5L 4s
| ↖ 4L 3s | ↑ 5L 3s | 6L 3s ↗ |
| ← 4L 4s | 5L 4s | 6L 4s → |
| ↙ 4L 5s | ↓ 5L 5s | 6L 5s ↘ |
┌╥╥┬╥┬╥┬╥┬┐ │║║│║│║│║││ │││││││││││ └┴┴┴┴┴┴┴┴┴┘
sLsLsLsLL
5L 4s, or semiquartal (in Inthar's MOS naming scheme), 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).
Semiquartal tunings can be divided into two major ranges:
- "hard-of-center" semiquartal, generated by semifourths flatter than 3\14 (257.14¢). This implies a diatonic fifth.
- The generator could be viewed as a 15/13, and the resulting "ultramajor" chords and "inframinor" triads could be viewed as approximating 10:13:15 and 26:30:39. See Arto and Tendo Theory.
- "soft-of-center" semiquartal, generated by semifourths sharper than 3\14 (257.14¢). This implies a "mavila" or superdiatonic fifth.
Notation
This article uses the convention JKLMNOPQR = LSLSLSLSL and pitch standard J = C4 = 261.6255653 Hz. The accidentals & and @ are used for raising and lowering by the chroma = L − S, respectively.
Tuning ranges
Hard-of-basic
These tunings satisfy the property that two semifourth generators make a diatonic (5L 2s) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢).
The sizes of the generator, large step and small step of 5L 4s are as follows in various semaphore tunings.
| 14edo | 19edo | 24edo | 29edo | |
|---|---|---|---|---|
| generator (g) | 3\14, 257.14 | 4\19, 252.63 | 5\24, 250.00 | 6\29, 248.28 |
| L (octave - 4g) | 171.43 | 189.47 | 200.00 | 206.90 |
| s (5g - octave) | 85.71 | 63.16 | 50.00 | 41.38 |
Parahard
One important sub-range of semaphore is given by stipulating that two semifourth generators must make a meantone fourth; i.e. that four fifths should approximate a 5/4 major third. This can be considered the 19edo (4\19)-to-24edo (5\24) range, i.e. parahard semiquartal, which also contains 43edo (9\43) and 62edo (13\62).
Soft-of-basic
These are tunings where two semifourth generators make a superdiatonic (7L 2s) fourth (i.e. 514.29¢ to 533.33¢), 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 this generator.
The sizes of the generator, large step and small step of 5L 4s are as follows in various soft-of-basic tunings.
| 23edo | 32edo | 37edo | |
|---|---|---|---|
| generator (g) | 5\23, 260.87 | 7\32, 262.50 | 8\37, 259.46 |
| L (octave - 4g) | 156.52 | 150.00 | 162.16 |
| s (5g - octave) | 104.35 | 112.50 | 97.30 |
Intervals
Modes
One can think of 5L 4s 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
- Rin's UFO Ride by Starshine
- File:Dream EP 14edo Sketch.mp3 is a short swing ditty in 14edo semiquartal, in the 212121221 mode.
- File:19edo Semaphore Fugue.mp3 is an unfinished fugue in 19edo semiquartal, in the 212121221 mode.
Scale tree
| Generator | Cents | L | s | L/s | Comments | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Chroma-positive | Chroma-negative | ||||||||||
| 7\9 | 933.333 | 266.667 | 1 | 1 | 1.000 | ||||||
| 39\50 | 936.000 | 264.000 | 6 | 5 | 1.200 | ||||||
| 32\41 | 936.585 | 263.415 | 5 | 4 | 1.250 | Septimin | |||||
| 57\73 | 936.986 | 263.014 | 9 | 7 | 1.286 | ||||||
| 25\32 | 937.500 | 262.500 | 4 | 3 | 1.333 | Beep | |||||
| 68\87 | 937.931 | 262.069 | 11 | 8 | 1.375 | ||||||
| 43\55 | 938.182 | 261.818 | 7 | 5 | 1.400 | ||||||
| 61\78 | 938.462 | 261.538 | 10 | 7 | 1.428 | ||||||
| 18\23 | 939.130 | 260.870 | 3 | 2 | 1.500 | L/s = 3/2, bug | |||||
| 65\83 | 939.759 | 260.241 | 11 | 7 | 1.571 | ||||||
| 47\60 | 940.000 | 260.000 | 8 | 5 | 1.600 | ||||||
| 76\97 | 940.206 | 259.794 | 13 | 8 | 1.625 | Golden bug | |||||
| 29\37 | 940.541 | 259.459 | 5 | 3 | 1.667 | ||||||
| 69\88 | 940.909 | 259.091 | 12 | 7 | 1.714 | ||||||
| 40\51 | 941.176 | 258.824 | 7 | 4 | 1.750 | ||||||
| 51\65 | 941.538 | 258.462 | 9 | 5 | 1.800 | ||||||
| 11\14 | 942.857 | 257.143 | 2 | 1 | 2.000 | Basic semiquartal (Generators smaller than this are proper) | |||||
| 48\61 | 944.262 | 255.738 | 9 | 4 | 2.250 | ||||||
| 37\47 | 944.681 | 255.319 | 7 | 3 | 2.333 | ||||||
| 63\80 | 945.000 | 255.000 | 12 | 5 | 2.400 | ||||||
| 26\33 | 945.455 | 254.545 | 5 | 2 | 2.500 | ||||||
| 67\85 | 945.882 | 254.118 | 13 | 5 | 2.600 | Unnamed golden tuning | |||||
| 41\52 | 946.154 | 253.846 | 8 | 3 | 2.667 | ||||||
| 56\71 | 946.479 | 253.521 | 11 | 4 | 2.750 | ||||||
| 15\19 | 947.368 | 252.632 | 3 | 1 | 3.000 | L/s = 3/1, godzilla | |||||
| 49\62 | 948.387 | 251.613 | 10 | 3 | 3.333 | ||||||
| 34\43 | 948.837 | 251.163 | 7 | 2 | 3.500 | ||||||
| 53\67 | 949.254 | 250.746 | 11 | 3 | 3.667 | Semaphore | |||||
| 19\24 | 950.000 | 250.000 | 4 | 1 | 4.000 | ||||||
| 42\53 | 950.943 | 249.057 | 9 | 2 | 4.500 | ||||||
| 23\29 | 951.724 | 248.276 | 5 | 1 | 5.000 | ||||||
| 27\34 | 952.941 | 247.059 | 6 | 1 | 6.000 | ||||||
| 4\5 | 960.000 | 240.000 | 1 | 0 | → inf | ||||||