Alpharabian tuning: Difference between revisions

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* The interval 33/32 is the standard Alpharabian quartertone due to not only being the simplest quartertone in the 2.3.11 subgroup, but also due to the fact that stacking three of these and subtracting the resulting interval from 9/8 yields 4096/3993 the simplest possible interval that can result from such as process; furthermore, modification of a Pythagorean interval by this quartertone generally results in an Alpharabian interval- the only two known exceptions to this being 11/8 and 16/11, which differ from 4/3 and 3/2 respectively by this interval.

* The rastma, [[243/242]], is functionally the simplest type of Alpharabian [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]], and, since three instances of 243/242 are almost equal to [[81/80]] in JI, one can closely approach just [[5-limit]] intervals simply by moving three rastmas away from Pythagorean intervals; furthermore, modification of a Pythagorean interval by this subchroma generally results in an Alpharabian interval- the only two known exceptions to this being 121/64 and 128/121, which differ from 243/128 and 256/243 respectively by this interval.

* The Parachromatic Semilimma, 1331/1296, is slightly over half of [[256/243]], the Pythagorean Limma, with the remainder being 4096/3993, and since 1331/1296 differs from 33/32 by the rastma, modification of a Pythagorean interval by this quartertone often results in an Alpharabian interval- the principle exceptions to this being 1331/1024 and 2048/1331, which differ from 81/64 and 128/81 respectively by this interval, though there are others.

* The Parachromatic Semilimma, [[1331/1296]], is slightly over half of [[256/243]], the Pythagorean Limma, with the remainder being 4096/3993, and since 1331/1296 differs from 33/32 by the rastma, modification of a Pythagorean interval by this quartertone often results in an Alpharabian interval- the principle exceptions to this being 1331/1024 and 2048/1331, which differ from 81/64 and 128/81 respectively by this interval, though there are others.

The following rules are directly derived from the above premises:

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 * The interval 33/32 is the standard Alpharabian quartertone due to not only being the simplest quartertone in the 2.3.11 subgroup, but also due to the fact that stacking three of these and subtracting the resulting interval from 9/8 yields 4096/3993 the simplest possible interval that can result from such as process; furthermore, modification of a Pythagorean interval by this quartertone generally results in an Alpharabian interval- the only two known exceptions to this being 11/8 and 16/11, which differ from 4/3 and 3/2 respectively by this interval.
 * The rastma, [[243/242]], is functionally the simplest type of Alpharabian [[Diatonic, Chromatic, Enharmonic, Subchromatic|subchroma]], and, since three instances of 243/242 are almost equal to [[81/80]] in JI, one can closely approach just [[5-limit]] intervals simply by moving three rastmas away from Pythagorean intervals; furthermore, modification of a Pythagorean interval by this subchroma generally results in an Alpharabian interval- the only two known exceptions to this being 121/64 and 128/121, which differ from 243/128 and 256/243 respectively by this interval.
-* The Parachromatic Semilimma, 1331/1296, is slightly over half of [[256/243]], the Pythagorean Limma, with the remainder being 4096/3993, and since 1331/1296 differs from 33/32 by the rastma, modification of a Pythagorean interval by this quartertone often results in an Alpharabian interval- the principle exceptions to this being 1331/1024 and 2048/1331, which differ from 81/64 and 128/81 respectively by this interval, though there are others.
+* The Parachromatic Semilimma, [[1331/1296]], is slightly over half of [[256/243]], the Pythagorean Limma, with the remainder being 4096/3993, and since 1331/1296 differs from 33/32 by the rastma, modification of a Pythagorean interval by this quartertone often results in an Alpharabian interval- the principle exceptions to this being 1331/1024 and 2048/1331, which differ from 81/64 and 128/81 respectively by this interval, though there are others.
 The following rules are directly derived from the above premises: