User:Moremajorthanmajor/Greater sephiroid

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
37\122	363.935	414.953	15	11	1.455
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
27\88	368.182	432.000	13	7	1.857
31\101	368.317	432.558	15	8	1.875
35\114	368.421	432.990	17	9	1.889
39\127	368.504	433.333	19	10	1.900
43\140	368.571	433.614	21	11	1.909
47\153	368.627	433.846	23	12	1.917
51\166	368.675	434.042	25	13	1.923
55\179	368.715	434.210	27	14	1.929
59\192	368.750	434.356	29	15	1.933
63\205	368.780	434.483	31	16	1.9375
67\218	368.807	434.595	33	17	1.941
71\231	368.831	434.694	35	18	1.944
75\244	368.852	434.782	37	19	1.947
79\257	368.872	434.862	39	20	1.950
83\270	368.889	434.834	41	21	1.953
87\283	368.905	435.000	43	22	1.955
4\13	369.231	436.364	2	1	2.000	Basic sephiroid (Generators smaller than this are proper)
77\250	369.600	437.915	39	19	1.053
73\237	369.620	438.000	37	18	1.056
69\224	369.643	438.085	35	17	1.059
65\211	369.668	438.202	33	16	1.0625
61\198	369.697	438.323	31	15	2.067
57\185	369.730	438.462	29	14	2.071
53\172	369.767	438.621	27	13	2.077
49\159	369.811	438.806	25	12	2.083
45\146	369.863	439.024	23	11	2.091
41\133	369.925	439.286	21	10	2.100
37\120	370.000	439.604	19	9	2.111
33\107	370.093	440.000	17	8	2.125
29\94	370.213	440.506	15	7	2.143
25\81	370.370	441.176	13	6	2.167
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
40\129	372.093	448.598	22	9	2.444
49\158	372.152	448.855	27	11	2.455
58\187	372.193	449.032	32	13	2.462
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
29\93	374.194	457.895	17	6	2.833
34\109	374.312	458.427	20	7	2.857
39\125	374.400	458.824	23	8	2.875
44\141	374.468	459.130	26	9	2.889
49\157	374.522	459.375	29	10	2.900
54\173	374.566	459.574	32	11	2.909
59\189	374.603	459.740	35	12	2.917
64\205	374.634	459.880	38	13	2.923
69\221	374.660	460.000	41	14	2.929
5\16	375.000	461.538	3	1	3.000	L/s = 3/1
76\243	375.309	462.944	46	15	3.067
71\227	375.330	463.043	43	14	3.071
66\211	375.355	463.158	40	13	3.077
61\195	375.385	463.291	37	12	3.083
56\179	375.419	463.448	34	11	3.091
51\163	375.460	463.636	31	10	3.100
46\147	375.510	463.866	28	9	3.111
41\131	375.573	464.151	25	8	3.125
36\115	375.652	464.516	22	7	3.143
31\99	375.756	465.000	19	6	3.167
26\83	375.904	465.672	16	5	3.200
21\67	376.119	466.667	13	4	3.250
37\118	376.271	457.368	23	7	3.286
16\51	376.471	468.293	10	3	3.333
27\86	376.744	469.565	17	5	3.400
38\121	376.860	470.103	24	7	3.429
11\35	377.143	471.429	7	2	3.500
39\124	377.419	472.727	25	7	3,571
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
29\92	378.261	476.712	19	5	3.800
35\111	378.378	477.273	23	6	3.833
41\130	378.462	477.670	27	7	3.857
47\149	378.523	477.966	31	8	3.875
53\168	378,571	478.195	35	9	3.889
59\187	378.607	478.378	39	10	3.900
65\206	378.641	478.528	43	11	3.909
71\225	378.667	478.652	47	12	3.917
77\244	378.689	478.756	51	13	3.923
83\263	378.707	478.846	55	14	3.929
89\282	378.723	478.924	59	15	3.933
95\301	378.738	478.992	63	16	3.9375
101\320	378.750	479.051	67	17	3.941
6\19	378.947	480.000	4	1	4.000	Magic/horcrux
25\79	379.747	483.871	17	4	4.250
44\139	379.856	484.404	30	7	4.286
19\60	380.000	485.105	13	3	4.333
32\101	380.198	486.076	22	5	4.400
13\41	380.488	487.500	9	2	4.500	Magic/witchcraft
33\104	380.769	488.889	23	5	4.600
20\63	380.952	489.769	14	3	4.667
27\85	381.176	490.908	19	4	4.750
34\107	381.308	491.566	24	5	4.800
41\129	381.395	492.000	29	6	4.833
48\151	381.456	492.308	34	7	4.857
55\173	381.503	492.537	39	8	4.875
62\195	381.538	492.715	44	9	4.889
69\217	381.567	492.857	49	10	4.900
76\239	381.590	492.973	54	11	4.909
83\261	381.609	493.069	59	12	4.917
7\22	381.818	494.118	5	1	5.000	Magic/telepathy
50\157	382.166	495.868	36	7	5.143
43\135	382.222	496.153	31	6	5.167
36\113	382.301	496.551	26	5	5.200
29\91	382.418	497.143	21	4	5.250
22\69	382.609	498.113	16	3	5.333
37\116	382.759	498.876	27	5	5.400
52\163	382.822	499.200	38	7	5.429
15\47	382.979	500.000	11	2	5.500
23\72	383.333	501.818	17	3	5.667
8\25	384.000	505.263	6	1	6.000	Würschmidt↓
49\153	384.314	506.897	37	6	6.167
41\128	384.375	507.216	31	5	6.200
33\103	384.466	507.692	25	4	6.250
25\78	384.615	508.475	19	3	6.333
42\131	384.733	509.091	32	5	6.400
17\53	384.906	510.000	13	2	6.500
9\28	385.714	514.286	7	1	7.000
1\3	400.000	600.000	1	0	→ inf

User:Moremajorthanmajor/Greater sephiroid

Contents

Modes

Scale tree

See also

Parents of Greater Luachoid

Upper tunings

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