Extended meantone notation: Difference between revisions
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revised symbols for diesis and kleisma alterations |
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<pre># — sharpen by meantone chromatic semitone, 7 fifths up | <pre># — sharpen by meantone chromatic semitone, 7 fifths up | ||
b — flatten by meantone chromatic semitone, 7 fifths down | b — flatten by meantone chromatic semitone, 7 fifths down | ||
^ — sharpen by meantone diesis, 12 fifths down | |||
v — flatten by meantone diesis, 12 fifths up | |||
+ — sharpen by meantone kleisma, 19 fifths up | |||
- — flatten by meantone kleisma, 19 fifths down</pre> | |||
A diesis plus a kleisma, added together, equals a meantone chromatic semitone. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone. | A diesis plus a kleisma, added together, equals a meantone chromatic semitone. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone. | ||
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There are of course notational equivalences. | There are of course notational equivalences. | ||
*B# | *B#^ and B##- are equal to C | ||
* | *C+^ is equal to C# (because the two semisharps add up) | ||
* | *Dbbv and Dbbb- are equal to C | ||
The meantone diesis can be considered to be [[36/35]], [[50/49]], [[64/63]], or [[128/125]], while the meantone kleisma is [[49/48]], [[245/243]], [[3125/3072]] or [[15625/15552]] assuming [[septimal meantone]]. An octave is made of 19 dieses and 12 kleisma. | The meantone diesis can be considered to be [[36/35]], [[50/49]], [[64/63]], or [[128/125]], while the meantone kleisma is [[49/48]], [[245/243]], [[3125/3072]] or [[15625/15552]] assuming [[septimal meantone]]. An octave is made of 19 dieses and 12 kleisma. | ||
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6/5 — Eb | 6/5 — Eb | ||
7/4 — A#, or | 7/4 — A#, or Bbv | ||
7/6 — D#, or | 7/6 — D#, or Ebv | ||
7/5 — F#, or | 7/5 — F#, or Gbv | ||
8/7 — Ebb, or | 8/7 — Ebb, or D^ | ||
12/7 — Bbb, or | 12/7 — Bbb, or A^ | ||
10/7 — Gb, or F# | 10/7 — Gb, or F#^ | ||
9/8 — D | 9/8 — D | ||
9/5 — Bb | 9/5 — Bb | ||
9/7 — Fb, or | 9/7 — Fb, or E^ | ||
16/9 — Bb | 16/9 — Bb | ||
10/9 — D | 10/9 — D | ||
14/9 — G#, or | 14/9 — G#, or Abv</pre> | ||
Two dieses or two kleismas cannot be stacked to produce a chromatic semitone. 11–limit and 13–limit notation can [[ | Two dieses or two kleismas cannot be stacked to produce a chromatic semitone. 11–limit and 13–limit notation can vary (see [[meantone vs meanpop]]). | ||
Extended meantone notation was created as a way to notate [[43edo]] with only a base letter with one symbol. | Extended meantone notation was created as a way to notate [[43edo]] with only a base letter with one symbol. | ||
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While Middle Eastern maqam music is far too complex in real life to be represented by either of these temperaments (although 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 by either of these temperaments (although 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. | ||
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 [[ | 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. | ||
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. | 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. | ||