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
					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
	4\13						369.231	436.364	2	1	2.000	Basic sephiroid (Generators smaller than this are proper)
						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
			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
		5\16					375.000	461.538	3	1	3.000	L/s = 3/1
						21\67	376.119	466.667	13	4	3.250
					16\51		376.471	468.293	10	3	3.333
						27\86	376.744	469.565	17	5	3.400
				11\35			377.143	471.429	7	2	3.500
						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
			6\19				378.947	480.000	4	1	4.000	Magic/horcrux
						19\60	380.000	485.105	13	3	4.333
					13\41		380.488	487.500	9	2	4.500	Magic/witchcraft
						20\63	380.952	489.769	14	3	4.667
				7\22			381.818	494.118	5	1	5.000	Magic/telepathy
						15\47	382.979	500.000	11	2	5.500
					8\25		384.000	505.263	6	1	6.000	Würschmidt↓
						9\28	385.714	514.286	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 (44/21-equivalent) - Neogothic minor ninth tuning

3L 7s (21/10-equivalent) - septimal chromatic minor ninth tuning

3L 7s (19/9-equivalent) - simplest ratio near highest tuning (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