Extended meantone notation: Difference between revisions

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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.

While Middle Eastern maqam music is far too complex in real life to be represented 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 Db are enharmonically equal, this 2D lattice collapses further to the 1D lattice of [[24edo]], which ~~has sometimes been~~ 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.

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.

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 a 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 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.

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 179: / Line 179: @@
 == 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.
-While Middle Eastern maqam music is far too complex in real life to be represented 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 Db are enharmonically equal, this 2D lattice collapses further to the 1D lattice of [[24edo]], which has sometimes been 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.
+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.
-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 a 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 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.
 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]]