User:Eliora/Concoctic scale

Concoctic scale (name proposed by Eliora) is a maximum eveness 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 definition

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:

[math]\displaystyle{ 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]\displaystyle{ aa \equiv 1\mod N }[/math],

which simply becomes

[math]\displaystyle{ 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

[math]\displaystyle{ 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.

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 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.

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]\displaystyle{ k = \frac{n}{4} }[/math] and rewrite the expression as [math]\displaystyle{ [(2k+1)/4k] }[/math];

[math]\displaystyle{ (2k+1)^2 = 4k^2 + 4k + 1 }[/math];

[math]\displaystyle{ 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.

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

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

N	Scale\EDO	MOS		Generator Size (cents)		Associated Ratio	Notes
		Chroma+	Chroma-	Below 1\2	Above 1\2
5	3\5			480	720	3/2
8	5\8	3L 2s	2L 1s	450	750	14/9
10	7\10	2L 1s	3L 4s	360	840	13/8
12	7\12	5L 2s	2L 3s	500	700	3/2	The scale predominantly in use in the world today.
13	8\13	3L 2s	5L 3s		738.461538		Forms the Oneirotonic scale.
15	11\15	4L 7s	3L 1s		880	5/3	Forms the Hanson.
16	9\16	7L 2s	2L 5s		675		Forms the Mavila.
17	13\17	1L 3s	4L 9s		917.647059	22/13	Forms Huxley and Lovecraft, but with a fair error.
20	11\20	9L 2s	2L 7s		660
21	13\21				742.857143
24	13\24, 17\24, 19\24				650, 850, 950
25	18\25				864
26	21\26	1L 4s	5L 16s			12/7	The 5-note scale itself is the slendric pentad.
28	15\28
29	17\29
30	19\30
32	17\32
33	23\33
34	21\34
35	29\35
36	19\36
37	31\37
39	25\39
40	21\40, 29\40, 31\40
41	32\41
53	30\53					40/27	One step short of 53edo's perfect fifth.
69	22\69					5/4
72	37\72, 53\72, 55\72					10/7, 5/3, 17/10	53\72 forms the Catakleismic.
80	41\80, 49\80				735	10/7, 26/17	49\80 forms the Semisept.
84	71\84	58L 13s			1014.285714
91	64\91	37L 27s			843.956043	13/8
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