5L 4s
5L 4s refers to the structure of MOS scales with generators ranging from 1\5 (one degree of 5edo = 240¢) to 2\9 (two degrees of 9edo = 266.7¢). In the case of 9edo, 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).
The familiar harmonic entropy minimum with this MOS pattern is godzilla, in which a generator is 8/7 or 7/6 (tempered to be the same interval, or even 37/32 if you like) so two of them make a 4/3. However, in addition to godzilla (tempering out 81/80) and the 2.3.7 temperament semaphore, there is also a weird scale called "pseudo-semaphore", in which two different flavors of 3/2 exist in the same scale: an octave minus two generators makes a sharp 3/2, and two octaves minus seven generators makes a flat 3/2.
Scale tree
| Generator | Cents | Comments | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1\5 | 240 | |||||||||||
| 12\59 | 244.068 | Pseudo-semaphore is around here | ||||||||||
| 11\54 | 244.444 | |||||||||||
| 10\49 | 244.898 | |||||||||||
| 9\44 | 245.455 | |||||||||||
| 8\39 | 246.154 | |||||||||||
| 7\34 | 247.059 | |||||||||||
| 6\29 | 248.276 | |||||||||||
| 11\53 | 249.057 | Semaphore is around here | ||||||||||
| 5\24 | 250 | L/s = 4 | ||||||||||
| 9\43 | 251.163 | |||||||||||
| 4\19 | 252.632 | Godzilla is around here
L/s = 3 | ||||||||||
| 11\52 | 253.813 | |||||||||||
| 29\137 | 254.015 | |||||||||||
| 76\359 | 254.039 | |||||||||||
| 199\940 | 254.043 | |||||||||||
| 123\581 | 254.045 | |||||||||||
| 47\222 | 254.054 | |||||||||||
| 18\85 | 254.118 | |||||||||||
| 7\33 | 254.5455 | |||||||||||
| 10\47 | 255.319 | |||||||||||
| 13\61 | 255.734 | |||||||||||
| 16\75 | 256.000 | |||||||||||
| 3\14 | 257.143 | Boundary of propriety (generators
larger than this are proper) | ||||||||||
| 11\51 | 258.8235 | |||||||||||
| 258.957 | ||||||||||||
| 8\37 | 259.459 | |||||||||||
| 21\97 | 259.794 | |||||||||||
| 55\254 | 259.843 | |||||||||||
| 144\665 | 259.850 | |||||||||||
| 233\1076 | 259.851 | Golden superpelog | ||||||||||
| 89\411 | 259.854 | |||||||||||
| 34\157 | 259.873 | |||||||||||
| 13\60 | 260 | |||||||||||
| 260.246 | ||||||||||||
| 5\23 | 260.870 | Optimum rank range (L/s=3/2) superpelog | ||||||||||
| 7\32 | 262.5 | |||||||||||
| 9\41 | 263.415 | |||||||||||
| 11\50 | 264 | |||||||||||
| 13\59 | 264.407 | |||||||||||
| 15\68 | 264.706 | |||||||||||
| 17\77 | 264.935 | |||||||||||
| 19\86 | 265.116 | |||||||||||
| 21\95 | 265.263 | |||||||||||
| 2\9 | 266.667 | |||||||||||
Tuning ranges and data
Semaphore
We can view semaphore as any 5L 4s tuning where two semifourth generators make a diatonic fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important tuning sub-range is given by stipulating that two semifourth generators must make a meantone fourth; i.e. that 81/80 be tempered out. This results in godzilla temperament, which is supported by 19edo and 24edo.
Bug
For convenience' sake, we can view bug as any 5L 4s tuning where two semifourth generators make a superdiatonic or mavila fourth, i.e. any tuning where the semifourth is between 3\14 (257.14¢) and 2\9 (266.67¢). 23edo's 5\23 (260.87¢) is an example of a bug generator.
Notation
Intervals
Modes
TODO: names
- LLsLsLsLs
- LsLLsLsLs
- LsLsLLsLs
- LsLsLsLLs
- LsLsLsLsL
- sLLsLsLsL
- sLsLLsLsL
- sLsLsLLsL
- sLsLsLsLL
One can think of godzilla[9] modes as being built from two pentachords (division of the perfect fourth into four intervals) plus a whole tone. The possible pentachords are LsLs, sLLs, and sLsL.
Chords
Primodal theory
Nejis
14nejis
- 95:100:105:110:116:122:128:135:141:148:156:164:172:180:190 (uses /19 prime family intervals while being pretty close to equal)
Samples
File:Dream EP 14edo Sketch.mp3 is a short swing ditty in 14edo semaphore[9], in the 212121221 mode.