3L 7s

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3L 7s
Pattern LssLssLsss
Period 2/1
Generator range 3\10 (360.0¢) to 1\3 (400.0¢)
Parent MOS 3L 4s
Daughter MOSes 10L 3s, 3L 10s
Sister MOS 7L 3s
TAMNAMS name sephiroid
Equal tunings
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¢)

3L 7s occupies the spectrum from 10edo (L = s) to 3edo (s = 0).

TAMNAMS calls this MOS pattern sephiroid (named after the abstract temperament sephiroth).

This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents). In the region of the spectrum around 23edo (L = 3, s = 2) , 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. Temperament using phi directly approximates the higher Fibonacci harmonics best.

If L = s, i.e. multiples of 10edo, the 13th harmonic becomes nearly perfect. 121edo seems to be the first to 'accurately' represent the comma (which might as well be represented accurately as it is quite small). Towards the other end, where the large and small steps are more contrasted, the comma 65/64 is liable to be tempered out, equating 8/5 and 13/8. In this category fall 13edo, 16edo, 19edo, 22edo, 29edo, and so on. This ends at s = 0 which gives multiples of 3edo.

Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10) 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. For more details see Kosmorsky's Tractatum de Modi Sephiratorum (Kosmorsky knows it should be "tractatus", but considers changing it is nothing but a bother.)

There are MODMOS as well, but Kosmorsky has not explored them yet. 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

s s s L s s L s s L - Keter

s s L s s L s s L s - Chesed

s L s s L s s L s s - Netzach

L s s L s s L s s s - Malkuth

s s L s s L s s s L - Binah

s L s s L s s s L s - Tiferet

L s s L s s s L s s - Yesod

s s L s s s L s s L - Chokmah

s L s s s L s s L s - Gevurah

L s s s L s s L s s - Hod

Scales tree

Generator Cents L s L/s Comments
3\10 360.000 1 1 1.000
16\53 362.264 6 5 1.200 Submajor
13\43 362.791 5 4 1.250
23\76 363.158 9 7 1.286
10\33 363.636 4 3 1.333
27\89 364.045 11 8 1.375
17\56 364.286 7 5 1.400
24\79 364.557 10 7 1.428
7\23 365.217 3 2 1.500 L/s = 3/2
25\82 365.854 11 7 1.571
18\59 366.102 8 5 1.600
29\95 366.316 13 8 1.625 Unnamed golden tuning
11\36 366.667 5 3 1.667
26\85 367.059 12 7 1.714
15\49 367.347 7 4 1.750
19\62 367.742 9 5 1.800
4\13 369.231 2 1 2.000 Basic sephiroid
(Generators smaller than this are proper)
17\55 370.909 9 4 2.250
13\42 371.429 7 3 2.333
22\71 371.831 12 5 2.400
9\29 372.414 5 2 2.500 Sephiroth
23\74 372.973 13 5 2.600 Golden sephiroth
14\45 373.333 8 3 2.667
19\61 373.770 11 4 2.750
5\16 375.000 3 1 3.000 L/s = 3/1
16\51 376.471 10 3 3.333
11\35 377.143 7 2 3.500
17\54 377.778 11 3 3.667 Muggles
6\19 378.947 4 1 4.000 Magic/horcrux
13\41 380.488 9 2 4.500 Magic/witchcraft
7\22 381.818 5 1 5.000 Magic/telepathy
8\25 384.000 6 1 6.000 Würschmidt↓
1\3 400.000 1 0 → inf