Dwarf: Difference between revisions

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A '''dwarf''' is a [[scale]] obtained by mapping odd [[harmonic]]s (1, 3, 5, 7...) using a [[regular temperament]].

The name "dwarf" refers to the fact that you are choosing for each degree the smallest [[Benedetti height]].

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== Construction ==

Suppose v is a [[~~Vals_and_Tuning_Space|~~val]] ~~<~~n ... | whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced [[wikipedia:Modulo_operation|modulo]] n, are distinct. An example would be ~~<~~12 19 28 34|; reduced mod 12 this is ~~<~~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. Examples may be found on the [[Scalesmith]] page.

Suppose v is a [[val]] {{val|n ...}} whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced [[wikipedia: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. Examples may be found on the [[Scalesmith]] page.

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

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

== See also ==

* [[Especially tempered dwarves]]

* [[Marvelous dwarves]]

[[Category:Benedetti]]

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[[Category:Math]]

[[Category:Scale]]

~~{{todo|reduce mathslang|improve readability|increase focus to lemma}}~~

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+{{Inaccessible}}
+A '''dwarf''' is a [[scale]] obtained by mapping odd [[harmonic]]s (1, 3, 5, 7...) using a [[regular temperament]].
 The name "dwarf" refers to the fact that you are choosing for each degree the smallest [[Benedetti height]].
@@ Line 4: / Line 7: @@
 == Construction ==
-Suppose v is a [[Vals_and_Tuning_Space|val]] &lt;n ... | whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced [[wikipedia:Modulo_operation|modulo]] n, are distinct. An example would be &lt;12 19 28 34|; reduced mod 12 this is &lt;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. Examples may be found on the [[Scalesmith]] page.
+Suppose v is a [[val]] {{val|n ...}} whose first coordinate is a positive integer n, and suppose the coordinates of the val, reduced [[wikipedia: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. Examples may be found on the [[Scalesmith]] page.
+Of particular interest are dwarf scales resulting from equal temperament vals which are [[Periodic_scale|epimorphic]] for the val v, but even vals far removed from an equal temperament will produce a scale.
-Of particular interest are dwarf scales resulting from equal temperament vals which are [[Periodic_scale|epimorphic]] for the val v, but even vals far removed from an equal temperament will produce a scale.
+== See also ==
+* [[Especially tempered dwarves]]
+* [[Marvelous dwarves]]
 [[Category:Benedetti]]
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 [[Category:Math]]
 [[Category:Scale]]
-{{todo|reduce mathslang|improve readability|increase focus to lemma}}