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 interval regions.

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 ¢

Generator chain

Generator chain of 5L 4s

Bright gens	Scale degree	Abbrev.
13	Augmented 1-cthondegree	A1ctd
12	Augmented 3-cthondegree	A3ctd
11	Augmented 5-cthondegree	A5ctd
10	Augmented 7-cthondegree	A7ctd
9	Augmented 0-cthondegree	A0ctd
8	Augmented 2-cthondegree	A2ctd
7	Major 4-cthondegree	M4ctd
6	Major 6-cthondegree	M6ctd
5	Major 8-cthondegree	M8ctd
4	Major 1-cthondegree	M1ctd
3	Major 3-cthondegree	M3ctd
2	Major 5-cthondegree	M5ctd
1	Perfect 7-cthondegree	P7ctd
0	Perfect 0-cthondegree Perfect 9-cthondegree	P0ctd P9ctd
−1	Perfect 2-cthondegree	P2ctd
−2	Minor 4-cthondegree	m4ctd
−3	Minor 6-cthondegree	m6ctd
−4	Minor 8-cthondegree	m8ctd
−5	Minor 1-cthondegree	m1ctd
−6	Minor 3-cthondegree	m3ctd
−7	Minor 5-cthondegree	m5ctd
−8	Diminished 7-cthondegree	d7ctd
−9	Diminished 9-cthondegree	d9ctd
−10	Diminished 2-cthondegree	d2ctd
−11	Diminished 4-cthondegree	d4ctd
−12	Diminished 6-cthondegree	d6ctd
−13	Diminished 8-cthondegree	d8ctd

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.

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 "inframinor" and "ultramajor" chords and triads could be viewed as approximating, respectively, 26:30:39 and 10:13:15 (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
					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	Supersoft 5L 4s 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	1.429
		18\23				939.130	260.870	3:2	1.500	Soft 5L 4s 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	Semisoft 5L 4s
					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 5L 4s Scales with tunings softer 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	Semihard 5L 4s
					67\85	945.882	254.118	13:5	2.600	Golden semaphore
				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	Hard 5L 4s 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	Superhard 5L 4s
					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	→ ∞	Collapsed 5L 4s

Gallery

An alternative diagram with branch depth = 5

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

Music

• Entropy, the Grandfather of Wind (broken link. 2011-03-04) In 14edo^{[dead link]}

Frédéric Gagné

• Whalectric (2022) – YouTube | score – In 51edo, 4|4 mode

Inthar

• Dream EP 14edo Sketch (2021) – A short swing ditty in 14edo, in the 212121221 mode
• 19edo Semaphore Fugue (2021) – An unfinished fugue in 19edo, in the 212121221 mode

Starshine

• Rin's UFO Ride (2020) – Semaphore[9] in 19edo

Sevish

• Desert Island Rain – Semaphore[9] in 313edo using 65\313 as the generator