Dwarf: Difference between revisions

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== Construction ==

~~Suppose~~ ''V'' is a [[val]] {{val| ''n'' … }} ~~whose first coordinate is a positive integer ''n''~~, ~~and~~ suppose the coordinates of the val, reduced {{w|~~Modulo~~ operation|modulo}} ''n'', are distinct. An example would be {{val| 12 19 28 34 }}; reduced mod 12 this is {{val| 0 7 4 10 }} and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers in order of increasing size, 1, 3, 5, 7, … and map them by the val ''V'', reducing the result ~~mod ''n''~~. If this number (from 0 to (''n'' - 1)) has not appeared before, add the odd positive integer to a set. When ''n'' values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(''V''), the dwarf scale resulting from the val ''V''.

For an [[equal temperament]] ''n''-et characterized by a [[val]] ''V'' = {{val| ''n'' … }}, suppose the coordinates of the val, reduced {{w|modulo operation|modulo}} ''n'', are distinct. An example would be {{val| 12 19 28 34 }}; reduced mod 12 this is {{val| 0 7 4 10 }} and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers within the [[subgroup]] of the equal temperament in order of increasing size, 1, 3, 5, 7, … and map them by the val ''V'', octave-reducing the result. If this number (from 0 to (''n'' - 1)) has not appeared before, add the odd positive integer to a set. When ''n'' values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(''V''), the dwarf scale resulting from the val ''V''.

== Example ==

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Of particular interest are dwarf scales resulting from equal temperament vals which are [[epimorphic]] for the val ''V'', but even vals far removed from an equal temperament will produce a scale.

Let us construct a JI dwarf by ~~the [[patent val]] ''V'' =~~ {{val| 12 19 28 34 }} of [[12edo|12et]]. As is shown above, 3/2 is mapped to 7\12, 5/4 to 4\12, and 7/4 to 10\12; we add these octave-reduced harmonics to the scale. We also have 9/8 mapped to 2\12. The next odd harmonic implied by the val is 15, which, after octave reduction, is mapped to 11\12. Follow the same process: 21/16 ~ 5\12, 25/16 ~ 7\12, and 27/16 ~ 8\12. Then there is 35/32 ~ 2\12, the same as ~9/8, so we reject it. Continuing on, we add 45/32 ~ 6\12, reject 49/32 ~ 8\12 which is the same as ~27/16, add 75/64 ~ 3\12, reject 81/64, 105/64, and 125/64, and add 135/128 ~ 1\12. With 135/128, we have added the last scale step of this 12-tone scale so the result is

Let us construct a JI dwarf by {{val| 12 19 28 34 }} of [[12edo|12et]]. As is shown above, 3/2 is mapped to 7\12, 5/4 to 4\12, and 7/4 to 10\12; we add these octave-reduced harmonics to the scale. We also have 9/8 mapped to 2\12. The next odd harmonic implied by the val is 15, which, after octave reduction, is mapped to 11\12. Follow the same process: 21/16 ~ 5\12, 25/16 ~ 7\12, and 27/16 ~ 8\12. Then there is 35/32 ~ 2\12, the same as ~9/8, so we reject it. Continuing on, we add 45/32 ~ 6\12, reject 49/32 ~ 8\12 which is the same as ~27/16, add 75/64 ~ 3\12, reject 81/64, 105/64, and 125/64, and add 135/128 ~ 1\12. With 135/128, we have added the last scale step of this 12-tone scale so the result is

: 135/128, 9/8, 75/64, 5/4, 21/16, 45/32, 3/2, 27/16, 7/4, 15/8, 2/1

And that is exactly [[Dwarf12 7]], the dwarf of 12et in the 7-limit.

For another example, consider {{val| 7 11 16 }} of 7et in the 5-limit. Here 3/2 is mapped to 4\7, 5/4 to 2\7, 9/8 to 1\7, 15/8 to 6\7, 25/16 to 4\7 (discarded), 27/16 to 5\7, and 45/32 to 3\7, so the scale is:

: 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1

In fact, this is the Lydian mode of [[Zarlino]], which exposes the fact that all the intervals of Zarlino Lydian can be expressed as octave-reduced harmonics.

