4L 2s: Difference between revisions
m →Temperament interpretations: This section name was "Low harmonic entropy scales" on other similarly formatted pages |
→Scale tree: Added scale tree template, transferred all entries (except for e and pi tunings) |
||
| Line 27: | Line 27: | ||
{{MOS modes}} | {{MOS modes}} | ||
==Scale tree== | ==Scale tree== | ||
{ | {{Scale tree|Comments=5/4: Antikythera; | ||
13/8: Golden lemba; | |||
7/4: Lemba is around here; | |||
2/1: Optimum rank range; | |||
6/1: Doublewide is around here}} | |||
Optimum rank range | |||
[[Category:6-tone scales]] | [[Category:6-tone scales]] | ||
Revision as of 19:19, 24 April 2024
| ↖ 3L 1s | ↑ 4L 1s | 5L 1s ↗ |
| ← 3L 2s | 4L 2s | 5L 2s → |
| ↙ 3L 3s | ↓ 4L 3s | 5L 3s ↘ |
┌╥╥┬╥╥┬┐ │║║│║║││ ││││││││ └┴┴┴┴┴┴┘
sLLsLL
4L 2s, named citric in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 2 small steps, with a period of 2 large steps and 1 small step that repeats every 600.0 ¢, or twice every octave. Generators that produce this scale range from 200 ¢ to 300 ¢, or from 300 ¢ to 400 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period. 4L 2s can be seen as a warped diatonic scale, where one large step of diatonic (5L 2s) is removed, or as the equal-tempered whole-tone scale (6edo), but with two "whole tones" that are smaller than the others.
Scales with the true MOS pattern are always proper, because there is only one small step per period. In addition, there are near-MOS patterns, such as LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents).
Name
TAMNAMS suggests the temperament-agnostic name citric for this scale.
Theory
Low harmonic entropy scales
There are three scales with this MOS pattern that are significant minima of harmonic entropy. The first is antikythera, or no-3's srutal/pajara, which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.
The second is decimal, in which two generators make a 4/3, and the third is Doublewide, in which the generator is 7/6 so the period minus the generator is 6/5.
Modes
| UDP | Cyclic order |
Step pattern |
|---|---|---|
| 4|0(2) | 1 | LLsLLs |
| 2|2(2) | 2 | LsLLsL |
| 0|4(2) | 3 | sLLsLL |