Dwarf - Revision history

FloraC: Re-install some changes that are valuable (2/2)

2025-11-12T10:15:49Z

Re-install some changes that are valuable (2/2)

← Older revision		Revision as of 10:15, 12 November 2025
Line 2:		Line 2:

	== Construction ==		== 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 ==		== Example ==
Line 9:		Line 9:
	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.		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		: 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.		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 ==		== Generalization ==

FloraC: Re-install some changes that are valuable (1/)

2025-11-12T09:57:51Z

Re-install some changes that are valuable (1/)

← Older revision		Revision as of 09:57, 12 November 2025
Line 1:		Line 1:
	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 are [[otonal]], with intervals expressible as [[octave reduction\|octave-reduced]] forms of [[harmonic]]s, and may be flipped to provide [[utonal]] versions of the same scales.

	== Construction ==		== Construction ==
Line 14:		Line 14:

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

			== Generalization ==
			=== Symmetrical dwarf ===
			For a symmetrical scale, we may consider for each odd harmonic both the octave reduction and octave complement, with equal priority. In the case of a tie, like in an even edo, either may be chosen. Unlike standard dwarves, which are strictly otonal, these are more balanced. For example, the symmetrical dwarf of 7et in the 5-limit is

			: 9/8, 5/4, 4/3, 3/2, 8/5, 16/9, 2/1

	== See also ==		== See also ==

FloraC: Too many problems in the last edit. We need to review it step by step

2025-11-12T08:50:55Z

Too many problems in the last edit. We need to review it step by step

← Older revision		Revision as of 08:50, 12 November 2025
Line 1:		Line 1:
	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~~.		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]].

	== Construction ==		== Construction ==
	~~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''.		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''.

	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.		== Example ==
			{{See also\| Scalesmith }}

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

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

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

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

VectorGraphics at 04:37, 12 November 2025

2025-11-12T04:37:01Z

← Older revision		Revision as of 04:37, 12 November 2025
Line 1:		Line 1:
	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 ==		== 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 ==		== See also ==

FloraC: +a long awaited example section. Note the periodicity and detemperedness and misc. clarifications. Benedetti height -> Tenney height (since the latter has better properties)

2024-11-01T10:43:12Z

+a long awaited example section. Note the periodicity and detemperedness and misc. clarifications. Benedetti height -> Tenney height (since the latter has better properties)

← Older revision		Revision as of 10:43, 1 November 2024
Line 1:		Line 1:
	~~{{Inaccessible}}~~		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 [[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]]. Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[otonality and utonality\|otonalities]] over [[otonality and utonality\|utonalities]].

	== Construction ==		== 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''. ~~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 {{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''.

			== Example ==
			{{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.		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

			: 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.

	== See also ==		== See also ==

FloraC: Style improvements

2024-11-01T09:46:14Z

Style improvements

← Older revision		Revision as of 09:46, 1 November 2024
Line 1:		Line 1:
	{{Inaccessible}}		{{Inaccessible}}
	A '''dwarf''' is a [[scale]] obtained by mapping odd [[harmonic]]s (1, 3, 5, 7~~...~~) using a [[regular temperament]].		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]]. Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[otonality and utonality\|otonalities]] over [[otonality and utonality\|utonalities]].

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

	Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[~~otonal~~]]~~ities~~ over [[~~utonal~~]]~~ities~~.

	== Construction ==		== Construction ==
	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.		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''. 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 [[epimorphic]] for the val ''V'', but even vals far removed from an equal temperament will produce a scale.

	== See also ==		== See also ==

FloraC: Recategorize

2023-12-13T17:04:14Z

Recategorize

← Older revision		Revision as of 17:04, 13 December 2023
Line 15:		Line 15:
	* [[Marvelous dwarves]]		* [[Marvelous dwarves]]

	[[Category:~~Benedetti]]~~		[[Category:Dwarf\| ]] <!-- main article -->
	~~[[Category:Dwarves~~]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Scale]]		[[Category:Scale]]

Fredg999: Improve lead section (slightly), misc. edits, mark as inaccessible

2023-07-08T14:54:50Z

Improve lead section (slightly), misc. edits, mark as inaccessible

← Older revision		Revision as of 14:54, 8 July 2023
Line 1:		Line 1:
			{{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]].		The name "dwarf" refers to the fact that you are choosing for each degree the smallest [[Benedetti height]].

Line 4:		Line 7:

	== Construction ==		== 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]]		[[Category:Benedetti]]
Line 12:		Line 19:
	[[Category:Math]]		[[Category:Math]]
	[[Category:Scale]]		[[Category:Scale]]
	~~{{todo\|reduce mathslang\|improve readability\|increase focus to lemma}}~~

BudjarnLambeth: Improved readability slightly, increased focus to lemma slightly

2023-07-07T02:33:37Z

Improved readability slightly, increased focus to lemma slightly

← Older revision		Revision as of 02:33, 7 July 2023
Line 1:		Line 1:
	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 [http://en.wikipedia.org/wiki/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\|Scalesmith~~]] ~~page~~.		The name "dwarf" refers to the fact that you are choosing for each degree the smallest [[Benedetti height]].

	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. Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[otonal\|otonal]]ities over [[utonal\|utonal]]ities. The name "dwarf" refers to the fact that you are choosing for each degree the smallest [[Benedetti_height\|Benedetti height]].		Dwarf scales often produce results which are rich harmonically, with a tendency to favor [[otonal]]ities over [[utonal]]ities.

			== 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.

			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.

	[[Category:Benedetti]]		[[Category:Benedetti]]

Fredg999 category edits: Moving from Category:Dwarf to Category:Dwarves using Cat-a-lot

2023-04-23T20:30:44Z

Moving from Category:Dwarf to Category:Dwarves using Cat-a-lot

← Older revision		Revision as of 20:30, 23 April 2023
Line 4:		Line 4:

	[[Category:Benedetti]]		[[Category:Benedetti]]
	[[Category:~~Dwarf~~]]		[[Category:Dwarves]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Scale]]		[[Category:Scale]]
	{{todo\|reduce mathslang\|improve readability\|increase focus to lemma}}		{{todo\|reduce mathslang\|improve readability\|increase focus to lemma}}