Extended meantone notation: Difference between revisions

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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 [[11/8|11th harmonic]] and [[13/8|13th harmonic]]) can vary (see [[meantone vs meanpop]]).

Two dieses or two kleismas cannot be stacked to produce a chromatic semitone except in 31edo, 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 ==

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 new tuning system consisting of a two-dimensional lattice 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 ~~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.

While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. 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 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.

The chain-of-neutral-thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any just 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 (resulting in 11/10 and 12/11, the greater and lesser neutral seconds, 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 ==

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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 [[11/8|11th harmonic]] and [[13/8|13th harmonic]]) can vary (see [[meantone vs meanpop]]).
+Two dieses or two kleismas cannot be stacked to produce a chromatic semitone except in 31edo, 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 ==
-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 new tuning system consisting of a two-dimensional lattice 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 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.
+While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. 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 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.
+The chain-of-neutral-thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any just 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 (resulting in 11/10 and 12/11, the greater and lesser neutral seconds, 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 ==