30aRbNtoIY ''Pitchblende''] (2023)
; [[Uncreative Name]]
* [https://www.youtube.com/watch?v=dcQe6ebpGFU ''Autumn''] (2024) – in Blackdye, 27edo tuning


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]
Line 1,224: Line 1,277:


; [[Xotla]]
; [[Xotla]]
* "Funkrotonal" from ''Microtonal Allsorts'' (2023) &ndash; [https://open.spotify.com/track/1zjNkbm8kIkuCxtodyFCL0 Spotify] | [https://xotla.bandcamp.com/track/funkrotonal-27edo Bandcamp] | [https://www.youtube.com/watch?v=7gt1BBJuJsE YouTube]
* "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]
 
== Notes ==
<references group="note" />


[[Category:Augene]]
[[Category:Augene]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Sensi]]
[[Category:Superpyth]]
[[Category:Superpyth]]
[[Category:Tetracot]]
[[Category:Tetracot]]
[[Category:Twentuning]]
[[Category:Twentuning]]
Retrieved from "https://en.xen.wiki/w/27edo"