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Concoctic scale (name proposed by Eliora) is a 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 today, is an example of such a scale.
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>.


a*x ≡ 1 (mod N),
<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:
where N is the period, and a is the note count. Therefore, a concoctic scale is defined for a given N:


a*a ≡ 1 (mod N),
<math>aa \equiv 1\mod N</math>,


which becomes
which simply becomes


a^2 ≡ 1 (mod N).
<math>a^2 \equiv 1\mod N \hspace{4cm} (1)</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.
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>.
 
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 ===
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 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 ===
A scale that is of the form (n/2+1)\n, where n is divisible by 4, is always orthoconcoctic. 12edo diatonic is also an example of such.
 
It can be shown as follows:
 
Let <math>k = \frac{n}{4}</math> and rewrite the expression as <math>[(2k+1)/4k]</math>;
 
<math>(2k+1)^2 = 4k^2 + 4k + 1</math>;
 
<math>4k^2</math> is divisible by 4 and k and thus by 4k;
 
4k being divisible by 4k is self-explanatory.
 
Therefore the remainder of +1 means that such a scale will always be orthoconcoctic. This type of scale, when used in keyboard making, produces two bundles of white keys whose numbers of black keys inside of them are 1 number apart, and so are the numbers of white keys themselves. The sequence goes as follows: 5\8, 7\12, 9\16, 11\20, etc.
 
=== 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.
 
