3L 7s

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3L+7s "Fair Mosh" (Modi Sephiratorum)

Fair Mosh is found in magic (chains of the 5th harmonic). It occupies the spectrum from 10edo (L=s) to 3edo (s=0).

This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents). In the region of the spectrum around 23 edo (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, ie. multiples of 10edo, the 13th harmonic becomes nearly perfect. 121 edo seems to be the first to 'accurately' represent the comma (which might as well be represented accurately as it's 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 5/4 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 http://ia600706.us.archive.org/23/items/TractatumDeModiSephiratorum/ModiSephiratorum.pdf

(I know it should be "tractatus", changing it is just a bother)

There are MODMOS as well, but I haven't 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".

(ascending)

s s s L s s L s s L - Mode 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

--

Generator Cents L s Comments
3\10 360 120 120
28\93 361.290 129.032 116.129
25\83 361.446 130.1205 115.663
22\73 361.644 131.507 115.0685
19\63 361.905 133.333 114.286
16\53 362.264 135.849 113.2075
13\43 362.791 139.535 111.628
10\33 363.636 145.455 109.091
7\23 365.217 156.522 104.348
365.848 160.937 102.456
18\59 366.102 162.712 104.29
47\154 366.234 163.636 101.299
123\403 366.253 163.771 101.241
322\1055 366.256 163.791 101.232
521\1707 366.257 163.796 101.230 Golden Sephiroth
199\652 366.258 163.804 101.227
76\249 366.265 163.855 101.205
29\95 366.316 164.2105 101.053
11\36 366.667 166.667 100
367.203 170.419 98.392
15\49 367.347 171.429 97.959
4\13 369.231 184.615 92.308 Boundary of propriety

(smaller generators are proper)

13\42 371.429 200 85.714
9\29 372.414 206.897 82.759
23\74 372.973 210.811 81.081
60\193 373.057 211.399 80.829
157\505 373.069 211.485 80.792
411\1322 373.071 211.498 80.787
254\817 373.072 211.5055 80.783
97\312 373.077 211.5385 80.769
37\119 373.109 211.765 80.672
14\45 373.333 213.333 80
374.870 224.090 79.183
5\16 375 225 75
375.130 225.910 73.06
11\35 377.143 240 68.571
6\19 378.947 252.632 63.158
19\60 380 260 60 Magic is around here
13\41 380.488 263.415 58.537
20\63 380.952 266.667 57.143
7\22 381.818 272.72 54.545
8\25 384 288 48
9\28 385.714 300 42.857
10\31 387.097 309.677 38.710
21\65 387.692 313.846 36.923 Würschmidt is around here
11\34 388.235 317.647 35.294
12\37 389.189 324.324 32.432
13\40 390 330 30
14\43 390.698 334.884 27,907
29\89 391.011 337.079 26.966 Amigo is around here
15\46 391.304 339.130 26.087
1\3 400 400 0

L=1 s=1 10edo

L=2 s=1 13edo

(L=3 s=1 16edo)

L=3 s=2 23edo

(L=4 s=1 19edo)

L=4 s=3 33edo

(L=5 s=1 22edo)

(L=5 s=2 29edo)

L=5 s=3 36edo

L=5 s=4 43edo

(L=6 s=1 25edo)

L=6 s=5 53edo

L=7 s=6 63edo

L=7 s=5 56edo

L=7 s=4 49edo

etc.