3L 7s (5/2-equivalent)

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↖ 2L 6s⟨5/2⟩ ↑ 3L 6s⟨5/2⟩ 4L 6s⟨5/2⟩ ↗
← 2L 7s⟨5/2⟩ 3L 7s (5/2-equivalent) 4L 7s⟨5/2⟩ →
↙ 2L 8s⟨5/2⟩ ↓ 3L 8s⟨5/2⟩ 4L 8s⟨5/2⟩ ↘ 
┌╥┬┬╥┬┬╥┬┬┬┐
│║││║││║││││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LssLssLsss
sssLssLssL
Equave 5/2 (1586.3 ¢)
Period 5/2 (1586.3 ¢)
Generator size(ed5/2)
Bright 3\10 to 1\3 (475.9 ¢ to 528.8 ¢)
Dark 2\3 to 7\10 (1057.5 ¢ to 1110.4 ¢)
Related MOS scales
Parent 3L 4s⟨5/2⟩
Sister 7L 3s⟨5/2⟩
Daughters 10L 3s⟨5/2⟩, 3L 10s⟨5/2⟩
Neutralized 6L 4s⟨5/2⟩
2-Flought 13L 7s⟨5/2⟩, 3L 17s⟨5/2⟩
Equal tunings(ed5/2)
Equalized (L:s = 1:1) 3\10 (475.9 ¢)
Supersoft (L:s = 4:3) 10\33 (480.7 ¢)
Soft (L:s = 3:2) 7\23 (482.8 ¢)
Semisoft (L:s = 5:3) 11\36 (484.7 ¢)
Basic (L:s = 2:1) 4\13 (488.1 ¢)
Semihard (L:s = 5:2) 9\29 (492.3 ¢)
Hard (L:s = 3:1) 5\16 (495.7 ¢)
Superhard (L:s = 4:1) 6\19 (500.9 ¢)
Collapsed (L:s = 1:0) 1\3 (528.8 ¢)

3L 7s(<5/2>) 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

Scale tree

Generator Cents Normalized Cents ed13\11 L s L/s Comments
3\10 360.000 400.000 390.000 1 1 1.000
16\53 362.264 408.511 392.453 6 5 1.200 Submajor
13\43 362.791 410.526 393.023 5 4 1.250
23\76 363.158 411.940 393.421 9 7 1.286
10\33 363.636 413.793 393.939 4 3 1.333
27\89 364.045 415.385 394.382 11 8 1.375
17\56 364.286 416.2365 394.643 7 5 1.400
24\79 364.557 417.319 394.937 10 7 1.428
7\23 365.217 420.000 395.652 3 2 1.500 L/s = 3/2
25\82 365.854 422.535 396.3415 11 7 1.571
18\59 366.102 423.529 396.610 8 5 1.600
29\95 366.316 424.390 396.842 13 8 1.625 Unnamed golden tuning
11\36 366.667 425.8065 396.667 5 3 1.667
26\85 367.059 427.397 397.647 12 7 1.714
15\49 367.347 428.571 397.959 7 4 1.750
19\62 367.742 430.189 398.387 9 5 1.800
4\13 369.231 436.364 400.000 2 1 2.000 Basic sephiroid
(Generators smaller than this are proper)
17\55 370.909 443.478 401.818 9 4 2.250
13\42 371.429 445.714 402.381 7 3 2.333
22\71 371.831 447.458 402.817 12 5 2.400
9\29 372.414 450.000 403.448 5 2 2.500 Sephiroth
23\74 372.973 452.459 404.054 13 5 2.600 Golden sephiroth
14\45 373.333 454.054 404.444 8 3 2.667
19\61 373.770 456.000 404.981 11 4 2.750
5\16 375.000 461.5385 406.250 3 1 3.000 L/s = 3/1
16\51 376.471 468.293 407.843 10 3 3.333
11\35 377.143 471.429 408.571 7 2 3.500
17\54 377.778 474.419 409.259 11 3 3.667 Muggles
6\19 378.947 480.000 410.526 4 1 4.000 Magic/horcrux
13\41 380.488 487.500 412.195 9 2 4.500 Magic/witchcraft
7\22 381.818 500.000 413.636 5 1 5.000 Magic/telepathy
8\25 384.000 505.263 416.000 6 1 6.000 Würschmidt↓
1\3 400.000 600.000 433.333 1 0 → inf
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