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Extended meantone notation: Difference between revisions

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[[Circle-of-fifths notation|Standard meantone notation]] uses 7 base note letters, plus sharps and flats.
[[Circle-of-fifths notation|Standard meantone notation]] uses 7 base note letters, plus sharps and flats.


... F𝄫 C𝄫 G𝄫 D𝄫 A𝄫 E𝄫 B𝄫 F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ F𝄪 C𝄪 G𝄪 D𝄪 A𝄪 E𝄪 B𝄪 ...
... F𝄫 – C𝄫 – G𝄫 – D𝄫 – A𝄫 – E𝄫 – B𝄫 – F♭ – C♭ – G♭ – D♭ – A♭ – E♭ – B♭ – F – C – G – D – A – E – B – F♯ – C♯ – G♯ – D♯ – A♯ – E♯ – B♯ – F𝄪 – C𝄪 – G𝄪 – D𝄪 – A𝄪 – E𝄪 – B𝄪 ...


However, when transferred onto a 31edo scale, it looks like this:
However, when transferred into a 31edo scale, the chromatic scale becomes:


C D𝄫 C♯ D♭ C𝄪 D E𝄫 D♯ E♭ D𝄪 E F♭ E♯ F G𝄫 F♯ G♭ F𝄪 G A𝄫 G♯ A♭ G𝄪 A B𝄫 A♯ B♭ A𝄪 B C♭ B♯ C
C – D𝄫 – C♯ – D♭ – C𝄪 – D – E𝄫 – D♯ – E♭ – D𝄪 – E – F♭ – E♯ – F – G𝄫 – F♯ – G♭ – F𝄪 – G – A𝄫 – G♯ – A♭ – G𝄪 – A – B𝄫 – A♯ – B♭ – A𝄪 – B – C♭ – B♯ – C


Note that the base note letters alternate.
Note that the base note letters alternate.


The 31edo sharp can be split in half, so in 31edo this is solved by semisharps and semiflats, sometimes notated with [[ups and Downs Notation|ups and downs]].
In 31edo, sharps can be split in half, so this is solved by semisharps and semiflats, sometimes notated with [[ups and Downs Notation|ups and downs]].


The meantone circle of fifths however, has no single semisharp/semiflat. In extended meantone notation, a sharp is split into 2 different parts that can be added to produce a sharp:
The meantone circle of fifths, however, has no single semisharp/semiflat. In extended meantone notation, a sharp is split into 2 different parts that can be added to produce a sharp:


<pre>
{| class="wikitable center-all"
♯ — sharpen by meantone chromatic semitone, 7 fifths up
! colspan="2" | Symbol
♭ — flatten by meantone chromatic semitone, 7 fifths down
! rowspan="2" | Interval
↑ — sharpen by meantone diesis, 12 fifths down
! rowspan="2" | Number of fifths<br>(move up to raise,<br>move down to lower)
↓ — flatten by meantone diesis, 12 fifths up
|-
+ — sharpen by meantone kleisma, 19 fifths up
! Raise
− — flatten by meantone kleisma, 19 fifths down
! Lower
</pre>
|-
| ♯
| ♭
| Chromatic semitone
| 7
|-
| ↑
| ↓
| Diesis
| 12
|-
| +
| &minus;
| Kleisma
| 19
|}


A diesis plus a kleisma, added together, equals a meantone chromatic semitone. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone.
A diesis plus a kleisma, added together, equals a meantone chromatic semitone. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone.
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Unlike a single semisharp/semiflat, this can be generalized to other meantone tunings:
Unlike a single semisharp/semiflat, this can be generalized to other meantone tunings:


*[[7edo]] (chromatic semitone is tempered out, diesis is positive, and kleisma is negative)
{| class="wikitable center-all"
*[[12edo]] (chromatic semitone is equal to kleisma, diesis is tempered out)
! rowspan="2" | EDO
*[[19edo]] (chromatic semitone is equal to diesis, kleisma is tempered out)
! rowspan="2" | Syntonic<br>comma fraction
*[[26edo]] (chromatic semitone is smaller than diesis, kleisma is negative)
! colspan="4" | Steps
*[[31edo]] (diesis is equal to kleisma)
! rowspan="2" | Explanation
*[[43edo]] and [[55edo]] (diesis is smaller than kleisma)
|-
*[[50edo]] (diesis is larger than kleisma)
! Chromatic<br>semitone
! Diatonic<br>semitone
! Diesis
! Kleisma
|-
| [[7edo|7]]
|
| 0
| 1
| 1
| &minus;1
| Chromatic semitone is tempered out, <br>diesis is positive, and kleisma is negative
|-
| [[12edo|12]]
| rowspan="3" | {{frac|11}} comma
| 1
| 1
| rowspan="3" | 0
| 1
| rowspan="3" | Chromatic semitone is equal to kleisma,<br>diesis is tempered out
|-
| [[24edo|24]]
| 2
| 2
| 2
|-
| [[36edo|36]]
| 3
| 3
| 3
|-
| [[19edo|19]]
| rowspan="2" | {{frac|3}} comma
| 1
| 2
| 1
| rowspan="2" | 0
| rowspan="2" | Chromatic semitone is equal to diesis,<br>kleisma is tempered out
|-
| [[38edo|38]]
| 2
| 4
| 2
|-
| [[26edo|26]]
|
| 1
| 3
| 2
| &minus;1
| rowspan="3" | [[Flattone]] tunings:<br>Diesis is larger than chromatic semitone,<br>kleisma is negative
|-
| [[33edo|33]]<br>(c mapping)
| {{frac|2}} comma
| 1
| 4
| 3
| &minus;2
|-
| [[45edo|45]]
| {{frac|2|5}} comma
| 2
| 5
| 3
| &minus;1
|-
| [[31edo|31]]
| {{frac|4}} comma
| 2
| 3
| 1
| 1
| Diesis is equal to kleisma
|-
| [[43edo|43]]
| {{frac|5}} comma
| 3
| 4
| 1
| 2
| rowspan="2" | Diesis is smaller than kleisma
|-
| [[55edo|55]]
| {{frac|6}} comma
| 4
| 5
| 1
| 3
|-
| [[50edo|50]]
| {{frac|2|7}} comma
| 3
| 5
| 2
| 1
| Diesis is larger than kleisma
|}


There are of course notational equivalences.
There are of course notational equivalences.
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