User:Eliora/Concoctic scale: Difference between revisions
Created page with "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 use..." |
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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 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<ref>https://individual.utoronto.ca/kalendis/leap/index.htm</ref>: | |||
a*x ≡ 1 (mod 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), | |||
which becomes | |||
a^2 ≡ 1 (mod N). | |||
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. | |||
== List == | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
Non-trivial concoctic scales (above 1\2) in EDOs up to 100 that have them | |||
!N | !N | ||
!Scale | !Scale | ||
| Line 14: | Line 27: | ||
!Generator Size (cents) | !Generator Size (cents) | ||
!Notes | !Notes | ||
|- | |||
|5 | |||
|3\5 | |||
| | |||
|720 | |||
| | |||
|- | |||
|8 | |||
|5\8 | |||
| | |||
|750 | |||
| | |||
|- | |||
|10 | |||
|7\10 | |||
| | |||
| | |||
| | |||
|- | |- | ||
|12 | |12 | ||
| Line 19: | Line 50: | ||
|5L 2s | |5L 2s | ||
|700 | |700 | ||
|The system predominantly in use in the world today. | |||
|- | |||
|13 | |||
|8\13 | |||
| | |||
| | |||
| | | | ||
|- | |- | ||
|15 | |15 | ||
| Line 37: | Line 68: | ||
|7L 2s | |7L 2s | ||
|675 | |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\16 | |||
| | |||
| | |||
|Forms the slendric pentad | |||
|- | |||
|28 | |||
|15\28 | |||
| | |||
| | |||
| | |||
|- | |||
|29 | |||
|17\29 | |||
| | |||
| | |||
| | | | ||
|- | |- | ||