Extended meantone notation: Difference between revisions

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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 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~~.

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. Mohajira is supported very well by 24edo and [[31edo]].

[[Category:Meantone]]

[[Category:Notation]]

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 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 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.
+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. Mohajira is supported very well by 24edo and [[31edo]].
 [[Category:Meantone]]
 [[Category:Notation]]