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27edo: Difference between revisions

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=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|27}}
{{Harmonics in equal|27}}
== Notation ==
{| class="wikitable center-all floatright"
|+ style="font-size: 105%;" | Circle of fifths in 27edo
|- style="white-space: nowrap;"
! rowspan="2" | [[Cent]]s !! colspan="6" | Note from C
|- style="white-space: nowrap;"
! colspan="2" | Standard<br />notation !! colspan="2" | Quarter tone<br />notation
|-
| 0
| colspan="2" | C
| colspan="2" | A{{sesquisharp2}}
|-
| 711.11
| colspan="2" | G
| colspan="2" | E{{sesquisharp2}}
|-
| 222.22
| colspan="2" | D
| B{{sesquisharp2}} || F{{sesquiflat2}}
|-
| 933.33
| colspan="2" | A
| colspan="2" | C{{sesquiflat2}}
|-
| 444.44
| colspan="2" | E
| colspan="2" | G{{sesquiflat2}}
|-
| 1155.55
| colspan="2" | B
| colspan="2" | D{{sesquiflat2}}
|-
| 666.66
| colspan="2" | F&#x266F;
| colspan="2" | A{{sesquiflat2}}
|-
| 177.77
| colspan="2" | C&#x266F;
| colspan="2" | E{{sesquiflat2}}
|-
| 888.88
| colspan="2" | G&#x266F;
| colspan="2" | B{{sesquiflat2}}
|-
| 400
| colspan="2" | D&#x266F;
| colspan="2" | F{{demiflat2}}
|-
| 1111.11
| colspan="2" | A&#x266F;
| colspan="2" | C{{demiflat2}}
|-
| 622.22
| colspan="2" | E&#x266F;
| colspan="2" | G{{demiflat2}}
|-
| 133.33
| B&#x266F;
| F&#x1D12B;
| colspan="2" | D{{demiflat2}}
|-
| 844.44
| F&#x1D12A;
| C&#x1D12B;
| colspan="2" | A{{demiflat2}}
|-
| 355.56
| C&#x1D12A;
| G&#x1D12B;
| colspan="2" | E{{demiflat2}}
|-
| 1066.67
| G&#x1D12A;
| D&#x1D12B;
| colspan="2" | B{{demiflat2}}
|-
| 577.78
| D&#x1D12A;
| A&#x1D12B;
| colspan="2" | F{{demisharp2}}
|-
| 88.89
| A&#x1D12A;
| E&#x1D12B;
| colspan="2" | C{{demisharp2}}
|-
| 800
| E&#x1D12A;
| B&#x1D12B;
| colspan="2" | G{{demisharp2}}
|-
| 311.11
| B&#x1D12A;
| F&#x266D;
| colspan="2" | D{{demisharp2}}
|-
| 1022.22
| colspan="2" | C&#x266D;
| colspan="2" | A{{demisharp2}}
|-
| 533.33
| colspan="2" | G&#x266D;
| colspan="2" | E{{demisharp2}}
|-
| 44.44
| colspan="2" | D&#x266D;
| colspan="2" | B{{demisharp2}}
|-
| 755.56
| colspan="2" | A&#x266D;
| colspan="2" | F{{sesquisharp2}}
|-
| 266.67
| colspan="2" | E&#x266D;
| colspan="2" | C{{sesquisharp2}}
|-
| 977.78
| colspan="2" | B&#x266D;
| colspan="2" | G{{sesquisharp2}}
|-
| 488.89
| colspan="2" | F
| colspan="2" | D{{sesquisharp2}}
|-
| 0
| colspan="2" | C
| 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.
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.
{{sharpness-sharp4}}
{{clear}}


== Intervals ==
== Intervals ==
Line 594: Line 457:
|}
|}
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;"
! rowspan="2" | [[Cent]]s !! colspan="6" | Note from C
|- style="white-space: nowrap;"
! colspan="2" | Standard<br />notation !! colspan="2" | Quarter tone<br />notation
|-
| 0
| colspan="2" | C
| colspan="2" | A{{sesquisharp2}}
|-
| 711.11
| colspan="2" | G
| colspan="2" | E{{sesquisharp2}}
|-
| 222.22
| colspan="2" | D
| B{{sesquisharp2}} || F{{sesquiflat2}}
|-
| 933.33
| colspan="2" | A
| colspan="2" | C{{sesquiflat2}}
|-
| 444.44
| colspan="2" | E
| colspan="2" | G{{sesquiflat2}}
|-
| 1155.55
| colspan="2" | B
| colspan="2" | D{{sesquiflat2}}
|-
| 666.66
| colspan="2" | F&#x266F;
| colspan="2" | A{{sesquiflat2}}
|-
| 177.77
| colspan="2" | C&#x266F;
| colspan="2" | E{{sesquiflat2}}
|-
| 888.88
| colspan="2" | G&#x266F;
| colspan="2" | B{{sesquiflat2}}
|-
| 400
| colspan="2" | D&#x266F;
| colspan="2" | F{{demiflat2}}
|-
| 1111.11
| colspan="2" | A&#x266F;
| colspan="2" | C{{demiflat2}}
|-
| 622.22
| colspan="2" | E&#x266F;
| colspan="2" | G{{demiflat2}}
|-
| 133.33
| B&#x266F;
| F&#x1D12B;
| colspan="2" | D{{demiflat2}}
|-
| 844.44
| F&#x1D12A;
| C&#x1D12B;
| colspan="2" | A{{demiflat2}}
|-
| 355.56
| C&#x1D12A;
| G&#x1D12B;
| colspan="2" | E{{demiflat2}}
|-
| 1066.67
| G&#x1D12A;
| D&#x1D12B;
| colspan="2" | B{{demiflat2}}
|-
| 577.78
| D&#x1D12A;
| A&#x1D12B;
| colspan="2" | F{{demisharp2}}
|-
| 88.89
| A&#x1D12A;
| E&#x1D12B;
| colspan="2" | C{{demisharp2}}
|-
| 800
| E&#x1D12A;
| B&#x1D12B;
| colspan="2" | G{{demisharp2}}
|-
| 311.11
| B&#x1D12A;
| F&#x266D;
| colspan="2" | D{{demisharp2}}
|-
| 1022.22
| colspan="2" | C&#x266D;
| colspan="2" | A{{demisharp2}}
|-
| 533.33
| colspan="2" | G&#x266D;
| colspan="2" | E{{demisharp2}}
|-
| 44.44
| colspan="2" | D&#x266D;
| colspan="2" | B{{demisharp2}}
|-
| 755.56
| colspan="2" | A&#x266D;
| colspan="2" | F{{sesquisharp2}}
|-
| 266.67
| colspan="2" | E&#x266D;
| colspan="2" | C{{sesquisharp2}}
|-
| 977.78
| colspan="2" | B&#x266D;
| colspan="2" | G{{sesquisharp2}}
|-
| 488.89
| colspan="2" | F
| colspan="2" | D{{sesquisharp2}}
|-
| 0
| colspan="2" | C
| 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.
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.
{{sharpness-sharp4}}
{{clear}}


== Approximation to JI ==
== Approximation to JI ==
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