5L 4s
↖4L 3s | ↑5L 3s | 6L 3s↗ |
←4L 4s | 5L 4s | 6L 4s→ |
↙4L 5s | ↓5L 5s | 6L 5s↘ |
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.
It is also equal to a degenerate form of diasem.
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¢).
Hypohard
The sizes of the generator, large step and small step of 5L 4s are as follows in various hypohard (2/1 ≤ L/s ≤ 3/1) tunings.
14edo (L/s = 2/1) | 47edo (L/s = 7/3) | 33edo (L/s = 5/2) | 52edo (L/s = 8/3) | 19edo (L/s = 3/1) | |
---|---|---|---|---|---|
generator (g) | 3\14, 257.14 | 10\47, 255.32 | 7\33, 254.54 | 11\52, 253.85 | 4\19, 252.63 |
L (octave - 4g) | 171.43 | 178.72 | 181.81 | 184.62 | 189.47 |
s (5g - octave) | 85.71 | 76.60 | 72.73 | 69.23 | 63.16 |
This range is notable for having many simple tunings that are close to being "eigentunings" (tunings that tune a certain JI interval exactly):
- 33edo semiquartal has close 7/5 (error -0.69¢), 9/5 (error -0.59¢) and 9/7 (error +1.28¢), thus can be used for the close 5:7:9 in the two Locrian-like modes 1|7 and 0|8
- 52edo semiquartal has close 22/19 (error +0.04¢)
- 19edo semiquartal has close 6/5 (error +0.15¢) and 28/27 (error +0.20¢)
However, for the more complex intervals such as 22/19 and 28/27, you might want to use the exact eigentuning for the full effect, unless you specifically need an edo for modulatory purposes.
Parahard and ultrahard
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). Parahard semiquartal can be given an RTT interpretation known as godzilla.
The sizes of the generator, large step and small step of 5L 4s are as follows in various hypohard (2/1 ≤ L/s ≤ 3/1) tunings.
19edo | 24edo | 29edo | |
---|---|---|---|
generator (g) | 4\19, 252.63 | 5\24, 250.00 | 6\29, 248.28 |
L (octave - 4g) | 189.47 | 200.00 | 206.90 |
s (5g - octave) | 63.16 | 50.00 | 41.38 |
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-thonstep". TAMNAMS discourages 1-indexed terms such as "mos(k+1)th" for non-diatonic mosses.
Modes
Inthar and cellularAutomaton have named 5L 4s modes after scientific names of various corvids.
Mode | UDP | Inthar's names as of 5/2/23 | Origin |
---|---|---|---|
LLsLsLsLs | 8|0 | Cristatan | Bluejay (Cyanocitta cristata) |
LsLLsLsLs | 7|1 | Pican | Magpie (Pica pica) |
LsLsLLsLs | 6|2 | Stellerian | Steller's jay (Cyanocitta stelleri) |
LsLsLsLLs | 5|3 | Cornician | Hooded crow (Corvus cornix) |
LsLsLsLsL | 4|4 | Nucifragan | Nutcracker (genus Nucifraga) |
sLLsLsLsL | 3|5 | Coracian | Common raven (Corvus corax) |
sLsLLsLsL | 2|6 | Frugilegian | Rook (Corvus frugilegus) |
sLsLsLLsL | 1|7 | Coloean | Jackdaw (genus Coloeus) |
sLsLsLsLL | 0|8 | Pyrrhian | Chough (genus Pyrrhocorax) |
Note that the darkest two modes have no diatonic or armotonic 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 |