Extended meantone notation: Difference between revisions
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! rowspan="2" style="width: 90px;" | [[Syntonic comma]] fraction | ! rowspan="2" style="width: 90px;" | [[Syntonic comma]] fraction | ||
! colspan="4" | Steps | ! colspan="4" | Steps | ||
! rowspan="2" style="width: | ! rowspan="2" style="width: 250px;" | Explanation | ||
|- | |- | ||
! style="width: 80px;" | Chromatic semitone | ! style="width: 80px;" | Chromatic semitone (e.g. C–C♯) | ||
! style="width: 80px;" | Diatonic semitone | ! style="width: 80px;" | Diatonic semitone (e.g. C–D♭) | ||
! Diesis | ! Diesis | ||
! Kleisma | ! Kleisma | ||
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| 1 | | 1 | ||
| −1 | | −1 | ||
| Chromatic semitone is tempered out, diesis is positive, and kleisma is negative | | Chromatic semitone is tempered out<ref group="note" name="chroma_note">In 7-tone equal temperament, the tempering out of the chromatic semitone means that sharps and flats cannot alter the pitch.</ref>, diesis is positive, and kleisma is negative<ref group="note" name="kleisma_note">A negative kleisma means that B♯ is lower in pitch than C♭ and E♯ is lower in pitch than F♭.</ref> | ||
|- | |- | ||
| [[12edo|12 (standard tuning)]] | | [[12edo|12 (standard tuning)]] | ||
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| 0 | | 0 | ||
| 1 | | 1 | ||
| Chromatic semitone is equal to kleisma, diesis is tempered out | | Chromatic semitone is equal to kleisma, diesis is tempered out<ref group="note" name="diesis_note">Having C♯ and D♭ be enharmonically equivalent is what most musicians are familiar with, but in a generalized meantone tuning, there are no enharmonic equivalents.</ref> | ||
|- | |- | ||
| [[19edo|19]] | | [[19edo|19]] | ||
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| 2 | | 2 | ||
| −1 | | −1 | ||
| rowspan="2" | Diesis is larger than chromatic semitone, kleisma is negative | | rowspan="2" | Diesis is larger than chromatic semitone, kleisma is negative<ref group="note" name="kleisma_note" /> | ||
|- | |- | ||
| [[33edo#Theory|33]] (c mapping) | | [[33edo#Theory|33]] (c mapping) | ||
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There are of course notational equivalences: | There are of course notational equivalences: | ||
*B♯↑ and B𝄪− are equal to C | * B♯↑ and B𝄪− are equal to C | ||
*C+↑ is equal to C♯ (because the two semisharps add up) | * C+↑ is equal to C♯ (because the two semisharps add up) | ||
*D𝄫↓ and D♭♭♭− are equal to C | * D𝄫↓ and D♭♭♭− are equal to C | ||
[[9–odd–limit]] intervals and their notation relative to C: | [[9–odd–limit]] intervals and their notation relative to C: | ||
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|} | |} | ||
Two dieses or two kleismas cannot be stacked to produce a chromatic semitone, and notation for 11-limit and 13-limit intervals (intervals involving the 11th and 13th | Two dieses or two kleismas cannot be stacked to produce a chromatic semitone, and notation for [[11-limit]] and [[13-limit]] intervals (intervals involving the [[11/8|11th harmonic]] and [[13/8|13th harmonic]]) can vary (see [[meantone vs meanpop]]). | ||
== True half-sharps and half-flats == | == True half-sharps and half-flats == | ||
If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a 2D new tuning system which is generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class. | If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a 2D new tuning system which is generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class. | ||
Real-world Arabic and Persian music often | Real-world Arabic and Persian music often involves many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibits significant regional variations, and as a result they are very difficult to notate exactly. However, they are commonly notated using half-sharps and-half flats. If we take these to be exactly equal to half of a chromatic semitone, then mathematically, this notation system results in a two-dimensional lattice that is generated by a neutral third and an octave. If adjacent sharps and flats, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into [[24edo]], which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different. | ||
The chain-of-neutral thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any JI interval mapping to it in the 7-limit. But, if we | The chain-of-neutral thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any JI interval mapping to it in the 7-limit. But, if we bring in the 11th harmonic, and decide that there should only be a single neutral second (i.e. 121/120 should be tempered out, resulting in the small neutral second of 12/11 and large neutral second of 11/10 both being mapped to a single equally-tempered interval), we obtain [[mohajira]], a very accurate 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth, resulting in [[243/242]] being tempered out. Furthermore, flattening a minor third by a half-flat results in an approximation of 7/6, while sharpening a major third by a half-sharp gives an approximation of 9/7. Mohajira is supported very well by 24edo and 31edo. | ||
== Notes == | == Notes == | ||