User talk:FloraC: Difference between revisions

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: Well, I took a look since a user on the Discord reported they had issues signing up. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:11, 15 December 2021 (UTC)

== Normalized mapping vs minimum generator ==

Why is the generator wider than a half octave in some temperaments? Why did you edit mappings to normalize?

* [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}] (generator: ~3 = 1896.5 cents) vs [{{val|1 2 4 7}}, {{val|0 -1 -4 -10}}] (generator: ~4/3 = 503.5 cents) in the [[Meantone family|meantone temperament]] (12&19)

* [{{val|1 0 -4 2}}, {{val|0 2 8 1}}] (generator: ~7/4 = 947.4 cents) vs [{{val|1 2 4 3}}, {{val|0 -2 -8 -1}}] (generator: ~7/6 = 252.6 cents) in the [[Meantone temperament #Godzilla|godzilla temperament]] (5&14c)

* [{{val|1 0 -13 -3}}, {{val|0 3 29 11}}] (generator: ~81/56 = 634.0 cents) vs [{{val|1 3 16 8}}, {{val|0 -3 -29 -11}}] (generator: ~112/81 = 566.0 cents) in the [[Tricot family|tricot temperament]] (53&70)

* [{{val|1 7 3 15}}, {{val|0 -8 -1 -18}}] (generator: ~8/5 = 812.6 cents) vs [{{val|1 -1 2 -3}}, {{val|0 8 1 18}}] (generator: ~5/4 = 387.4 cents) in the [[Würschmidt family|würschmidt temperament]] (31&96)

* [{{val|1 12 56 -2}}, {{val|0 -13 -67 6}}] (generator: ~256/147 = 961.4 cents) vs [{{val|1 -1 -11 4}}, {{val|0 13 67 -6}}] (generator: ~147/128 = 238.6 cents) in the [[Wizmic microtemperaments #Tokko|tokko temperament]] (5&166)

* [{{val|1 16 32 -15}}, {{val|0 -17 -35 21}}] (generator: ~9/5 = 1017.5 cents) vs [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}] (generator: ~10/9 = 182.5 cents) in the [[Minortonic family #Mitonic|mitonic temperament]] (46&125)

* [{{val|1 25 -31 -8}}, {{val|0 -26 37 12}}] (generator: ~28/15 = 1080.7 cents) vs [{{val|1 -1 6 4}}, {{val|0 26 -37 -12}}] (generator: ~15/14 = 119.3 cents) in the [[Breedsmic temperaments #Septidiasemi|septidiasemi temperament]] (10&161)

* [{{val|1 17 9 10}}, {{val|0 -30 -13 -14}}] (generator: ~10/7 = 616.6 cents) vs [{{val|1 -13 -4 -4}}, {{val|0 30 13 14}}] (generator: ~7/5 = 583.4 cents) in the [[Breedsmic temperaments #Cotritone|cotritone temperament]] (37&72)

* [{{val|2 0 11 31}}, {{val|0 1 -2 -8}}] (generator: ~3 = 1903.7 cents) vs [{{val|2 3 5 7}}, {{val|0 1 -2 -8}}] (generator: ~16/15 = 103.7 cents) in the [[Diaschismic family #Diaschismic|diaschismic temperament]] (46&58)

* [{{val|2 1 9 -2}}, {{val|0 2 -4 7}}] (generator: ~35/24 = 652.8 cents) vs [{{val|2 3 5 5}}, {{val|0 2 -4 7}}] (generator: ~36/35 = 52.8 cents) in the [[Diaschismic family #Shrutar|shrutar temperament]] (22&46)

* [{{val|3 0 7 18}}, {{val|0 1 0 -2}}] (generator: ~3 = 1909.3 cents) vs [{{val|3 5 7 8}}, {{val|0 -1 0 2}}] (generator: ~16/15 = 90.7 cents) in [[Augmented family #Augene|augene temperament]] (12&15)

* [{{val|9 1 1 12}}, {{val|0 2 3 2}}] (generator: ~5/3 = 884.3 cents) vs [{{val|9 15 22 26}}, {{val|0 -2 -3 -2}}] (generator: ~36/35 = 49.0 cents) in the [[Ragismic microtemperaments #Ennealimmal|ennealimmic temperament]] (27&45)

There are an infinite of mappings of each temperaments including normalized form (left) and minimum generator form (right). In the normalized form, ''a2'' in the mapping [{{val|a1 a2 a3 …}}, {{val|0 b2 b3 …}}] takes 0 ≤ ''a2'' < abs(''b2'') if ''b2'' ≠ 0. The minimum generator form ("Reduced Mapping" in the [http://x31eq.com/temper Temperament finding scripts] by [[Graham Breed]], taking 0 ≤ ''g'' ≤ ''p''/2 where ''p'' is the period and ''g'' is the generator) can be yielded by Euclidean algorithm. Which form are you favor? --[[User:Xenllium|Xenllium]] ([[User talk:Xenllium|talk]]) 13:48, 29 January 2022 (UTC)

