5L 4s

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). One suggested name for this pattern is hemidiatessaric or hemitess (from Ancient Greek hemi 'half' + diatessaron, the term for the perfect fourth in Ancient Greek music).

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, or even 37/32 if you like) 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.

There are not really "good" temperament interpretations for 5L 4s scales except for semaphore or godzilla, but 5L 4s tunings can be divided into two major ranges:

1. Semaphore 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. Superpelog, or bug, generated by semifourths sharper than 3\14 (257.14¢). This implies a "mavila" or superdiatonic fifth.

Scale tree

Generator											Cents	Comments
1\5											240
										12\59	244.068	Pseudo-semaphore is around here
									11\54		244.444
								10\49			244.898
							9\44				245.455
						8\39					246.154
					7\34						247.059
				6\29							248.276
					11\53						249.057	Semaphore is around here
			5\24								250	L/s = 4
					9\43						251.163
		4\19									252.632	Godzilla is around here L/s = 3
				11\52							253.813
						29\137					254.015
								76\359			254.039
										199\940	254.043
									123\581		254.045
							47\222				254.054
					18\85						254.118
			7\33								254.5455
				10\47							255.319
					13\61						255.734
						16\75					256.000
	3\14										257.143	Boundary of propriety (generators larger than this are proper)
				11\51							258.8235
											258.957
			8\37								259.459
					21\97						259.794
							55\254				259.843
									144\665		259.850
										233\1076	259.851	Golden superpelog
								89\411			259.854
						34\157					259.873
				13\60							260
											260.246
		5\23									260.870	Optimum rank range (L/s=3/2) superpelog
			7\32								262.5
				9\41							263.415
					11\50						264
						13\59					264.407
							15\68				264.706
								17\77			264.935
									19\86		265.116
										21\95	265.263
2\9											266.667

Tuning ranges

Semaphore

We can view semaphore as any 5L 4s tuning where two semifourth generators make a diatonic (5L 2s) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important sub-range of semaphore 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 results in godzilla temperament, which is supported by 19edo and 24edo.

The sizes of the generator, large step and small step of 5L 4s are as follows in various semaphore tunings.

	14edo	19edo	24edo	29edo
generator (g)	3\14, 257.14	4\19, 252.63	5\24, 250.	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

Superpelog

For convenience's sake, we can view superpelog as any 5L 4s tuning where two semifourth generators make a superdiatonic (7L 2s) fourth, 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 a superpelog generator.

The sizes of the generator, large step and small step of 5L 4s are as follows in various superpelog tunings.

	23edo	32edo	37edo
generator (g)	5\23, 252.63	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

Notation

This article uses diamond MOS notation, with the convention JKLMNOPQR = LLSLSLSLS and pitch standard J = C4 = 261.6255653 Hz. The accidentals & and @ are used for raising and lowering by the chroma = L − S, respectively.

Modes

TODO: names

• LLsLsLsLs
• LsLLsLsLs
• LsLsLLsLs
• LsLsLsLLs
• LsLsLsLsL
• sLLsLsLsL
• sLsLLsLsL
• sLsLsLLsL
• sLsLsLsLL

One can think of 5L 4s modes as being built from two pentachords (division of the perfect fourth into four intervals) plus a whole tone. The possible pentachords are LsLs, sLLs, and sLsL.

Primodal theory

Nejis

14nejis

1. 95:100:105:110:116:122:128:135:141:148:156:164:172:180:190 (uses /19 prime family intervals while being pretty close to equal)

Samples

File:Dream EP 14edo Sketch.mp3 is a short swing ditty in 14edo semaphore[9], in the 212121221 mode.