3L 7s

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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.

Intervals

This article assumes TAMNAMS for naming step ratios, mossteps, and mosdegrees.
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¢

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 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".

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

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
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

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