22L 1s: Difference between revisions
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22L 1s is the scale that is most commonly produced by stacking the interval of [[33/32]]. If it had a name, it would most probably be '''quartismoid''', since its harmonic entropy minimum corresponds to tempering out the [[quartisma]] - five 33/32s being equated with 7/6. | 22L 1s is the scale that is most commonly produced by stacking the interval of [[33/32]]. If it had a name, it would most probably be '''quartismoid''', since its harmonic entropy minimum corresponds to tempering out the [[quartisma]] - five 33/32s being equated with 7/6. | ||
==Relation to equal divisions== | |||
From 1\22 to 4\91, 13 steps amount to a diatonic fifth. Between 4\91 and 1\23, 13 steps amount to a pelog / mavila fifth. | From 1\22 to 4\91, 13 steps amount to a diatonic fifth. Between 4\91 and 1\23, 13 steps amount to a pelog / mavila fifth. | ||
Further breaking down the categories, when the step ratio is greater than 4.472, then 13 generators amount to a superpyth fifth and the tuning approaches [[22edo]]. | |||
6 steps act as a pseudo-6/5, and when they actually act as 6/5 along with 5 steps being equal to 7/6, [[385/384]] is tempered out. If one were to instead tune in favour of 6/5 instead of 7/6, the resulting hardness would be aroun 1.233. | |||
== Scale tree == | == Scale tree == | ||
Revision as of 14:59, 25 September 2022
| ← 21L 1s | 22L 1s | 23L 1s → |
| ↙ 21L 2s | ↓ 22L 2s | 23L 2s ↘ |
┌╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥╥┬┐ │║║║║║║║║║║║║║║║║║║║║║║││ │││││││││││││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
sLLLLLLLLLLLLLLLLLLLLLL
22L 1s is the scale that is most commonly produced by stacking the interval of 33/32. If it had a name, it would most probably be quartismoid, since its harmonic entropy minimum corresponds to tempering out the quartisma - five 33/32s being equated with 7/6.
Relation to equal divisions
From 1\22 to 4\91, 13 steps amount to a diatonic fifth. Between 4\91 and 1\23, 13 steps amount to a pelog / mavila fifth.
Further breaking down the categories, when the step ratio is greater than 4.472, then 13 generators amount to a superpyth fifth and the tuning approaches 22edo.
6 steps act as a pseudo-6/5, and when they actually act as 6/5 along with 5 steps being equal to 7/6, 385/384 is tempered out. If one were to instead tune in favour of 6/5 instead of 7/6, the resulting hardness would be aroun 1.233.
Scale tree
| Generator | L | s | L/s | Comments | |||||
|---|---|---|---|---|---|---|---|---|---|
| 1\23 | 1 | 1 | 1.000 | ||||||
| 6\137 | 6 | 5 | 1.200 | ||||||
| 5\114 | 5 | 4 | 1.250 | ||||||
| 9\205 | 9 | 7 | 1.286 | ||||||
| 4\91 | 4 | 3 | 1.333 | 13 steps adding to lower bound of diatonic fifths (684.17c) is here | |||||
| 11\250 | 11 | 8 | 1.375 | ||||||
| 7\159 | 7 | 5 | 1.400 | ||||||
| 10\227 | 10 | 7 | 1.428 | ||||||
| 3\68 | 3 | 2 | 1.500 | Stretched 23edo is in this range | |||||
| 11\249 | 11 | 7 | 1.571 | ||||||
| 8\181 | 8 | 5 | 1.600 | ||||||
| 13\294 | 13 | 8 | 1.625 | ||||||
| 5\113 | 5 | 3 | 1.667 | ||||||
| 12\271 | 12 | 7 | 1.714 | ||||||
| 7\158 | 7 | 4 | 1.750 | ||||||
| 9\203 | 9 | 5 | 1.800 | ||||||
| 2\45 | 2 | 1 | 2.000 | Basic quartismoid | |||||
| 9\202 | 9 | 4 | 2.250 | ||||||
| 7\157 | 7 | 3 | 2.333 | ||||||
| 12\269 | 12 | 5 | 2.400 | ||||||
| 5\112 | 5 | 2 | 2.500 | 13 steps adding to 1/4 comma meantone fifth
is around here | |||||
| 13\291 | 13 | 5 | 2.600 | ||||||
| 8\179 | 8 | 3 | 2.667 | ||||||
| 11\246 | 11 | 4 | 2.750 | ||||||
| 3\67 | 3 | 1 | 3.000 | ||||||
| 10\223 | 10 | 3 | 3.333 | ||||||
| 7\156 | 7 | 2 | 3.500 | ||||||
| 11\245 | 11 | 3 | 3.667 | ||||||
| 4\89 | 4 | 1 | 4.000 | ||||||
| 9\200 | 9 | 2 | 4.500 | 13 steps adding to 3/2 perfect fifth is around here | |||||
| 5\111 | 5 | 1 | 5.000 | ||||||
| 6\133 | 6 | 1 | 6.000 | ||||||
| 1\22 | 1 | 0 | → inf | ||||||