User:Eliora/Concoctic scale: Difference between revisions

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Concoctic scale (name proposed by Eliora) is a [[Maximal evenness|~~maximum eveness~~]] scale which has the same number of notes as its MOS generator.

Concoctic scale (name proposed by Eliora) is a [[Maximal evenness|maximally even]] scale which has the same number of notes as its MOS generator.

12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.

== Mathematical derivation ==

The length of a ~~maximum evenness~~ scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>.

The length of a maximally even scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>.

<math>ax \equiv 1\mod N</math>,

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12edo keyboard layout predominantly in use in the world today features 7 white keys and 5 black keys. In direction-conscious manner, the diatonic scale of 7 keys is obtained by stacking the generator, 7\12 fifth 7 times. Likewise, the pentatonic of black keys is obtained by stacking the 5\12 perfect fourth 5 times. And such scale is generated with the first formula.

On the other hand, in [[25edo]], stacking 18\25 will lead to ~~maximum evenness~~ scale of 7 note "black keys", and stacking 7\25 will result in a 18-note scale of "white keys". This is the EDO that only has the scale through the second formula.

On the other hand, in [[25edo]], stacking 18\25 will lead to maximally even scale of 7 note "black keys", and stacking 7\25 will result in a 18-note scale of "white keys". This is the EDO that only has the scale through the second formula.

=== Observations ===

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-Concoctic scale (name proposed by Eliora) is a [[Maximal evenness|maximum eveness]] scale which has the same number of notes as its MOS generator.
+Concoctic scale (name proposed by Eliora) is a [[Maximal evenness|maximally even]] scale which has the same number of notes as its MOS generator.
 edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.
 == Mathematical derivation ==
-The length of a maximum evenness scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>.
+The length of a maximally even scale's generator can be determined through a '''modular multiplicative inverse''' of the note amount and the tuning size<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>.
 <math>ax \equiv 1\mod N</math>,
@@ Line 25: / Line 25: @@
 edo keyboard layout predominantly in use in the world today features 7 white keys and 5 black keys. In direction-conscious manner, the diatonic scale of 7 keys is obtained by stacking the generator, 7\12 fifth 7 times. Likewise, the pentatonic of black keys is obtained by stacking the 5\12 perfect fourth 5 times. And such scale is generated with the first formula.
-On the other hand, in [[25edo]], stacking 18\25 will lead to maximum evenness scale of 7 note "black keys", and stacking 7\25 will result in a 18-note scale of "white keys". This is the EDO that only has the scale through the second formula.
+On the other hand, in [[25edo]], stacking 18\25 will lead to maximally even scale of 7 note "black keys", and stacking 7\25 will result in a 18-note scale of "white keys". This is the EDO that only has the scale through the second formula.
 === Observations ===