5L 4s

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Scale structure
Step pattern LLsLsLsLs
sLsLsLsLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 7\9 to 4\5 (933.3¢ to 960.0¢)
Dark 1\5 to 2\9 (240.0¢ to 266.7¢)
TAMNAMS information
Name semiquartal
Prefix cthon-
Abbrev. ct
Related MOS scales
Parent 4L 1s
Sister 4L 5s
Daughters 9L 5s, 5L 9s
Neutralized 1L 8s
2-Flought 14L 4s, 5L 13s
Equal tunings
Equalized (L:s = 1:1) 7\9 (933.3¢)
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¢)
Collapsed (L:s = 1:0) 4\5 (960.0¢)

5L 4s, named semiquartal in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 4 small steps, repeating every octave. Generators that produce this scale range from 933.3¢ to 960¢, or from 240¢ to 266.7¢. 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.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for diatonic interval categories.

Intervals

Intervals of 5L 4s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-cthonstep Perfect 0-cthonstep P0cts 0 0.0¢
1-cthonstep Minor 1-cthonstep m1cts s 0.0¢ to 133.3¢
Major 1-cthonstep M1cts L 133.3¢ to 240.0¢
2-cthonstep Perfect 2-cthonstep P2cts L + s 240.0¢ to 266.7¢
Augmented 2-cthonstep A2cts 2L 266.7¢ to 480.0¢
3-cthonstep Minor 3-cthonstep m3cts L + 2s 240.0¢ to 400.0¢
Major 3-cthonstep M3cts 2L + s 400.0¢ to 480.0¢
4-cthonstep Minor 4-cthonstep m4cts 2L + 2s 480.0¢ to 533.3¢
Major 4-cthonstep M4cts 3L + s 533.3¢ to 720.0¢
5-cthonstep Minor 5-cthonstep m5cts 2L + 3s 480.0¢ to 666.7¢
Major 5-cthonstep M5cts 3L + 2s 666.7¢ to 720.0¢
6-cthonstep Minor 6-cthonstep m6cts 3L + 3s 720.0¢ to 800.0¢
Major 6-cthonstep M6cts 4L + 2s 800.0¢ to 960.0¢
7-cthonstep Diminished 7-cthonstep d7cts 3L + 4s 720.0¢ to 933.3¢
Perfect 7-cthonstep P7cts 4L + 3s 933.3¢ to 960.0¢
8-cthonstep Minor 8-cthonstep m8cts 4L + 4s 960.0¢ to 1066.7¢
Major 8-cthonstep M8cts 5L + 3s 1066.7¢ to 1200.0¢
9-cthonstep Perfect 9-cthonstep P9cts 5L + 4s 1200.0¢

Modes

Scale degrees of the modes of 5L 4s 
UDP Cyclic
order		 Step
pattern		 Scale degree (cthondegree)
0 1 2 3 4 5 6 7 8 9
8|0 1 LLsLsLsLs Perf. Maj. Aug. Maj. Maj. Maj. Maj. Perf. Maj. Perf.
7|1 8 LsLLsLsLs Perf. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Perf.
6|2 6 LsLsLLsLs Perf. Maj. Perf. Maj. Min. Maj. Maj. Perf. Maj. Perf.
5|3 4 LsLsLsLLs Perf. Maj. Perf. Maj. Min. Maj. Min. Perf. Maj. Perf.
4|4 2 LsLsLsLsL Perf. Maj. Perf. Maj. Min. Maj. Min. Perf. Min. Perf.
3|5 9 sLLsLsLsL Perf. Min. Perf. Maj. Min. Maj. Min. Perf. Min. Perf.
2|6 7 sLsLLsLsL Perf. Min. Perf. Min. Min. Maj. Min. Perf. Min. Perf.
1|7 5 sLsLsLLsL Perf. Min. Perf. Min. Min. Min. Min. Perf. Min. Perf.
0|8 3 sLsLsLsLL Perf. Min. Perf. Min. Min. Min. Min. Dim. Min. Perf.

Inthar and cellularAutomaton have proposed mode names based on scientific names of various corvids. The names as of 5/2/23⁠ ⁠[clarification needed ] are as follows.

Modes of 5L 4s
UDP Cyclic
order		 Step
pattern		 Mode names Name Origin
8|0 1 LLsLsLsLs Cristatan Bluejay (Cyanocitta cristata)
7|1 8 LsLLsLsLs Pican Magpie (Pica pica)
6|2 6 LsLsLLsLs Stellerian Steller's jay (Cyanocitta stelleri)
5|3 4 LsLsLsLLs Podocian Ground jay (genus Podoces)
4|4 2 LsLsLsLsL Nucifragan Nutcracker (genus Nucifraga)
3|5 9 sLLsLsLsL Coracian Common raven (Corvus corax)
2|6 7 sLsLLsLsL Frugilegian Rook (Corvus frugilegus)
1|7 5 sLsLsLLsL Temnurial Ratchet-tailed treepie (genus Temnurus)
0|8 3 sLsLsLsLL Pyrrhian Chough (genus Pyrrhocorax)

Note that the darkest two modes have no diatonic or armotonic fifth on the root in nonextreme semiquartal tunings.

