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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.