5L 5s
5L 5s refers to the structure of octave-equivalent MOS scales with period 1\5 (one degree of 5edo = 240¢) and generators ranging from 1\10 (one degree of 10edo = 120¢) to 1\5 (240¢). In the case of 10edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). There is only one significant harmonic entropy minimum with this MOS pattern: blackwood, in which intervals of the prime numbers 3 and 7 are all represented using steps of 5edo, and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5. Hence from a regular temperament perspective, this MOS pattern is essentially synonymous with blackwood.
The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being proper (because 1\15 is in the middle of the range of good blackwood generators).
| Generator | Cents | Comments | ||||
|---|---|---|---|---|---|---|
| 0\5 | 0 | |||||
| 1\30 | 40 | |||||
| 1\25 | 48 | |||||
| 240/(1+pi) | ||||||
| 1\20 | 60 | |||||
| 240/(1+e) | ||||||
| 2\35 | 68.57 | |||||
| 3\50 | 72 | |||||
| 1\15 | 80 | Blackwood is around here
Optimum rank range (L/s=2/1) for MOS | ||||
| 240/(1+sqrt(3)) | ||||||
| 3\40 | 90 | |||||
| 5\65 | 92.31 | Golden blackwood | ||||
| 240/(1+pi/2) | ||||||
| 2\25 | 96 | |||||
| 3\35 | 102.86 | |||||
| 4\45 | 103.33 | |||||
| 1\10 | 120 | |||||