3L 7s

↖ 2L 6s	↑ 3L 6s	4L 6s ↗
← 2L 7s	3L 7s	4L 7s →
↙ 2L 8s	↓ 3L 8s	4L 8s ↘

┌╥┬┬╥┬┬╥┬┬┬┐ │║││║││║││││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LssLssLsss sssLssLssL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	3\10 to 1\3 (360.0 ¢ to 400.0 ¢)
Dark	2\3 to 7\10 (800.0 ¢ to 840.0 ¢)
TAMNAMS information
Name	sephiroid
Prefix	seph-
Abbrev.	sp
Related MOS scales
Parent	3L 4s
Sister	7L 3s
Daughters	10L 3s, 3L 10s
Neutralized	6L 4s
2-Flought	13L 7s, 3L 17s
Equal tunings
Equalized (L:s = 1:1)	3\10 (360.0 ¢)
Supersoft (L:s = 4:3)	10\33 (363.6 ¢)
Soft (L:s = 3:2)	7\23 (365.2 ¢)
Semisoft (L:s = 5:3)	11\36 (366.7 ¢)
Basic (L:s = 2:1)	4\13 (369.2 ¢)
Semihard (L:s = 5:2)	9\29 (372.4 ¢)
Hard (L:s = 3:1)	5\16 (375.0 ¢)
Superhard (L:s = 4:1)	6\19 (378.9 ¢)
Collapsed (L:s = 1:0)	1\3 (400.0 ¢)

3L 7s, named sephiroid in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 7 small steps, repeating every octave. Generators that produce this scale range from 360 ¢ to 400 ¢, or from 800 ¢ to 840 ¢.

Name

TAMNAMS suggests the temperament-agnostic name sephiroid for this scale, in reference to Kosmorsky's Tracatum de Modi Sephiratorum.

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 3L 7s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-sephstep	Perfect 0-sephstep	P0sps	0	0.0 ¢
1-sephstep	Minor 1-sephstep	m1sps	s	0.0 ¢ to 120.0 ¢
	Major 1-sephstep	M1sps	L	120.0 ¢ to 400.0 ¢
2-sephstep	Minor 2-sephstep	m2sps	2s	0.0 ¢ to 240.0 ¢
	Major 2-sephstep	M2sps	L + s	240.0 ¢ to 400.0 ¢
3-sephstep	Diminished 3-sephstep	d3sps	3s	0.0 ¢ to 360.0 ¢
	Perfect 3-sephstep	P3sps	L + 2s	360.0 ¢ to 400.0 ¢
4-sephstep	Minor 4-sephstep	m4sps	L + 3s	400.0 ¢ to 480.0 ¢
	Major 4-sephstep	M4sps	2L + 2s	480.0 ¢ to 800.0 ¢
5-sephstep	Minor 5-sephstep	m5sps	L + 4s	400.0 ¢ to 600.0 ¢
	Major 5-sephstep	M5sps	2L + 3s	600.0 ¢ to 800.0 ¢
6-sephstep	Minor 6-sephstep	m6sps	L + 5s	400.0 ¢ to 720.0 ¢
	Major 6-sephstep	M6sps	2L + 4s	720.0 ¢ to 800.0 ¢
7-sephstep	Perfect 7-sephstep	P7sps	2L + 5s	800.0 ¢ to 840.0 ¢
	Augmented 7-sephstep	A7sps	3L + 4s	840.0 ¢ to 1200.0 ¢
8-sephstep	Minor 8-sephstep	m8sps	2L + 6s	800.0 ¢ to 960.0 ¢
	Major 8-sephstep	M8sps	3L + 5s	960.0 ¢ to 1200.0 ¢
9-sephstep	Minor 9-sephstep	m9sps	2L + 7s	800.0 ¢ to 1080.0 ¢
	Major 9-sephstep	M9sps	3L + 6s	1080.0 ¢ to 1200.0 ¢
10-sephstep	Perfect 10-sephstep	P10sps	3L + 7s	1200.0 ¢

Generator chain

Generator chain of 3L 7s

Bright gens	Scale degree	Abbrev.
12	Augmented 6-sephdegree	A6spd
11	Augmented 3-sephdegree	A3spd
10	Augmented 0-sephdegree	A0spd
9	Augmented 7-sephdegree	A7spd
8	Major 4-sephdegree	M4spd
7	Major 1-sephdegree	M1spd
6	Major 8-sephdegree	M8spd
5	Major 5-sephdegree	M5spd
4	Major 2-sephdegree	M2spd
3	Major 9-sephdegree	M9spd
2	Major 6-sephdegree	M6spd
1	Perfect 3-sephdegree	P3spd
0	Perfect 0-sephdegree Perfect 10-sephdegree	P0spd P10spd
−1	Perfect 7-sephdegree	P7spd
−2	Minor 4-sephdegree	m4spd
−3	Minor 1-sephdegree	m1spd
−4	Minor 8-sephdegree	m8spd
−5	Minor 5-sephdegree	m5spd
−6	Minor 2-sephdegree	m2spd
−7	Minor 9-sephdegree	m9spd
−8	Minor 6-sephdegree	m6spd
−9	Diminished 3-sephdegree	d3spd
−10	Diminished 10-sephdegree	d10spd
−11	Diminished 7-sephdegree	d7spd
−12	Diminished 4-sephdegree	d4spd

