3L 7s

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.