User:Eliora/Concoctic scale: Difference between revisions

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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>:

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

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

where N is the period, and a is the note count. Therefore, a concoctic scale is defined for a given N:

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

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

which simply becomes

<math>a^2 \equiv 1 \mod N</math>.

<math>a^2 \equiv 1\mod N</math>.

~~Paraconcoctic scales~~ are ~~those, which in a pure sense are the octave inversions of one another. For example, a {7/10}'s generator is 3, and of {3/10} is 7. Since octave-inverting the MOS generator has no impact on the scale,~~ paraconcoctic scales ~~are identical to their usual counterparts. However~~, ~~the difference is pronounced in keyboard making~~ - ~~in paraconcoctic~~ scales~~, white keys' generator will be the amount of black keys and vice versa~~. The formula for such a scale is

There are also paraconcoctic scales, or chroma-negative concoctic scales. The formula for such a scale is

<math>a^2 \equiv -1 \mod N</math>

<math>a^2 \equiv -1\mod N</math>.

Since octave-inverting the MOS generator has no impact on the scale, paraconcoctic scales are identical to their usual, orthoconcoctic counterparts. However, the difference is pronounced in keyboard making - in terms of chroma direction, the white keys' generator will be the amount of black keys and vice versa.

=== Example ===

12edo keyboard layout predominantly in use in the world today features 7 white keys and 5 black keys. 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 it 5 times.

== List ==

@@ Line 6: / Line 6: @@
 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>:
-<math>ax \equiv 1 \mod N</math>,
+<math>ax \equiv 1\mod N</math>,
 where N is the period, and a is the note count. Therefore, a concoctic scale is defined for a given N:
-<math>aa \equiv 1 \mod N</math>,
+<math>aa \equiv 1\mod N</math>,
 which simply becomes
-<math>a^2 \equiv 1 \mod N</math>.
+<math>a^2 \equiv 1\mod N</math>.
-Paraconcoctic scales are those, which in a pure sense are the octave inversions of one another. For example, a {7/10}'s generator is 3, and of {3/10} is 7. Since octave-inverting the MOS generator has no impact on the scale, paraconcoctic scales are identical to their usual counterparts. However, the difference is pronounced in keyboard making - in paraconcoctic scales, white keys' generator will be the amount of black keys and vice versa. The formula for such a scale is
+There are also paraconcoctic scales, or chroma-negative concoctic scales. The formula for such a scale is
-<math>a^2 \equiv -1 \mod N</math>
+<math>a^2 \equiv -1\mod N</math>.
+Since octave-inverting the MOS generator has no impact on the scale, paraconcoctic scales are identical to their usual, orthoconcoctic counterparts. However, the difference is pronounced in keyboard making - in terms of chroma direction, the white keys' generator will be the amount of black keys and vice versa.
+=== Example ===
+edo keyboard layout predominantly in use in the world today features 7 white keys and 5 black keys. 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 it 5 times.
 == List ==