Dwarf: Difference between revisions

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A '''dwarf''' is a [[period]]ic [[scale]] obtained by sequentially mapping odd [[harmonic]]s (1, 3, 5, 7, …) using a [[regular temperament]]. A dwarf is a kind of [[detempering|detempered scale]]. The name ''dwarf'' refers to the fact that you are choosing for each degree the smallest [[Tenney height]]. Dwarf scales often produce results which are rich harmonically~~, with a tendency to favor~~ [[~~otonality and utonality|otonalities~~]] ~~over~~ [[~~otonality and utonality|utonalities~~]].

A '''dwarf''' is a [[period]]ic [[scale]] obtained by sequentially mapping odd [[harmonic]]s (1, 3, 5, 7, …) using a [[regular temperament]]. A dwarf is a kind of [[detempering|detempered scale]]. The name ''dwarf'' refers to the fact that you are choosing for each degree the smallest [[Tenney height]]. Dwarf scales often produce results which are rich harmonically. They exclusively contain [[otonal]] intervals, expressible as octave-reduced forms of harmonics, and may be flipped to provide [[utonal]] versions of the same scales.

== 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, 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 temper them to the equal temperament, 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 ==~~

The dwarf will ultimately end up being in a subgroup where no two primes map to the same interval when octave-reduced, as ultimately the simpler prime would always be chosen due to the method of the scale's construction.

~~{{See also| Scalesmith }}~~

~~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~~.

For example, let us consider [[7edo]] in the 5-limit: {{val| 7 11 16}}, reduced to {{val| 0 4 2}}.

~~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~~

* 1 is mapped to 0 steps (or 2/1 to the octave)

* 3 is mapped to 4 steps

* 5 is mapped to 2 steps

* 9 is mapped to a single step

* 15 is mapped to 6 steps

* 25 is mapped to 4 steps (but 3 is already chosen for 4 steps, so 25 can be ignored)

* 27 is mapped to 5 steps

* 45 is mapped to 3 steps

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

At this point, we have have 7 intervals, so the dwarf is complete: [9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1]. In fact, this is the Lydian mode of the [[zarlino]] diatonic scale, which exposes the fact that all the intervals of Lydian can be expressed as otonalities.

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

== Symmetrical dwarf ==

For a symmetrical scale, we may consider for each odd both the reduced interval and its octave complement, with the same priority (in the case of a tie, like in an even edo, either may be chosen). Unlike standard dwarves, which contain exclusively otonalities or utonalities, these contain a balanced number of both. Again taking 7edo in the 5-limit as an example:

* 1 is mapped to 0 steps (or 2/1 to the octave)

* 3 is mapped to 4 steps

* 4/3 is mapped to 3 steps

* 5 is mapped to 2 steps

* 8/5 is mapped to 5 steps

* 9 is mapped to a single step

* 16/9 is mapped to 6 steps

So the result is [9/8, 5/4, 4/3, 3/2, 8/5, 16/9, 2/1].

== See also ==

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-A '''dwarf''' is a [[period]]ic [[scale]] obtained by sequentially mapping odd [[harmonic]]s (1, 3, 5, 7, …) using a [[regular temperament]]. A dwarf is a kind of [[detempering|detempered scale]]. The name ''dwarf'' refers to the fact that you are choosing for each degree the smallest [[Tenney height]]. Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[otonality and utonality|otonalities]] over [[otonality and utonality|utonalities]].
+A '''dwarf''' is a [[period]]ic [[scale]] obtained by sequentially mapping odd [[harmonic]]s (1, 3, 5, 7, …) using a [[regular temperament]]. A dwarf is a kind of [[detempering|detempered scale]]. The name ''dwarf'' refers to the fact that you are choosing for each degree the smallest [[Tenney height]]. Dwarf scales often produce results which are rich harmonically. They exclusively contain [[otonal]] intervals, expressible as octave-reduced forms of harmonics, and may be flipped to provide [[utonal]] versions of the same scales.
 == 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, 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 temper them to the equal temperament, 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 ==
+The dwarf will ultimately end up being in a subgroup where no two primes map to the same interval when octave-reduced, as ultimately the simpler prime would always be chosen due to the method of the scale's construction.
-{{See also| Scalesmith }}
-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.
+For example, let us consider [[7edo]] in the 5-limit: {{val| 7 11 16}}, reduced to {{val| 0 4 2}}.
-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
+* 1 is mapped to 0 steps (or 2/1 to the octave)
+* 3 is mapped to 4 steps
+* 5 is mapped to 2 steps
+* 9 is mapped to a single step
+* 15 is mapped to 6 steps
+* 25 is mapped to 4 steps (but 3 is already chosen for 4 steps, so 25 can be ignored)
+* 27 is mapped to 5 steps
+* 45 is mapped to 3 steps
-: 135/128, 9/8, 75/64, 5/4, 21/16, 45/32, 3/2, 27/16, 7/4, 15/8, 2/1
+At this point, we have have 7 intervals, so the dwarf is complete: [9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1]. In fact, this is the Lydian mode of the [[zarlino]] diatonic scale, which exposes the fact that all the intervals of Lydian can be expressed as otonalities.
-And that is exactly [[Dwarf12 7]], the dwarf of 12et in the 7-limit.
+== Symmetrical dwarf ==
+For a symmetrical scale, we may consider for each odd both the reduced interval and its octave complement, with the same priority (in the case of a tie, like in an even edo, either may be chosen). Unlike standard dwarves, which contain exclusively otonalities or utonalities, these contain a balanced number of both. Again taking 7edo in the 5-limit as an example:
+* 1 is mapped to 0 steps (or 2/1 to the octave)
+* 3 is mapped to 4 steps
+* 4/3 is mapped to 3 steps
+* 5 is mapped to 2 steps
+* 8/5 is mapped to 5 steps
+* 9 is mapped to a single step
+* 16/9 is mapped to 6 steps
+So the result is [9/8, 5/4, 4/3, 3/2, 8/5, 16/9, 2/1].
 == See also ==