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 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 ===
Notation: c.II means contorted order 2, etc for other Roman numerals.
{| class="wikitable"
{| class="wikitable"
|+
|+
Non-trivial concoctic scales (above 1\2) in EDOs up to 100 that have them
Non-trivial concoctic scales in EDOs up to 100 that have them
!N
! rowspan="2" |N
!Scale
! rowspan="2" |Scale\EDO
!Mos type
! colspan="2" |MOS
!Generator Size (cents)
! colspan="2" |Generator Size (cents)
! rowspan="2" |Associated
5-limit comma
! rowspan="2" |Associated
other commas
!Notes
!Notes
|-
!Chroma+
!Chroma-
!Below 1\2
!Above 1\2
!
|-
|-
|5
|5
|3\5
|3\5
|
|
|
|480
|720
|720
|[[16/15]]
|
|
|
|-
|-
|8
|8
|5\8
|5\8
|
|[[3L 2s]]
|2L 1s
|450
|750
|750
|16/15
|
|
|Forms the [[Father]].
|-
|-
|10
|10
|7\10
|7\10
|2L 1s
|[[3L 4s]]
|360
|840
|[[25/24]]
|
|
|
|Forms the [[Dicot]].
|
|-
|-
|12
|12
|7\12
|7\12
|5L 2s
|[[5L 2s]]
|[[2L 3s]]
|500
|700
|700
|The system predominantly in use in the world today.
|[[81/80]]
|
|The scale predominantly in use in the world today.
|-
|-
|13
|13
|8\13
|8\13
|[[3L 2s]]
|[[5L 3s]]
|
|
|738.461538
|[[2560/2187]]
|
|
|
|Forms the [[Oneirotonic]] scale.
|-
|-
|15
|15
|11\15
|11\15
|3L 1s
|[[Tetrad 3L 1s|4L 7s]]
|[[Tetrad 3L 1s|3L 1s]]
|
|880
|880
|[[15625/15552]]*
|
|
|*Forms the [[Hanson]] (11b & 15)
|-
|-
|16
|16
|9\16
|9\16
|7L 2s
|[[7L 2s]]
|[[2L 5s]]
|
|675
|675
|[[135/128]]
|
|
|Forms the [[Mavila]].
|-
|-
|17
|17
|13\17
|13\17
|[[1L 3s]]
|[[4L 9s]]
|
|
|917.647059
|[[25/24]] c.II
|
|
|
|Forms [[Lovecraft]], [[Huxley]] and [[Subklei]], but with a fair error.
|-
|-
|20
|20
|11\20
|11\20
|[[9L 2s]]
|[[2L 7s]]
|
|
|660
|[[34171875/33554432|[-25, 7, 6⟩]] c.II
|
|
|
|
Line 85: Line 167:
|13\21
|13\21
|
|
|
|
|742.857143
|[39, -7, -12⟩
|
|
|
|
Line 91: Line 177:
|13\24, 17\24, 19\24
|13\24, 17\24, 19\24
|
|
|350, 650, 850
|
|
|
|650, 850, 950
|262144/253125 c.II,
32805/32768 c.II,
[[Godzilla|81/80 c.II]]
|
|Contorted [[Passion]], contorted [[Helmholtz (temperament)|Helmholtz]] and [[Godzilla]].
|-
|-
|25
|25
Line 99: Line 192:
|
|
|
|
|864
|3125/2916
|
|Forms the [[Sixix]].
|-
|-
|26
|26
|21\16
|21\26
|[[1L 4s]]
|[[5L 16s]]
|
|
|
|
|Forms the slendric pentad
|[<nowiki/>[[597871125/536870912|-29, 14, 3]]⟩
|
|The 5-note scale itself is the [[slendric pentad]].
|-
|-
|28
|28
|15\28
|15\28
|
|
|
|
|
|[20, 5, -12⟩
|
|
|
|
Line 117: Line 222:
|
|
|
|
|
|[[32805/32768]]
|
|Forms the [[Helmholtz (temperament)|Helmholtz]].
|-
|30
|19\30
|
|
|
|
|15625/15552 c.II
|
|
|-
|32
|17\32
|
|
|
|
|64000/59049
|
|Forms the [[Satriyo]].
|-
|33
|23\33
|
|
|
|
|177147/160000 c.II
|
|
|-
|34
|21\34
|
|
|
|
|[39, -7, -12⟩
|
|
|-
|35
|29\35
|
|
|
|
|[-41, 4, 15⟩
|
|
|-
|36
|19\36
|
|
|
|
|81/80 c.III
|2.3.7 [[177147/175616]]
|In the 2.3.7, forms [[Liese]].
|-
|37
|31\37
|
|
|
|
|393216/390625 c.II
|
|
|-
|39
|25\39
|
|
|
|
|[44, -13, -10⟩
|
|
|-
|40
|21\40, 29\40, 31\40
|
|
|
|
|273375/262144,
[-57, 17, 13⟩,
[[Orson|[-21, 3, 7⟩]]
|
|31\40 forms the [[Orwell]] or [[Orson]].
|-
|41
|32\41
|
|
|
|
|[-35, 6, 11⟩
|
|
|-
|53
|30\53
|
|
|
|
|15625/15552 c.IV
|
|One step short of [[53edo]]'s perfect fifth.
|-
|55
|34\55
|
|
|
|
|[39, -7, -12⟩
|
|
|-
|69
|22\69
|
|
|
|
|[-41, 1, 17⟩
|
|
|-
|72
|37\72, 53\72, 55\72
|
|
|
|
|
|
|53\72 forms the [[Catakleismic]].
|-
|80
|41\80, 49\80
|
|
|
|735
|
|
|49\80 forms the [[Semisept]].
|-
|-
|84
|84
|71\84
|71\84
|58L 13s
|58L 13s
|
|
|1014.285714
|1014.285714
|
|
|
|
|-
|-
Line 127: Line 394:
|64\91
|64\91
|37L 27s
|37L 27s
|
|
|843.956043
|843.956043
|
|
|
|-
|93
|61\93
|
|
|
|
|
|
|
|-
|100
|51\100
|
|
|
|
|
|
|
|
|}
|}
=== Concoctic scales by note count ===
{| class="wikitable"
|+Concoctic scales sorted by note count in MOSes that have them
!N
!MOS (chroma+)
!Generator
!Cents
|-
| rowspan="5" |7
|1L 6s
|
|
|-
|2L 5s
|7\16
|
|-
|3L 4s
|7\10
|
|-
|5L 2s
|7\12
|
|-
|6L 1s
|
|
|}
== References ==
[[Category:Scale]]
Retrieved from "https://en.xen.wiki/w/User:Eliora/Concoctic_scale"