== Generalization ==

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 == Construction ==
-Suppose ''V'' is a [[val]] {{val| ''n'' … }} whose first coordinate is a positive integer ''n'', and suppose the coordinates of the val, reduced {{w|Modulo operation|modulo}} ''n'', are distinct. An example would be {{val| 12 19 28 34 }}; reduced mod 12 this is {{val| 0 7 4 10 }} and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers in order of increasing size, 1, 3, 5, 7, … and map them by the val ''V'', reducing the result mod ''n''. If this number (from 0 to (''n'' - 1)) has not appeared before, add the odd positive integer to a set. When ''n'' values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(''V''), the dwarf scale resulting from the val ''V''.
+For an [[equal temperament]] ''n''-et characterized by a [[val]] ''V'' = {{val| ''n'' … }}, suppose the coordinates of the val, reduced {{w|modulo operation|modulo}} ''n'', are distinct. An example would be {{val| 12 19 28 34 }}; reduced mod 12 this is {{val| 0 7 4 10 }} and 0, 7, 4, and 10 are all distinct. Starting from 1, take the odd positive integers within the [[subgroup]] of the equal temperament in order of increasing size, 1, 3, 5, 7, … and map them by the val ''V'', octave-reducing the result. If this number (from 0 to (''n'' - 1)) has not appeared before, add the odd positive integer to a set. When ''n'' values have been obtained and no further additions are possible, take the resulting set and reduce its elements to an octave. The result is Dwarf(''V''), the dwarf scale resulting from the val ''V''.
 == Example ==
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 Of particular interest are dwarf scales resulting from equal temperament vals which are [[epimorphic]] for the val ''V'', but even vals far removed from an equal temperament will produce a scale.
-Let us construct a JI dwarf by the [[patent val]] ''V'' = {{val| 12 19 28 34 }} of [[12edo|12et]]. As is shown above, 3/2 is mapped to 7\12, 5/4 to 4\12, and 7/4 to 10\12; we add these octave-reduced harmonics to the scale. We also have 9/8 mapped to 2\12. The next odd harmonic implied by the val is 15, which, after octave reduction, is mapped to 11\12. Follow the same process: 21/16 ~ 5\12, 25/16 ~ 7\12, and 27/16 ~ 8\12. Then there is 35/32 ~ 2\12, the same as ~9/8, so we reject it. Continuing on, we add 45/32 ~ 6\12, reject 49/32 ~ 8\12 which is the same as ~27/16, add 75/64 ~ 3\12, reject 81/64, 105/64, and 125/64, and add 135/128 ~ 1\12. With 135/128, we have added the last scale step of this 12-tone scale so the result is
+Let us construct a JI dwarf by {{val| 12 19 28 34 }} of [[12edo|12et]]. As is shown above, 3/2 is mapped to 7\12, 5/4 to 4\12, and 7/4 to 10\12; we add these octave-reduced harmonics to the scale. We also have 9/8 mapped to 2\12. The next odd harmonic implied by the val is 15, which, after octave reduction, is mapped to 11\12. Follow the same process: 21/16 ~ 5\12, 25/16 ~ 7\12, and 27/16 ~ 8\12. Then there is 35/32 ~ 2\12, the same as ~9/8, so we reject it. Continuing on, we add 45/32 ~ 6\12, reject 49/32 ~ 8\12 which is the same as ~27/16, add 75/64 ~ 3\12, reject 81/64, 105/64, and 125/64, and add 135/128 ~ 1\12. With 135/128, we have added the last scale step of this 12-tone scale so the result is
 : 135/128, 9/8, 75/64, 5/4, 21/16, 45/32, 3/2, 27/16, 7/4, 15/8, 2/1
 And that is exactly [[Dwarf12 7]], the dwarf of 12et in the 7-limit.
+For another example, consider {{val| 7 11 16 }} of 7et in the 5-limit. Here 3/2 is mapped to 4\7, 5/4 to 2\7, 9/8 to 1\7, 15/8 to 6\7, 25/16 to 4\7 (discarded), 27/16 to 5\7, and 45/32 to 3\7, so the scale is:
+: 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1
+In fact, this is the Lydian mode of [[Zarlino]], which exposes the fact that all the intervals of Zarlino Lydian can be expressed as octave-reduced harmonics.
 == Generalization ==