5L 4s

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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:

  1. 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.
  2. 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)

14edo, basic semiquartal

19edo, hard semiquartal

23edo, soft semiquartal

24edo, superhard semiquartal

33edo, semihard semiquartal

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

Scale tree

An alternative diagram with branch depth = 5
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