Extended meantone notation: Difference between revisions

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The meantone circle of fifths, however, has no single semisharp or semiflat. In extended meantone notation, a sharp is split into 2 different parts, the diesis and the kleisma.

=== Generalizing accidentals ===

== Generalizing accidentals ==

Sharps and flats denote raising and lowering by a given number of chromatic semitones. The diesis is the difference between adjacent accidentals (e.g. C♯–D♭ and D♯–E♭), while the kleisma is the amount by which B♯ exceeds C♭ and E♯ exceeds F♭.

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== True half-sharps and half-flats ==

If true half-sharps and half-flats are desired, which exactly bisect the chromatic semitone, ~~this mathematically implies that~~ 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.

~~While Middle Eastern maqam~~ music ~~is far too complex in real life~~ to ~~be represented~~ exactly ~~by either of these temperaments~~, ~~(one can certainly try—see [[Maqamat in maqamic temperament]]), it is~~ commonly notated using half-sharps and-half flats. If we take these to be exactly equal to ~~1/2~~ of a chromatic semitone, then mathematically, this notation system results in a 2D lattice that is generated by a neutral third and an octave. If ~~we furthermore decide that~~ C♯ and D♭ are enharmonically ~~equal~~, this 2D lattice ~~collapses~~ further ~~to the 1D lattice of~~ [[24edo]], which is often suggested as a simplified framework for ~~maqam~~ music. But the usual written notation typically lets ~~you notate them~~ as two distinct entities if ~~you want, so if we instead decide to leave them unequal, we get the 2D lattice above~~.

Real-world Arabic and Persian music often involve many fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit 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 go to the 11-limit, and ~~add~~ 121/120 to ~~the kernel~~, we obtain [[mohajira]], an exceptionally good 11-limit temperament. The neutral third ~~becomes equal to~~ 11/9, and two of them make ~~3/2~~. Furthermore, ~~if you take~~ a minor third ~~and ''flatten'' it~~ by a half-flat~~, you obtain a good representation~~ of 7/6~~. Conversely if you take~~ a major third ~~and ''sharpen'' it~~ by a half-sharp~~, you obtain a good representation for~~ 9/7. [[31edo]] ~~is another very good tuning for mohajira.~~

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 go to the 11-limit, 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]], an exceptionally good 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth. 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. In both 24edo and [[31edo]], their closest approximations to the 7th and 11th harmonics both allow them to support mohajira.

~~Although mohajira may not be a great tuning~~ to ~~reflect~~ the ~~way maqam music is played in practice, which often uses multiple unequal neutral thirds~~ and ~~exhibits significant regional variations, it is a highly interesting regular temperament of its own merit, and deserves further study~~.

[[Category:Meantone]]

[[Category:Notation]]

@@ Line 15: / Line 15: @@
 The meantone circle of fifths, however, has no single semisharp or semiflat. In extended meantone notation, a sharp is split into 2 different parts, the diesis and the kleisma.
-=== Generalizing accidentals ===
+== Generalizing accidentals ==
 Sharps and flats denote raising and lowering by a given number of chromatic semitones. The diesis is the difference between adjacent accidentals (e.g. C♯&ndash;D♭ and D♯&ndash;E♭), while the kleisma is the amount by which B♯ exceeds C♭ and E♯ exceeds F♭.
@@ Line 185: / Line 185: @@
 == True half-sharps and half-flats ==
-If true half-sharps and half-flats are desired, which exactly bisect the chromatic semitone, this mathematically implies that 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.
-While Middle Eastern maqam music is far too complex in real life to be represented exactly by either of these temperaments, (one can certainly try&mdash;see [[Maqamat in maqamic temperament]]), it is commonly notated using half-sharps and-half flats. If we take these to be exactly equal to 1/2 of a chromatic semitone, then mathematically, this notation system results in a 2D lattice that is generated by a neutral third and an octave. If we furthermore decide that C♯ and D♭ are enharmonically equal, this 2D lattice collapses further to the 1D lattice of [[24edo]], which is often suggested as a simplified framework for maqam music. But the usual written notation typically lets you notate them as two distinct entities if you want, so if we instead decide to leave them unequal, we get the 2D lattice above.
+Real-world Arabic and Persian music often involve many fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit 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 go to the 11-limit, and add 121/120 to the kernel, we obtain [[mohajira]], an exceptionally good 11-limit temperament. The neutral third becomes equal to 11/9, and two of them make 3/2. Furthermore, if you take a minor third and ''flatten'' it by a half-flat, you obtain a good representation of 7/6. Conversely if you take a major third and ''sharpen'' it by a half-sharp, you obtain a good representation for 9/7. [[31edo]] is another very good tuning for mohajira.
+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 go to the 11-limit, 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]], an exceptionally good 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth. 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. In both 24edo and [[31edo]], their closest approximations to the 7th and 11th harmonics both allow them to support mohajira.
-Although mohajira may not be a great tuning to reflect the way maqam music is played in practice, which often uses multiple unequal neutral thirds and exhibits significant regional variations, it is a highly interesting regular temperament of its own merit, and deserves further study.
 [[Category:Meantone]]
 [[Category:Notation]]