3L 7s

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↖ 2L 6s ↑3L 6s 4L 6s ↗
← 2L 7s3L 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. seph
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 Average of HE
(from HE Calc)
Min of HE
Generic[1] Specific[2] Abbrev.[3]
0-sephstep Perfect 0-sephstep P0ms 0 0.0¢ ~2.4654 nats ~2.4654 nats
1-sephstep Minor 1-sephstep m1ms s 0.0¢ to 120.0¢ ~4.7039 nats ~4.6644 nats
Major 1-sephstep M1ms L 120.0¢ to 400.0¢ ~4.5991 nats ~4.5794 nats
2-sephstep Minor 2-sephstep m2ms 2s 0.0¢ to 240.0¢ ~4.6036 nats ~4.5842 nats
Major 2-sephstep M2ms L + s 240.0¢ to 400.0¢ ~4.5661 nats ~4.5379 nats
3-sephstep Diminished 3-sephstep d3ms 3s 0.0¢ to 360.0¢ ~4.5705 nats ~4.5385 nats
Perfect 3-sephstep P3ms L + 2s 360.0¢ to 400.0¢ ~4.5561 nats ~4.5066 nats
4-sephstep Minor 4-sephstep m4ms L + 3s 400.0¢ to 480.0¢ ~4.6041 nats ~4.5769 nats
Major 4-sephstep M4ms 2L + 2s 480.0¢ to 800.0¢ ~4.5670 nats ~4.4205 nats
5-sephstep Minor 5-sephstep m5ms L + 4s 400.0¢ to 600.0¢ ~4.5634 nats ~4.3884 nats
Major 5-sephstep M5ms 2L + 3s 600.0¢ to 800.0¢ ~4.5520 nats ~4.1942 nats
6-sephstep Minor 6-sephstep m6ms L + 5s 400.0¢ to 720.0¢ ~4.5470 nats ~4.2657 nats
Major 6-sephstep M6ms 2L + 4s 720.0¢ to 800.0¢ ~4.6051 nats ~4.5145 nats
7-sephstep Perfect 7-sephstep P7ms 2L + 5s 800.0¢ to 840.0¢ ~4.5912 nats ~4.5754 nats
Augmented 7-sephstep A7ms 3L + 4s 840.0¢ to 1200.0¢ ~4.5302 nats ~4.4212 nats
8-sephstep Minor 8-sephstep m8ms 2L + 6s 800.0¢ to 960.0¢ ~4.5619 nats ~4.4208 nats
Major 8-sephstep M8ms 3L + 5s 960.0¢ to 1200.0¢ ~4.5863 nats ~4.5502 nats
9-sephstep Minor 9-sephstep m9ms 2L + 7s 800.0¢ to 1080.0¢ ~4.5845 nats ~4.5338 nats
Major 9-sephstep M9ms 3L + 6s 1080.0¢ to 1200.0¢ ~4.6360 nats ~4.6123 nats
10-sephstep Perfect 10-sephstep P10ms 3L + 7s 1200.0¢ ~3.3273 nats ~3.3273 nats

  1. Generic intervals are denoted solely by the number of steps they subtend.
  2. Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
  3. Abbreviations can be further shortened to 'ms' if context allows.

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

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 Rotational 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