Alpharabian tuning: Difference between revisions

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* Intervals that are in the 2.11 subgroup are all considered Axirabian intervals as 2.11 forms a core navigational axis of Alpharabian tuning.

* The intervals [[3/2]], [[4/3]], [[9/8]], [[16/9]], and so forth, have the same functions as in [[Pythagorean tuning]].

* 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 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 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 following rules are directly derived from the above premises:

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:* Dimunition of a Pythagorean Major interval by 33/32 results in a Tendoneutral interval

The following rules ~~are more questionable as they rely on the aforementioned questionable premise~~:

The following rules have not yet been finalized in their entirety due to lack of details:

* Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take ~~either~~ the '~~super~~' or '~~sub~~' prefixes, ~~with these prefixes generally being stacked where multiple such modifications occur~~, however, there are some significant caveats...

* Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take both the 'ultra' or 'infra' prefixes along with the "semilimmic" modifier before the word with the prefix, since 1331/1296 is the chromatic half of a Pythagorean Limma, however, there are some significant caveats...

:* Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a Supraminor interval, but a second such augmentation results in a Betarabian Major interval due to said interval differing from the nearby Alpharabian Major (covered under modifications by 1089/1024) by a rastma.

:* Dimunition of a Pythagorean Major interval by a single 1331/1296 results in a Submajor interval, but a second such dimunition results in a Betarabian Minor interval due to said interval differing from the nearby Alpharabian Minor (covered under modifications by 1089/1024) by a rastma.

[[Category:Tuning]]

[[Category:Alpharabian| ]]

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 * Intervals that are in the 2.11 subgroup are all considered Axirabian intervals as 2.11 forms a core navigational axis of Alpharabian tuning.
 * The intervals [[3/2]], [[4/3]], [[9/8]], [[16/9]], and so forth, have the same functions as in [[Pythagorean tuning]].
-* 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 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 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 following rules are directly derived from the above premises:
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 :* Dimunition of a Pythagorean Major interval by 33/32 results in a Tendoneutral interval
-The following rules are more questionable as they rely on the aforementioned questionable premise:
+The following rules have not yet been finalized in their entirety due to lack of details:
-* Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take either the 'super' or 'sub' prefixes, with these prefixes generally being stacked where multiple such modifications occur, however, there are some significant caveats...
+* Generally, intervals that result that result from the modification of a Pythagorean interval by 1331/1296 take both the 'ultra' or 'infra' prefixes along with the "semilimmic" modifier before the word with the prefix, since 1331/1296 is the chromatic half of a Pythagorean Limma, however, there are some significant caveats...
 :* Augmentation of a Pythagorean Minor interval by a single 1331/1296 results in a Supraminor interval, but a second such augmentation results in a Betarabian Major interval due to said interval differing from the nearby Alpharabian Major (covered under modifications by 1089/1024) by a rastma.
 :* Dimunition of a Pythagorean Major interval by a single 1331/1296 results in a Submajor interval, but a second such dimunition results in a Betarabian Minor interval due to said interval differing from the nearby Alpharabian Minor (covered under modifications by 1089/1024) by a rastma.
 [[Category:Tuning]]
 [[Category:Alpharabian| ]] <!-- main article -->