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⟩ →
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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\12 L s L/s Comments
3\10 360.000 360.000 390.000 1 1 1.000
19\63 361.905 367.742 392.0635 7 6 1.167
16\53 362.264 369.231 392.453 6 5 1.200 Submajor
29\96 362.500 370.213 392.708 11 9 1.222
13\43 362.791 371.429 393.023 5 4 1.250
23\76 363.158 372.973 393.421 9 7 1.286
33\109 363.303 373.585 393.578 13 10 1.300
10\33 363.636 375.000 393.939 4 3 1.333
27\89 364.045 376.744 394.382 11 8 1.375
17\56 364.286 377.778 394.643 7 5 1.400
24\79 364.557 378.947 394.937 10 7 1.428
31\102 364.706 379.592 395.098 13 9 1.444
7\23 365.217 381.818 395.652 3 2 1.500 L/s = 3/2
32\105 365.714 384.000 396.1905 14 9 1.556
25\82 365.854 384.615 396.3415 11 7 1.571
18\59 366.102 385.714 396.610 8 5 1.600
29\95 366.316 386.667 396.842 13 8 1.625 Unnamed golden tuning
40\131 366.412 387.097 396.947 18 11 1.636
11\36 366.667 388.235 397.222 5 3 1.667
37\121 366.942 389.474 397.521 17 10 1.700
26\85 367.059 390.000 397.647 12 7 1.714
15\49 367.347 391.304 397.959 7 4 1.750
19\62 367.742 393.103 398.387 9 5 1.800
23\75 368.000 394.286 398.667 11 6 1.833
4\13 369.231 400.000 400.000 2 1 2.000 Basic sephiroid
(Generators smaller than this are proper)
21\68 370.588 406.452 401.471 11 5 2.200
17\55 370.909 408.000 401.818 9 4 2.250
30\97 371.134 409.091 402.062 16 7 2.286
13\42 371.429 410.526 402.381 7 3 2.333
35\113 371.681 411.765 402.655 19 8 2.375
22\71 371.831 412.500 402.817 12 5 2.400
31\100 372.000 413.333 403.000 17 7 2.429
9\29 372.414 415.385 403.448 5 2 2.500 Sephiroth
32\103 372.8155 417.391 403.8835 18 7 2.571
23\74 372.973 418.182 404.054 13 5 2.600
37\119 373.109 418.868 404.202 21 8 2.625 Golden sephiroth
14\45 373.333 420.000 404.444 8 3 2.667
33\106 373.585 421.277 404.717 19 7 2.714
19\61 373.770 422.222 404.981 11 4 2.750
24\77 374.000 423.529 405.195 14 5 2.800
5\16 375.000 428.571 406.250 3 1 3.000 L/s = 3/1
21\67 376.119 434.483 407.463 13 4 3.250
16\51 376.471 436.364 407.843 10 3 3.333
27\86 376.744 437.838 408.1395 17 5 3.400
11\35 377.143 440.000 408.571 7 2 3.500
28\89 377.528 442.105 408.989 18 5 3.600
17\54 377.778 443.478 409.259 11 3 3.667 Muggles
23\73 378.082 445.161 409.589 15 4 3.750
6\19 378.947 450.000 410.526 4 1 4.000 Magic/horcrux
19\60 380.000 456.000 411.667 13 3 4.333
13\41 380.488 458.8235 412.195 9 2 4.500 Magic/witchcraft
20\63 380.952 461.5385 412.698 14 3 4.667
7\22 381.818 466.667 413.636 5 1 5.000 Magic/telepathy
15\47 382.979 473.684 414.894 11 2 5.500
8\25 384.000 480.000 416.000 6 1 6.000 Würschmidt↓
9\28 385.714 490.909 417.857 7 1 7.000
1\3 400.000 600.000 433.333 1 0 → inf
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