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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>,
Line 16: Line 17:
<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 ===
Line 41: Line 42:


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.
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 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.
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\EDO
! rowspan="2" |Scale\EDO
!MOS
! colspan="2" |MOS
(chroma+)
! colspan="2" |Generator Size (cents)
!MOS
! rowspan="2" |Associated
(chroma-)
5-limit comma
!Generator Size (cents)
! rowspan="2" |Associated
!Associated Ratio
other commas
!Notes
!Notes
|-
!Chroma+
!Chroma-
!Below 1\2
!Above 1\2
!
|-
|-
|5
|5
Line 63: Line 78:
|
|
|
|
|480
|720
|720
|[[3/2]]
|[[16/15]]
|
|
|
|-
|-
Line 71: Line 88:
|[[3L 2s]]
|[[3L 2s]]
|2L 1s
|2L 1s
|450
|750
|750
|[[14/9]]
|16/15
|
|
|Forms the [[Father]].
|-
|-
|10
|10
Line 79: Line 98:
|2L 1s
|2L 1s
|[[3L 4s]]
|[[3L 4s]]
|360
|840
|840
|[[13/8]]
|[[25/24]]
|
|
|Forms the [[Dicot]].
|-
|-
|12
|12
Line 87: Line 108:
|[[5L 2s]]
|[[5L 2s]]
|[[2L 3s]]
|[[2L 3s]]
|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 95: Line 118:
|[[3L 2s]]
|[[3L 2s]]
|[[5L 3s]]
|[[5L 3s]]
|
|738.461538
|738.461538
|[[2560/2187]]
|
|
|Forms the [[Oneirotonic]] scale.
|Forms the [[Oneirotonic]] scale.
Line 103: Line 128:
|[[Tetrad 3L 1s|4L 7s]]
|[[Tetrad 3L 1s|4L 7s]]
|[[Tetrad 3L 1s|3L 1s]]
|[[Tetrad 3L 1s|3L 1s]]
|
|880
|880
|[[5/3]]
|[[15625/15552]]*
|Forms the [[Hanson]].
|
|*Forms the [[Hanson]] (11b & 15)
|-
|-
|16
|16
Line 111: Line 138:
|[[7L 2s]]
|[[7L 2s]]
|[[2L 5s]]
|[[2L 5s]]
|
|675
|675
|[[135/128]]
|
|
|Forms the [[Mavila]].
|Forms the [[Mavila]].
Line 119: Line 148:
|[[1L 3s]]
|[[1L 3s]]
|[[4L 9s]]
|[[4L 9s]]
|
|917.647059
|917.647059
|[[22/13]]
|[[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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|[[9L 2s]]
|[[9L 2s]]
|[[2L 7s]]
|[[2L 7s]]
|
|660
|660
|[[34171875/33554432|[-25, 7, 6⟩]] c.II
|
|
|
|
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|21
|21
|13\21
|13\21
|
|
|
|
|
|742.857143
|742.857143
|[39, -7, -12⟩
|
|
|
|
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|24
|24
|13\24, 17\24, 19\24
|13\24, 17\24, 19\24
|
|
|
|
|
|650, 850, 950
|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
|18\25
|18\25
|
|
|
|
|
|864
|864
|3125/2916
|
|
|
|Forms the [[Sixix]].
|-
|-
|26
|26
Line 160: Line 202:
|[[5L 16s]]
|[[5L 16s]]
|
|
|[[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]].
|-
|-
Line 168: Line 212:
|
|
|
|
|
|[20, 5, -12⟩
|
|
|
|
Line 177: Line 223:
|
|
|
|
|[[32805/32768]]
|
|
|Forms the [[Helmholtz (temperament)|Helmholtz]].
|-
|-
|30
|30
Line 184: Line 232:
|
|
|
|
|
|15625/15552 c.II
|
|
|
|
Line 193: Line 243:
|
|
|
|
|64000/59049
|
|
|Forms the [[Satriyo]].
|-
|-
|33
|33
Line 200: Line 252:
|
|
|
|
|
|177147/160000 c.II
|
|
|
|
Line 208: Line 262:
|
|
|
|
|
|[39, -7, -12⟩
|
|
|
|
Line 216: Line 272:
|
|
|
|
|
|[-41, 4, 15⟩
|
|
|
|
Line 224: Line 282:
|
|
|
|
|
|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⟩
|
|
|
|
Line 232: Line 336:
|
|
|
|
|[[40/27]]
|
|One step short of 53edo's perfect fifth.
|15625/15552 c.IV
|
|One step short of [[53edo]]'s perfect fifth.
|-
|55
|34\55
|
|
|
|
|[39, -7, -12⟩
|
|
|-
|-
|69
|69
Line 240: Line 356:
|
|
|
|
|[[5/4]]
|
|[-41, 1, 17⟩
|
|
|
|-
|72
|37\72, 53\72, 55\72
|
|
|
|
|
|
|53\72 forms the [[Catakleismic]].
|-
|-
|80
|80
|41\80, 49\80
|41\80, 49\80
|
|
|
|
|
|735
|735
|[[10/7]], [[26/17]]
|
|
|49\80 forms the [[Semisept]].
|49\80 forms the [[Semisept]].
|-
|-
Line 254: Line 384:
|71\84
|71\84
|58L 13s
|58L 13s
|
|
|
|1014.285714
|1014.285714
|
|
|
|
|
Line 262: Line 394:
|64\91
|64\91
|37L 27s
|37L 27s
|
|
|
|843.956043
|843.956043
|[[13/8]]
|
|
|
|-
|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
|
|
|
|}
|}
Line 270: Line 454:
== References ==
== References ==


[[Category:Scale theory]]
[[Category:Scale]]
Retrieved from "https://en.xen.wiki/w/User:Eliora/Concoctic_scale"