5L 4s
| Pattern | LsLsLsLsL | |
| Period | 2/1 | |
| Generator range | 1\5 (240.0¢) to 2\9 (266.7¢) | |
| Parent MOS | 4L 1s | |
| Daughter MOSes | 9L 5s, 5L 9s | |
| Sister MOS | 4L 5s | |
| TAMNAMS name | semiquartal | |
| Equal tunings | ||
| Supersoft (L:s = 4:3) | 7\32 (262.5¢) | |
| Soft (L:s = 3:2) | 5\23 (260.9¢) | |
| Semisoft (L:s = 5:3) | 8\37 (259.5¢) | |
| Basic (L:s = 2:1) | 3\14 (257.1¢) | |
| Semihard (L:s = 5:2) | 7\33 (254.5¢) | |
| Hard (L:s = 3:1) | 4\19 (252.6¢) | |
| Superhard (L:s = 4:1) | 5\24 (250.0¢) | |
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).
5L 4s tunings can be divided into two major ranges:
- hard-of-basic 5L 4s, 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-basic 5L 4s, generated by semifourths sharper than 3\14 (257.14¢). This implies a "mavila" or superdiatonic fifth.
Names
The TAMNAMS convention, used by this article, uses semiquartal (derived from 'half a fourth') for the 5L 4s pattern. Another attested name is hemifourths.
Notation
This article uses the convention JKLMNOPQR = LsLsLsLsL. The accidentals & and @ are used for raising and lowering by the chroma = L − s, respectively.
Temperaments
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) 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. The 2.3.13/5 barbados temperament is another possible interpretation.
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¢) and 3\14 (257.14¢).
The sizes of the generator, large step and small step of 5L 4s are as follows in various hard-of-basic 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 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). This range has an RTT interpretation known as godzilla.
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 |
Tuning examples
An example in the Diasem Lydian mode LSLSLMLSLM with M and S equated. (score)
Intervals
Note: In TAMNAMS, a k-step interval class in semiquartal may be called a "k-step", "k-mosstep", or "k-sequarstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
Modes
| Mode | UDP |
| LLsLsLsLs | 8|0 |
| LsLLsLsLs | 7|1 |
| LsLsLLsLs | 6|2 |
| LsLsLsLLs | 5|3 |
| LsLsLsLsL | 4|4 |
| sLLsLsLsL | 3|5 |
| sLsLLsLsL | 2|6 |
| sLsLsLLsL | 1|7 |
| sLsLsLsLL | 0|8 |
Note that the darkest two modes have no fifth on the root in nonextreme semiquartal tunings.
Approaches
Music
- Rin's UFO Ride by Starshine, in 19edo
- Dream EP 14edo Sketch by Inthar, a short swing ditty in 14edo, in the 212121221 mode
- 19edo Semaphore Fugue by Inthar, an unfinished fugue in 19edo, in the 212121221 mode
- qt mode chord prog.mp3
- Entropy, the Grandfather of Wind (broken link. 2011-03-04) in 14edo [dead link]
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 | ||||||