Concoctic scale

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.

Mathematical derivation

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.

[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:

[math]aa \equiv 1\mod N[/math],

which simply becomes

[math]a^2 \equiv 1\mod N \hspace{4cm} (1)[/math].

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

The sequence of EDOs which have concoctic scales of any kind appears to be 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.

Concoctic scales in EDOs

Notation: c.II means contorted order 2, etc for other Roman numerals.

Non-trivial concoctic scales in EDOs up to 100 that have them

N	Scale\EDO	MOS		Generator Size (cents)		Associated 5-limit comma	Associated other commas	Notes
		Chroma+	Chroma-	Below 1\2	Above 1\2
5	3\5			480	720	16/15
8	5\8	3L 2s	2L 1s	450	750	16/15		Forms the Father.
10	7\10	2L 1s	3L 4s	360	840	25/24		Forms the Dicot.
12	7\12	5L 2s	2L 3s	500	700	81/80		The scale predominantly in use in the world today.
13	8\13	3L 2s	5L 3s		738.461538	2560/2187		Forms the Oneirotonic scale.
15	11\15	4L 7s	3L 1s		880	15625/15552*		*Forms the Hanson (11b & 15)
16	9\16	7L 2s	2L 5s		675	135/128		Forms the Mavila.
17	13\17	1L 3s	4L 9s		917.647059	25/24 c.II		Forms Huxley and Lovecraft, but with a fair error.
20	11\20	9L 2s	2L 7s		660	[-25, 7, 6⟩ c.II
21	13\21				742.857143	[39, -7, -12⟩
24	13\24, 17\24, 19\24				650, 850, 950	262144/253125 c.II, 32805/32768 c.II, 81/80 c.II		Contorted Passion, contorted Helmholtz and Godzilla.
25	18\25				864	3125/2916		Forms the Sixix.
26	21\26	1L 4s	5L 16s			[-29, 14, 3⟩		The 5-note scale itself is the slendric pentad.
28	15\28					[20, 5, -12⟩
29	17\29					32805/32768		Forms the 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⟩, [-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	71\84	58L 13s			1014.285714
91	64\91	37L 27s			843.956043
93	61\93
100	51\100

Concoctic scales by note count

Concoctic scales sorted by note count in MOSes that have them

N	MOS (chroma+)	Generator	Cents
7	1L 6s
	2L 5s	7\16
	3L 4s	7\10
	5L 2s	7\12
	6L 1s