User:Eliora/Concoctic scale: Difference between revisions
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Concoctic scale (name proposed by Eliora) is a maximum eveness scale which has the same number of notes as its MOS generator. | Concoctic scale (name proposed by Eliora) is a [[Maximal evenness|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. | 12edo 5L2s diatonic scale, the predominantly used scale in the world today, is an example of such a scale. | ||
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== List == | == List == | ||
The sequence of EDOs which have concoctic scales of any kind appears to be [[oeis:A172019|A172019]]. | The sequence of EDOs which have concoctic scales of any kind appears to be [[oeis:A172019|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. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| Line 49: | Line 51: | ||
!N | !N | ||
!Scale | !Scale | ||
!Mos | !Mos | ||
(chroma+) | |||
!Mos | |||
(chroma-) | |||
!Generator Size (cents) | !Generator Size (cents) | ||
!Notes | !Notes | ||
| Line 55: | Line 60: | ||
|5 | |5 | ||
|3\5 | |3\5 | ||
| | |||
| | | | ||
|720 | |720 | ||
| Line 61: | Line 67: | ||
|8 | |8 | ||
|5\8 | |5\8 | ||
| | |[[3L 2s]] | ||
|2L 1s | |||
|750 | |750 | ||
| | | | ||
| Line 67: | Line 74: | ||
|10 | |10 | ||
|7\10 | |7\10 | ||
| | |2L 1s | ||
| | |[[3L 4s]] | ||
|840 | |||
| | | | ||
|- | |- | ||
|12 | |12 | ||
|7\12 | |7\12 | ||
|5L 2s | |[[5L 2s]] | ||
|[[2L 3s]] | |||
|700 | |700 | ||
|The | |The scale predominantly in use in the world today. | ||
|- | |- | ||
|13 | |13 | ||
|8\13 | |8\13 | ||
| | |[[3L 2s]] | ||
| | |[[5L 3s]] | ||
| | |738.461538 | ||
|Forms the [[Oneirotonic]] scale. | |||
|- | |- | ||
|15 | |15 | ||
|11\15 | |11\15 | ||
|3L 1s | |[[Tetrad 3L 1s|4L 7s]] | ||
|[[Tetrad 3L 1s|3L 1s]] | |||
|880 | |880 | ||
| | |Forms the [[Hanson]]. | ||
|- | |- | ||
|16 | |16 | ||
|9\16 | |9\16 | ||
|7L 2s | |[[7L 2s]] | ||
|[[2L 5s]] | |||
|675 | |675 | ||
| | |Forms the [[Mavila]]. | ||
|- | |- | ||
|17 | |17 | ||
|13\17 | |13\17 | ||
| | |[[1L 3s]] | ||
| | |[[4L 9s]] | ||
|917.647059 | |||
| | | | ||
|- | |- | ||
|20 | |20 | ||
|11\20 | |11\20 | ||
| | |[[9L 2s]] | ||
| | |[[2L 7s]] | ||
|660 | |||
| | | | ||
|- | |- | ||
| Line 111: | Line 125: | ||
| | | | ||
| | | | ||
|742.857143 | |||
| | | | ||
|- | |- | ||
|24 | |24 | ||
|13\24, 17\24, 19\24 | |13\24, 17\24, 19\24 | ||
| | |||
| | | | ||
|350, 650, 850 | |350, 650, 850 | ||
| Line 123: | Line 139: | ||
| | | | ||
| | | | ||
|864 | |||
| | | | ||
|- | |- | ||
| Line 129: | Line 146: | ||
| | | | ||
| | | | ||
|Forms the slendric pentad | | | ||
|Forms the slendric pentad. | |||
|- | |- | ||
|28 | |28 | ||
|15\28 | |15\28 | ||
| | |||
| | | | ||
| | | | ||
| Line 139: | Line 158: | ||
|29 | |29 | ||
|17\29 | |17\29 | ||
| | |||
| | |||
| | |||
| | |||
|- | |||
|30 | |||
|19\30 | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
|32 | |||
|17\32 | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
|33 | |||
|23\33 | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
|34 | |||
|21\34 | |||
| | |||
| | | | ||
| | | | ||
| | | | ||
|- | |||
|35 | |||
|29\35 | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
|36 | |||
|19\36 | |||
| | |||
| | |||
| | |||
| | |||
|- | |||
|53 | |||
|30\53 | |||
| | |||
| | |||
| | |||
|One step short of 53edo's perfect fifth. | |||
|- | |- | ||
|84 | |84 | ||
|71\84 | |71\84 | ||
|58L 13s | |58L 13s | ||
| | |||
|1014.285714 | |1014.285714 | ||
| | | | ||
| Line 152: | Line 222: | ||
|64\91 | |64\91 | ||
|37L 27s | |37L 27s | ||
| | |||
|843.956043 | |843.956043 | ||
| | | | ||