Theory

The harmonic entropy minimum with this MOS pattern is godzilla, in which the generator tempers 8/7 or 7/6 to be the same interval, and two generators is 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

Hard-of-basic tunings have semifourths as generators, between 1\5 (240¢) and 3\14 (257.14¢), where two of them create a diatonic 4th. 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.

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

Soft-of-basic tunings have semifourths that are between 3\14 (257.14¢) and 2\9 (266.67¢), creating a "mavila" or "superdiatonic" 4th. 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

Scale tree

Scale Tree and Tuning Spectrum of 5L 4s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
7\9 933.333 266.667 1:1 1.000 Equalized 5L 4s
46\59 935.593 264.407 7:6 1.167
39\50 936.000 264.000 6:5 1.200
71\91 936.264 263.736 11:9 1.222
32\41 936.585 263.415 5:4 1.250 Septimin
89\114 936.842 263.158 14:11 1.273
57\73 936.986 263.014 9:7 1.286
82\105 937.143 262.857 13:10 1.300
25\32 937.500 262.500 4:3 1.333 Supersoft 5L 4s
Beep
93\119 937.815 262.185 15:11 1.364
68\87 937.931 262.069 11:8 1.375
111\142 938.028 261.972 18:13 1.385
43\55 938.182 261.818 7:5 1.400
104\133 938.346 261.654 17:12 1.417
61\78 938.462 261.538 10:7 1.429
79\101 938.614 261.386 13:9 1.444
18\23 939.130 260.870 3:2 1.500 Soft 5L 4s
Bug
83\106 939.623 260.377 14:9 1.556
65\83 939.759 260.241 11:7 1.571
112\143 939.860 260.140 19:12 1.583
47\60 940.000 260.000 8:5 1.600
123\157 940.127 259.873 21:13 1.615
76\97 940.206 259.794 13:8 1.625 Golden bug
105\134 940.299 259.701 18:11 1.636
29\37 940.541 259.459 5:3 1.667 Semisoft 5L 4s
98\125 940.800 259.200 17:10 1.700
69\88 940.909 259.091 12:7 1.714
109\139 941.007 258.993 19:11 1.727
40\51 941.176 258.824 7:4 1.750
91\116 941.379 258.621 16:9 1.778
51\65 941.538 258.462 9:5 1.800
62\79 941.772 258.228 11:6 1.833
11\14 942.857 257.143 2:1 2.000 Basic 5L 4s
Scales with tunings softer than this are proper
59\75 944.000 256.000 11:5 2.200
48\61 944.262 255.738 9:4 2.250
85\108 944.444 255.556 16:7 2.286
37\47 944.681 255.319 7:3 2.333
100\127 944.882 255.118 19:8 2.375
63\80 945.000 255.000 12:5 2.400
89\113 945.133 254.867 17:7 2.429
26\33 945.455 254.545 5:2 2.500 Semihard 5L 4s
93\118 945.763 254.237 18:7 2.571
67\85 945.882 254.118 13:5 2.600 Golden semaphore
108\137 945.985 254.015 21:8 2.625
41\52 946.154 253.846 8:3 2.667
97\123 946.341 253.659 19:7 2.714
56\71 946.479 253.521 11:4 2.750
71\90 946.667 253.333 14:5 2.800
15\19 947.368 252.632 3:1 3.000 Hard 5L 4s
Godzilla
64\81 948.148 251.852 13:4 3.250
49\62 948.387 251.613 10:3 3.333
83\105 948.571 251.429 17:5 3.400
34\43 948.837 251.163 7:2 3.500
87\110 949.091 250.909 18:5 3.600
53\67 949.254 250.746 11:3 3.667 Semaphore
72\91 949.451 250.549 15:4 3.750
19\24 950.000 250.000 4:1 4.000 Superhard 5L 4s
61\77 950.649 249.351 13:3 4.333
42\53 950.943 249.057 9:2 4.500
65\82 951.220 248.780 14:3 4.667
23\29 951.724 248.276 5:1 5.000
50\63 952.381 247.619 11:2 5.500
27\34 952.941 247.059 6:1 6.000
31\39 953.846 246.154 7:1 7.000
4\5 960.000 240.000 1:0 → ∞ Collapsed 5L 4s

Gallery

An alternative diagram with branch depth = 5

A voice-leading sketch in 24edo by Jacob Barton:

Music

Frédéric Gagné
Inthar
Starshine
Sevish
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