Modes

Scale degrees of the modes of 3L 7s

UDP	Cyclic order	Step pattern	Scale degree (sephdegree)
			0	1	2	3	4	5	6	7	8	9	10
9|0	1	LssLssLsss	Perf.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Aug.	Maj.	Maj.	Perf.
8|1	4	LssLsssLss	Perf.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Perf.
7|2	7	LsssLssLss	Perf.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.	Maj.	Maj.	Perf.
6|3	10	sLssLssLss	Perf.	Min.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.	Maj.	Maj.	Perf.
5|4	3	sLssLsssLs	Perf.	Min.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Perf.
4|5	6	sLsssLssLs	Perf.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Perf.	Min.	Maj.	Perf.
3|6	9	ssLssLssLs	Perf.	Min.	Min.	Perf.	Min.	Min.	Maj.	Perf.	Min.	Maj.	Perf.
2|7	2	ssLssLsssL	Perf.	Min.	Min.	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Perf.
1|8	5	ssLsssLssL	Perf.	Min.	Min.	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Perf.
0|9	8	sssLssLssL	Perf.	Min.	Min.	Dim.	Min.	Min.	Min.	Perf.	Min.	Min.	Perf.

Proposed Names

Mode names are described by Kosmorsky, which use names from the Sefirot (or sephiroth). Kosmorsky describes the mode Keter to be akin to the lydian mode of 5L 2s, and the mode Malkuth like the locrian mode.

Modes of 3L 7s

UDP	Cyclic order	Step pattern	Mode names
9|0	1	LssLssLsss	Malkuth
8|1	4	LssLsssLss	Yesod
7|2	7	LsssLssLss	Hod
6|3	10	sLssLssLss	Netzach
5|4	3	sLssLsssLs	Tiferet
4|5	6	sLsssLssLs	Gevurah
3|6	9	ssLssLssLs	Chesed
2|7	2	ssLssLsssL	Binah
1|8	5	ssLsssLssL	Chokmah
0|9	8	sssLssLssL	Keter

Theory

The modi sephiratorum

This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents).

With sephiroid scales with a soft-of-basic step ratio (around {nowrap|L:s = 3:2}}, or 23edo), the 17th and 21st harmonics are tempered toward most accurately, which together are a stable harmony. This is the major chord of the modi sephiratorum.

Scales approaching an equalized step ratio (L:s = 1:1, or 10edo) contain a 13th harmonic that's nearly perfect. 121edo seems to be the first to 'accurately' represent the comma^{[clarification needed]}. Scales approaching a collapsed step ratio (L:s = 1:0, or 3edo) have the comma 65/64 liable to be tempered out, thus equating 8/5 and 13/8. Edos include 13edo, 16edo, 19edo, 22edo, 29edo, and others.

Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10)^{[clarification needed]} is symmetrical – not ascending but rather descending, and so reminiscent of ancient Greek practice. These scales, and their truncated heptatonic forms referenced below, are strikingly linear in several ways and so seem suited to a similar outlook as traditional western music (modality, baroque tonality, classical tonality, etc. progressing to today) but with higher harmonics.

There are MODMOS as well, but Kosmorsky has not explored them yet, as "there's enough undiscovered harmonic resources already in these to last me a while!" Taking this approach to the 13th harmonic also yields heptatonic MOS with similar properties: 4s+3L "mish" in the form of modes of ssLsLsL "led".

Scale tree

Scale tree and tuning spectrum of 3L 7s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
3\10						360.000	840.000	1:1	1.000	Equalized 3L 7s
					16\53	362.264	837.736	6:5	1.200	Submajor
				13\43		362.791	837.209	5:4	1.250
					23\76	363.158	836.842	9:7	1.286
			10\33			363.636	836.364	4:3	1.333	Supersoft 3L 7s
					27\89	364.045	835.955	11:8	1.375
				17\56		364.286	835.714	7:5	1.400
					24\79	364.557	835.443	10:7	1.429
		7\23				365.217	834.783	3:2	1.500	Soft 3L 7s
					25\82	365.854	834.146	11:7	1.571
				18\59		366.102	833.898	8:5	1.600
					29\95	366.316	833.684	13:8	1.625	Unnamed golden tuning
			11\36			366.667	833.333	5:3	1.667	Semisoft 3L 7s
					26\85	367.059	832.941	12:7	1.714
				15\49		367.347	832.653	7:4	1.750
					19\62	367.742	832.258	9:5	1.800
	4\13					369.231	830.769	2:1	2.000	Basic 3L 7s Scales with tunings softer than this are proper
					17\55	370.909	829.091	9:4	2.250
				13\42		371.429	828.571	7:3	2.333
					22\71	371.831	828.169	12:5	2.400
			9\29			372.414	827.586	5:2	2.500	Semihard 3L 7s Sephiroth
					23\74	372.973	827.027	13:5	2.600	Golden sephiroth
				14\45		373.333	826.667	8:3	2.667
					19\61	373.770	826.230	11:4	2.750
		5\16				375.000	825.000	3:1	3.000	Hard 3L 7s
					16\51	376.471	823.529	10:3	3.333
				11\35		377.143	822.857	7:2	3.500
					17\54	377.778	822.222	11:3	3.667	Muggles
			6\19			378.947	821.053	4:1	4.000	Superhard 3L 7s Magic / horcrux
					13\41	380.488	819.512	9:2	4.500	Magic / witchcraft / necromancy
				7\22		381.818	818.182	5:1	5.000	Magic / telepathy
					8\25	384.000	816.000	6:1	6.000	Würschmidt ↓
1\3						400.000	800.000	1:0	→ ∞	Collapsed 3L 7s

External links

• Tractatum de Modi Sephiratorum by Kosmorsky