User:Eliora/Concoctic scale: Difference between revisions
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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 | |||
| | |||
| | |||
| | | | ||
|- | |- | ||
Revision as of 20:36, 19 December 2021
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]:
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
| 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\16 | Forms the slendric pentad | ||
| 28 | 15\28 | |||
| 29 | 17\29 | |||
| 84 | 71\84 | 58L 13s | 1014.285714 | |
| 91 | 64\91 | 37L 27s | 843.956043 |