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 : Well, I took a look since a user on the Discord reported they had issues signing up. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:11, 15 December 2021 (UTC)
+== Normalized mapping vs minimum generator ==
+Why is the generator wider than a half octave in some temperaments? Why did you edit mappings to normalize?
+* [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}] (generator: ~3 = 1896.5 cents) vs [{{val|1 2 4 7}}, {{val|0 -1 -4 -10}}] (generator: ~4/3 = 503.5 cents) in the [[Meantone family|meantone temperament]] (12&amp;19)
+* [{{val|1 0 -4 2}}, {{val|0 2 8 1}}] (generator: ~7/4 = 947.4 cents) vs [{{val|1 2 4 3}}, {{val|0 -2 -8 -1}}] (generator: ~7/6 = 252.6 cents) in the [[Meantone temperament #Godzilla|godzilla temperament]] (5&amp;14c)
+* [{{val|1 0 -13 -3}}, {{val|0 3 29 11}}] (generator: ~81/56 = 634.0 cents) vs [{{val|1 3 16 8}}, {{val|0 -3 -29 -11}}] (generator: ~112/81 = 566.0 cents) in the [[Tricot family|tricot temperament]] (53&amp;70)
+* [{{val|1 7 3 15}}, {{val|0 -8 -1 -18}}] (generator: ~8/5 = 812.6 cents) vs [{{val|1 -1 2 -3}}, {{val|0 8 1 18}}] (generator: ~5/4 = 387.4 cents) in the [[Würschmidt family|würschmidt temperament]] (31&amp;96)
+* [{{val|1 12 56 -2}}, {{val|0 -13 -67 6}}] (generator: ~256/147 = 961.4 cents) vs [{{val|1 -1 -11 4}}, {{val|0 13 67 -6}}] (generator: ~147/128 = 238.6 cents) in the [[Wizmic microtemperaments #Tokko|tokko temperament]] (5&amp;166)
+* [{{val|1 16 32 -15}}, {{val|0 -17 -35 21}}] (generator: ~9/5 = 1017.5 cents) vs [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}] (generator: ~10/9 = 182.5 cents) in the [[Minortonic family #Mitonic|mitonic temperament]] (46&amp;125)
+* [{{val|1 25 -31 -8}}, {{val|0 -26 37 12}}] (generator: ~28/15 = 1080.7 cents) vs [{{val|1 -1 6 4}}, {{val|0 26 -37 -12}}] (generator: ~15/14 = 119.3 cents) in the [[Breedsmic temperaments #Septidiasemi|septidiasemi temperament]] (10&amp;161)
+* [{{val|1 17 9 10}}, {{val|0 -30 -13 -14}}] (generator: ~10/7 = 616.6 cents) vs [{{val|1 -13 -4 -4}}, {{val|0 30 13 14}}] (generator: ~7/5 = 583.4 cents) in the [[Breedsmic temperaments #Cotritone|cotritone temperament]] (37&amp;72)
+* [{{val|2 0 11 31}}, {{val|0 1 -2 -8}}] (generator: ~3 = 1903.7 cents) vs [{{val|2 3 5 7}}, {{val|0 1 -2 -8}}] (generator: ~16/15 = 103.7 cents) in the [[Diaschismic family #Diaschismic|diaschismic temperament]] (46&amp;58)
+* [{{val|2 1 9 -2}}, {{val|0 2 -4 7}}] (generator: ~35/24 = 652.8 cents) vs [{{val|2 3 5 5}}, {{val|0 2 -4 7}}] (generator: ~36/35 = 52.8 cents) in the [[Diaschismic family #Shrutar|shrutar temperament]] (22&amp;46)
+* [{{val|3 0 7 18}}, {{val|0 1 0 -2}}] (generator: ~3 = 1909.3 cents) vs [{{val|3 5 7 8}}, {{val|0 -1 0 2}}] (generator: ~16/15 = 90.7 cents) in [[Augmented family #Augene|augene temperament]] (12&amp;15)
+* [{{val|9 1 1 12}}, {{val|0 2 3 2}}] (generator: ~5/3 = 884.3 cents) vs [{{val|9 15 22 26}}, {{val|0 -2 -3 -2}}] (generator: ~36/35 = 49.0 cents) in the [[Ragismic microtemperaments #Ennealimmal|ennealimmic temperament]] (27&amp;45)
+There are an infinite of mappings of each temperaments including normalized form (left) and minimum generator form (right). In the normalized form, ''a<sub>2</sub>'' in the mapping [{{val|a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> …}}, {{val|0 b<sub>2</sub> b<sub>3</sub> …}}] takes 0 &le; ''a<sub>2</sub>'' &lt; abs(''b<sub>2</sub>'') if ''b<sub>2</sub>'' &ne; 0. The minimum generator form ("Reduced Mapping" in the [http://x31eq.com/temper Temperament finding scripts] by [[Graham Breed]], taking 0 &le; ''g'' &le; ''p''/2 where ''p'' is the period and ''g'' is the generator) can be yielded by Euclidean algorithm. Which form are you favor? --[[User:Xenllium|Xenllium]] ([[User talk:Xenllium|talk]]) 13:48, 29 January 2022 (UTC)