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	367.742	7	6	1.167
					16\53		362.264	369.231	6	5	1.200	Submajor
						29\96	362.500	370.213	11	9	1.222
				13\43			362.791	371.429	5	4	1.250
					23\76		363.158	372.973	9	7	1.286
						33\109	363.303	373.585	13	10	1.300
			10\33				363.636	375.000	4	3	1.333
					27\89		364.045	376.744	11	8	1.375
				17\56			364.286	377.778	7	5	1.400
					24\79		364.557	378.947	10	7	1.428
						31\102	364.706	379.592	13	9	1.444
		7\23					365.217	381.818	3	2	1.500	L/s = 3/2
						32\105	365.714	384.000	14	9	1.556
					25\82		365.854	384.615	11	7	1.571
				18\59			366.102	385.714	8	5	1.600
					29\95		366.316	386.667	13	8	1.625	Unnamed golden tuning
						40\131	366.412	387.097	18	11	1.636
			11\36				366.667	388.235	5	3	1.667
						37\121	366.942	389.474	17	10	1.700
					26\85		367.059	390.000	12	7	1.714
				15\49			367.347	391.304	7	4	1.750
					19\62		367.742	393.103	9	5	1.800
						23\75	368.000	394.286	11	6	1.833
	4\13						369.231	400.000	2	1	2.000	Basic sephiroid (Generators smaller than this are proper)
						21\68	370.588	406.452	11	5	2.200
					17\55		370.909	408.000	9	4	2.250
						30\97	371.134	409.091	16	7	2.286
				13\42			371.429	410.526	7	3	2.333
						35\113	371.681	411.765	19	8	2.375
					22\71		371.831	412.500	12	5	2.400
						31\100	372.000	413.333	17	7	2.429
			9\29				372.414	415.385	5	2	2.500	Sephiroth
						32\103	372.8155	417.391	18	7	2.571
					23\74		372.973	418.182	13	5	2.600
						37\119	373.109	418.868	21	8	2.625	Golden sephiroth
				14\45			373.333	420.000	8	3	2.667
						33\106	373.585	421.277	19	7	2.714
					19\61		373.770	422.222	11	4	2.750
						24\77	374.000	423.529	14	5	2.800
		5\16					375.000	428.571	3	1	3.000	L/s = 3/1
						21\67	376.119	434.483	13	4	3.250
					16\51		376.471	436.364	10	3	3.333
						27\86	376.744	437.838	17	5	3.400
				11\35			377.143	440.000	7	2	3.500
						28\89	377.528	442.105	18	5	3.600
					17\54		377.778	443.478	11	3	3.667	Muggles
						23\73	378.082	445.161	15	4	3.750
			6\19				378.947	450.000	4	1	4.000	Magic/horcrux
						19\60	380.000	456.000	13	3	4.333
					13\41		380.488	458.8235	9	2	4.500	Magic/witchcraft
						20\63	380.952	461.5385	14	3	4.667
				7\22			381.818	466.667	5	1	5.000	Magic/telepathy
						15\47	382.979	473.684	11	2	5.500
					8\25		384.000	480.000	6	1	6.000	Würschmidt↓
						9\28	385.714	490.909	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