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=== Temperaments ===
=== Temperaments ===
Since maximal evenness scales can be used to generate a temperament by merging the note count in the period and the period cardinality, in this case being 1 octave, an array of concoctic temperaments can be defined through such mergers. For example, temperament taken this way from 12edo, 7 & 12, is meantone, and is predominantly in use in the world's music today.
Since maximal evenness scales can be used to generate a temperament by merging the note count in the period and the period cardinality, in this case being 1 octave, an array of concoctic temperaments can be defined through such mergers. For example, temperament taken this way from 12edo, 7 & 12, is meantone, and is predominantly in use in the world's music today.
In addition, this also means that every concoctic scale has a 5-limit comma attached to it, and also an infinite array of 3-number subgroup commas.


== List ==
== List ==
The sequence of EDOs which have concoctic scales of any kind appears to be [[oeis:A172019|A172019]]. This implies that in order for an EDO to have a concoctic scale, it's number of coprime distinct generators must be divisible by 4. The reason for this is yet to be investigated.
The sequence of EDOs which have concoctic scales of any kind appears to be [[oeis:A172019|A172019]]. This implies that in order for an EDO to have a concoctic scale, it's number of coprime distinct generators must be divisible by 4. The reason for this is yet to be investigated.


The sequence has the asymptotic density 1, meaning that as EDOs grow increasingly large, they are significantly more likely to have a concoctic scale than not to. As a result, it may be better to refer to A097987, a set of numbers which lack a concoctic scale.
The sequence has the asymptotic density 1, meaning that as EDOs grow increasingly large, they are significantly more likely to have a concoctic scale than not to. As a result, it may be better to refer to [[oeis:A097987|A097987]], a set of numbers which lack a concoctic scale.


=== Concoctic scales in EDOs ===
=== Concoctic scales in EDOs ===
Notation: c.II means contorted order 2, etc for other Roman numerals.
{| class="wikitable"
{| class="wikitable"
|+
|+
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! colspan="2" |MOS
! colspan="2" |MOS
! colspan="2" |Generator Size (cents)
! colspan="2" |Generator Size (cents)
! rowspan="2" |Associated Ratio
! rowspan="2" |Associated  
5-limit comma
!Notes
!Notes
|-
|-
Line 73: Line 77:
|480
|480
|720
|720
|[[3/2]]
|[[16/15]]
|
|
|-
|-
Line 82: Line 86:
|450
|450
|750
|750
|[[14/9]]
|16/15
|
|Forms the Father.
|-
|-
|10
|10
Line 91: Line 95:
|360
|360
|840
|840
|[[13/8]]
|[[25/24]]
|
|Forms the Dicot.
|-
|-
|12
|12
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|500
|500
|700
|700
|[[3/2]]
|[[81/80]]
|The scale predominantly in use in the world today.
|The scale predominantly in use in the world today.
|-
|-
Line 109: Line 113:
|
|
|738.461538
|738.461538
|
|[[2560/2187]]
|Forms the [[Oneirotonic]] scale.
|Forms the [[Oneirotonic]] scale.
|-
|-
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|
|
|880
|880
|[[5/3]]
|[[15625/15552]]*
|Forms the [[Hanson]].
|Forms the [[Hanson]].
|-
|-
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|
|
|675
|675
|
|[[135/128]]
|Forms the [[Mavila]].
|Forms the [[Mavila]].
|-
|-
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|
|
|917.647059
|917.647059
|[[22/13]]
|[[25/24]] c.II
|Forms Huxley and Lovecraft, but with a fair error.
|Forms Huxley and Lovecraft, but with a fair error.
|-
|-
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|
|
|660
|660
|
|[[34171875/33554432|[-25, 7, 6⟩]] c.II
|
|
|-
|-
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|
|
|742.857143
|742.857143
|
|[39, -7, -12⟩
|
|
|-
|-
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|
|
|650, 850, 950
|650, 850, 950
|
|262144/253125 c.II,
|
32805/32768 c.II,
 
[[Godzilla|81/80 c.II]]
|Contorted Passion, contorted Helmholtz and Godzilla.
|-
|-
|25
|25
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|
|
|864
|864
|
|3125/2916
|Forms the [[Sixix]].
|Forms the [[Sixix]].
|-
|-
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|
|
|
|
|[[12/7]]
|[<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⟩
|
|
|-
|-
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|
|
|
|
|
|[[32805/32768]]
|
|Forms the Helmholtz.
|-
|-
|30
|30
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|
|
|
|
|
|15625/15552 c.II
|
|
|-
|-
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|
|
|
|
|
|64000/59049
|
|Forms the Satriyo.
|-
|-
|33
|33
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|
|
|
|
|
|177147/160000 c.II
|
|
|-
|-
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|
|
|
|
|
|[39, -7, -12⟩
|
|
|-
|-
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|
|
|
|
|
|[-41, 4, 15⟩
|
|
|-
|-
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|
|
|
|
|
|81/80 c.III
|
|
|-
|-
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|
|
|
|
|
|393216/390625 c.II
|
|
|-
|-
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|
|
|
|
|
|[44, -13, -10⟩
|
|
|-
|-
Line 280: Line 287:
|
|
|
|
|
|273375/262144,
|
 
[-57, 17, 13⟩,
 
[[Orson|[-21, 3, 7⟩]]
|31\40 forms the [[Orwell]] or Orson.
|-
|-
|41
|41
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|
|
|
|
|
|[-35, 6, 11⟩
|
|
|-
|-
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|
|
|
|
|[[40/27]]
|15625/15552 c.IV
|One step short of 53edo's perfect fifth.
|One step short of 53edo's perfect fifth.
|-
|-
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|
|
|
|
|[[5/4]]
|[-41, 1, 17⟩
|
|
|-
|-
Line 316: Line 327:
|
|
|
|
|10/7, [[5/3]], [[17/10]]
|
|53\72 forms the [[Catakleismic]].
|53\72 forms the [[Catakleismic]].
|-
|-
Line 325: Line 336:
|
|
|735
|735
|[[10/7]], [[26/17]]
|
|49\80 forms the [[Semisept]].
|49\80 forms the [[Semisept]].
|-
|-
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|
|
|843.956043
|843.956043
|[[13/8]]
|
|
|
|-
|-
Retrieved from "https://en.xen.wiki/w/User:Eliora/Concoctic_scale"