5L 4s

From Xenharmonic Wiki
(Redirected from Semiquartal)
Jump to navigation Jump to search
5L 4s
Brightest mode LLsLsLsLs
Period 2/1
Range for bright generator 7\9 (933.3¢) to 4\5 (960.0¢)
Range for dark generator 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) 25\32 (937.5¢)
Soft (L:s = 3:2) 18\23 (939.1¢)
Semisoft (L:s = 5:3) 29\37 (940.5¢)
Basic (L:s = 2:1) 11\14 (942.9¢)
Semihard (L:s = 5:2) 26\33 (945.5¢)
Hard (L:s = 3:1) 15\19 (947.4¢)
Superhard (L:s = 4:1) 19\24 (950.0¢)

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¢).

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)

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-thonstep". TAMNAMS discourages 1-indexed terms such as "mos(k+1)th" for non-diatonic mosses.

Modes

Inthar has named 5L 4s modes after scientific names of various corvids.

Mode UDP Inthar's names as of 11/4/22 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 Nutcrackers (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 Pyrrhocoracian, Pyrrhian Chough (genus Pyrrhocorax)

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