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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.
{{Editable user page}}
A '''concoctic scale''' (name proposed by Eliora) is a [[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.
12edo 5L2s diatonic scale, the predominantly used scale in the world's music today, is an example.


== Mathematical derivation ==
== 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>,
<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.
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 ===
=== Observations ===
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|16/15
|16/15
|
|
|Forms the Father.
|Forms the [[Father]].
|-
|-
|10
|10
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|[[25/24]]
|[[25/24]]
|
|
|Forms the Dicot.
|Forms the [[Dicot]].
|-
|-
|12
|12
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|[[25/24]] c.II
|[[25/24]] c.II
|
|
|Forms Huxley and Lovecraft, but with a fair error.
|Forms [[Lovecraft]], [[Huxley]] and [[Subklei]], but with a fair error.
|-
|-
|20
|20
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[[Godzilla|81/80 c.II]]
[[Godzilla|81/80 c.II]]
|
|
|Contorted Passion, contorted Helmholtz and Godzilla.
|Contorted [[Passion]], contorted [[Helmholtz (temperament)|Helmholtz]] and [[Godzilla]].
|-
|-
|25
|25
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|[[32805/32768]]
|[[32805/32768]]
|
|
|Forms the Helmholtz.
|Forms the [[Helmholtz (temperament)|Helmholtz]].
|-
|-
|30
|30
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|64000/59049
|64000/59049
|
|
|Forms the Satriyo.
|Forms the [[Satriyo]].
|-
|-
|33
|33
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[[Orson|[-21, 3, 7⟩]]
[[Orson|[-21, 3, 7⟩]]
|
|
|31\40 forms the [[Orwell]] or Orson.
|31\40 forms the [[Orwell]] or [[Orson]].
|-
|-
|41
|41
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|15625/15552 c.IV
|15625/15552 c.IV
|
|
|One step short of 53edo's perfect fifth.
|One step short of [[53edo]]'s perfect fifth.
|-
|-
|55
|55
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