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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 definition ==
== 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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<math>a^2 \equiv 1\mod N \hspace{4cm} (1)</math>.
<math>a^2 \equiv 1\mod N \hspace{4cm} (1)</math>.


There are also paraconcoctic scales, or chroma-negative concoctic scales. The formula for such a scale is  
A scale is called '''orthoconcoctic''', if the generator corresponding to note amount is the chroma-positive generator, for example - the 12edo diatonic 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 \hspace{4cm} (2)</math>.
<math>a^2 \equiv -1\mod N \hspace{4cm} (2)</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.  
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 terms of modal brightness.  


=== Example ===
=== Example ===
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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! rowspan="2" |Associated  
! rowspan="2" |Associated  
5-limit comma
5-limit comma
! rowspan="2" |Associated
other commas
!Notes
!Notes
|-
|-
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|720
|720
|[[16/15]]
|[[16/15]]
|
|
|
|-
|-
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|750
|750
|16/15
|16/15
|Forms the Father.
|
|Forms the [[Father]].
|-
|-
|10
|10
Line 96: Line 101:
|840
|840
|[[25/24]]
|[[25/24]]
|Forms the Dicot.
|
|Forms the [[Dicot]].
|-
|-
|12
|12
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|700
|700
|[[81/80]]
|[[81/80]]
|
|The scale predominantly in use in the world today.
|The scale predominantly in use in the world today.
|-
|-
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|738.461538
|738.461538
|[[2560/2187]]
|[[2560/2187]]
|
|Forms the [[Oneirotonic]] scale.
|Forms the [[Oneirotonic]] scale.
|-
|-
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|880
|880
|[[15625/15552]]*
|[[15625/15552]]*
|Forms the [[Hanson]].
|
|*Forms the [[Hanson]] (11b & 15)
|-
|-
|16
|16
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|675
|675
|[[135/128]]
|[[135/128]]
|
|Forms the [[Mavila]].
|Forms the [[Mavila]].
|-
|-
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|917.647059
|917.647059
|[[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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|660
|660
|[[34171875/33554432|[-25, 7, 6⟩]] c.II
|[[34171875/33554432|[-25, 7, 6⟩]] c.II
|
|
|
|-
|-
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|742.857143
|742.857143
|[39, -7, -12⟩
|[39, -7, -12⟩
|
|
|
|-
|-
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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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|864
|864
|3125/2916
|3125/2916
|
|Forms the [[Sixix]].
|Forms the [[Sixix]].
|-
|-
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|
|
|[<nowiki/>[[597871125/536870912|-29, 14, 3]]⟩
|[<nowiki/>[[597871125/536870912|-29, 14, 3]]⟩
|
|The 5-note scale itself is the [[slendric pentad]].
|The 5-note scale itself is the [[slendric pentad]].
|-
|-
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|
|
|[20, 5, -12⟩
|[20, 5, -12⟩
|
|
|
|-
|-
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|
|
|[[32805/32768]]
|[[32805/32768]]
|Forms the Helmholtz.
|
|Forms the [[Helmholtz (temperament)|Helmholtz]].
|-
|-
|30
|30
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|
|
|15625/15552 c.II
|15625/15552 c.II
|
|
|
|-
|-
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|
|
|64000/59049
|64000/59049
|Forms the Satriyo.
|
|Forms the [[Satriyo]].
|-
|-
|33
|33
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|
|
|177147/160000 c.II
|177147/160000 c.II
|
|
|
|-
|-
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|
|
|[39, -7, -12⟩
|[39, -7, -12⟩
|
|
|
|-
|-
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|
|
|[-41, 4, 15⟩
|[-41, 4, 15⟩
|
|
|
|-
|-
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|
|
|81/80 c.III
|81/80 c.III
|
|2.3.7 [[177147/175616]]
|In the 2.3.7, forms [[Liese]].
|-
|-
|37
|37
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|
|
|393216/390625 c.II
|393216/390625 c.II
|
|
|
|-
|-
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|
|
|[44, -13, -10⟩
|[44, -13, -10⟩
|
|
|
|-
|-
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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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|
|
|[-35, 6, 11⟩
|[-35, 6, 11⟩
|
|
|
|-
|-
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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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|
|
|[39, -7, -12⟩
|[39, -7, -12⟩
|
|
|
|-
|-
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|
|
|[-41, 1, 17⟩
|[-41, 1, 17⟩
|
|
|
|-
|-
|72
|72
|37\72, 53\72, 55\72
|37\72, 53\72, 55\72
|
|
|
|
|
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|
|
|735
|735
|
|
|
|49\80 forms the [[Semisept]].
|49\80 forms the [[Semisept]].
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|
|
|1014.285714
|1014.285714
|
|
|
|
|
Line 363: Line 397:
|
|
|843.956043
|843.956043
|
|
|
|
|
Line 368: Line 403:
|93
|93
|61\93
|61\93
|
|
|
|
|
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|100
|100
|51\100
|51\100
|
|
|
|
|
Retrieved from "https://en.xen.wiki/w/User:Eliora/Concoctic_scale"