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 today, is an example of such a scale.
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[1]:
[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 k = n/4 and rewrite the expression as (2k+1)\4k;
(2k+1)^2 = 4k^2 + 4k + 1;
4k^2 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.
| N | Scale | Mos type | Generator Size (cents) | Notes |
|---|---|---|---|---|
| 5 | 3\5 | 720 | ||
| 8 | 5\8 | 750 | ||
| 10 | 7\10 | |||
| 12 | 7\12 | 5L 2s | 700 | The system predominantly in use in the world today. |
| 13 | 8\13 | |||
| 15 | 11\15 | 3L 1s | 880 | |
| 16 | 9\16 | 7L 2s | 675 | |
| 17 | 13\17 | |||
| 20 | 11\20 | |||
| 21 | 13\21 | |||
| 24 | 13\24, 17\24, 19\24 | 350, 650, 850 | ||
| 25 | 18\25 | |||
| 26 | 21\26 | Forms the slendric pentad | ||
| 28 | 15\28 | |||
| 29 | 17\29 | |||
| 84 | 71\84 | 58L 13s | 1014.285714 | |
| 91 | 64\91 | 37L 27s | 843.956043 |