User:Moremajorthanmajor/Greater sephiroid: Difference between revisions

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|400.000||2||1||2.000||Basic sephiroid<br>(Generators smaller than this are proper)
|436.364||2||1||2.000||Basic sephiroid<br>(Generators smaller than this are proper)
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Revision as of 14:33, 21 June 2023

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 L s L/s Comments
3\10 360.000 360.000 1 1 1.000
19\63 361.905 407.143 7 6 1.167
16\53 362.264 408.511 6 5 1.200 Submajor
29\96 362.500 409.412 11 9 1.222
13\43 362.791 410.526 5 4 1.250
23\76 363.158 411.940 9 7 1.286
33\109 363.303 412.500 13 10 1.300
10\33 363.636 413.793 4 3 1.333
27\89 364.045 415.385 11 8 1.375
17\56 364.286 416.327 7 5 1.400
24\79 364.557 417.391 10 7 1.428
31\102 364.706 417.978 13 9 1.444
7\23 365.217 420.000 3 2 1.500 L/s = 3/2
32\105 365.714 421.978 14 9 1.556
25\82 365.854 422.535 11 7 1.571
18\59 366.102 423.529 8 5 1.600
29\95 366.316 424.390 13 8 1.625 Unnamed golden tuning
11\36 366.667 425.806 5 3 1.667
37\121 366.942 426.923 17 10 1.700
26\85 367.059 427.397 12 7 1.714
15\49 367.347 428.571 7 4 1.750
19\62 367.742 430.189 9 5 1.800
23\75 368.000 431.250 11 6 1.833
4\13 369.231 436.364 2 1 2.000 Basic sephiroid
(Generators smaller than this are proper)
21\68 370.588 442.105 11 5 2.200
17\55 370.909 443.478 9 4 2.250
30\97 371.134 444.444 16 7 2.286
13\42 371.429 445.714 7 3 2.333
35\113 371.681 446.809 19 8 2.375
22\71 371.831 447.458 12 5 2.400
31\100 372.000 448.193 17 7 2.429
9\29 372.414 450.000 5 2 2.500 Sephiroth
32\103 372.816 451.765 18 7 2.571
23\74 372.973 452.459 13 5 2.600
37\119 373.109 453.061 21 8 2.625 Golden sephiroth
14\45 373.333 454.054 8 3 2.667
33\106 373.585 455.172 19 7 2.714
19\61 373.770 456.000 11 4 2.750
24\77 374.000 457.143 14 5 2.800
5\16 375.000 461.538 3 1 3.000 L/s = 3/1
21\67 376.119 466.667 13 4 3.250
16\51 376.471 468.293 10 3 3.333
27\86 376.744 469.565 17 5 3.400
11\35 377.143 471.429 7 2 3.500
28\89 377.528 473.239 18 5 3.600
17\54 377.778 474.419 11 3 3.667 Muggles
23\73 378.082 475.862 15 4 3.750
6\19 378.947 480.000 4 1 4.000 Magic/horcrux
19\60 380.000 485.105 13 3 4.333
13\41 380.488 487.500 9 2 4.500 Magic/witchcraft
20\63 380.952 489.769 14 3 4.667
7\22 381.818 494.118 5 1 5.000 Magic/telepathy
15\47 382.979 500.000 11 2 5.500
8\25 384.000 505.263 6 1 6.000 Würschmidt↓
9\28 385.714 514.286 7 1 7.000
1\3 400.000 600.000 1 0 → inf

See also

Parents of Greater Luachoid

3L 7s (33/16-equivalent) - harmonic subminor ninth tuning

3L 7s (21/10-equivalent) - septimal chromatic minor ninth tuning

3L 7s (19/9-equivalent) - simplest ratio near median edo interval (13\12)

Upper tunings

3L 7s (15/7-equivalent) - septimal diatonic minor ninth tuning

3L 7s (11/5-equivalent) - undecimal neutral ninth tuning

3L 7s (9/4-equivalent) - major ninth tuning

3L 7s (7/3-equivalent) - septimal minor tenth tuning

3L 7s (5/2-equivalent) - major tenth tuning

3L 7s (8/3-equivalent) - Anti-Choralic

3L 7s (11/4-equivalent) - undecimal augmented eleventh/diminished twelfth tuning

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