User:Moremajorthanmajor/Greater dicoid

7L 3s refers to the structure of moment of symmetry scales built from a 10-tone chain of neutral or major thirds (assuming a period of an octave up to 10/9edo):

L s L L L s L L s L

Graham Breed has a page on his website dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:

t q t t t q t t q t

Names

This MOS is called Greater dicoid (from dicot, an exotemperament) in TAMNAMS.

Intervals

The generator (g) will fall between 343 cents (2\7 - two degrees of 7edo and 400 cents (3\9 - three degrees of 9edo), hence a neutral or major third.

2g, then, will fall between 686 cents (4\7) and 800 cents (2\3), the range of diatonic fifths and subminor sixths.

The "large step" will fall between 171 cents (1\7) and 120 cents (1\10), ranging from a submajor second to a sinaic.

The "small step" will fall between 0 cents and 120 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.

The most frequent interval, then is the neutral third (and its inversion, the neutral sixth), followed by the perfect fourth and fifth. Thus, 7+3 combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals. They are compatible with Arabic and Turkish scales, but not with traditional Western ones.

Note: In TAMNAMS, a k-step interval class in dicoid may be called a "k-step", "k-mosstep", or "k-dicostep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

# generators up	Notation (1/1 = 0)	name	In L's and s's	# generators up	Notation of 15/7 inverse	name	In L's and s's
The 10-note MOS has the following intervals (from some root):
0	0	perfect unison	0	0	0	perfect 10-step	7L+3s
1	7	perfect 7-step	5L+2s	-1	3	perfect 3-step	2L+1s
2	4	major 4-step	3L+1s	-2	6v	minor 6-step	4L+2s
3	1	major (1-)step	1L	-3	9v	minor 9-step	6L+3s
4	8	major 8-step	6L+2s	-4	2v	minor 2-step	1L+1s
5	5	major 5-step	4L+1s	-5	5v	minor 5-step	3L+2s
6	2	major 2-step	2L	-6	8v	minor 8-step	5L+3s
7	9	major 9-step	7L+2s	-7	1v	minor (1-)step	1s
8	6	major 6-step	5L+1s	-8	4v	minor 4-step	2L+2s
9	3^	augmented 3-step	3L	-9	7v	diminished 7-step	4L+3s
10	0^	augmented unison	1L-1s	-10	0v	diminished 10-step	6L+4s
11	7^	augmented 7-step	6L+1s	-11	3v	diminished 3-step	1L+2s
The chromatic 17-note MOS (either 7L 10s, 10L 7s, or 17edIX) also has the following intervals (from some root):
12	4^	augmented 4-step	4L	-12	6v	diminished 6-step	3L+3s
13	1^	augmented (1-)step	2L-1s	-13	9w	diminished 9-step	5L+4s
14	8^	augmented 8-step	8L+1s	-14	2w	diminished 2-step	2s
15	5^	augmented 5-step	5L	-15	5w	diminished 5-step	2L+3s
16	2^	augmented 2-step	3L-1s	-16	8w	diminished 8-step	4L+4s

Scale tree

The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edIX would be (3+2)\(10+7) = 5\17 – five degrees of 17edIX:

Generator											Normalized Cents		L	s	L/s	Comments
											Bright	Dark
7\10											933.333	400.000	1	1	1.000
					40\57						923.077	392.307	6	5	1.200	Restles↑
						73\104					922.105	391.579	11	9	1.222
							106\151				921.739	391.304	16	13	1.231
				33\47							920.930	390.698	5	4	1.250
						92\131					920.000	390.000	13	11	1.273
					59\84						919.481	389.610	9	7	1.286
						85\121					918.919	389.189	13	10	1.000
							111\158				918.621	388.966	17	13	1.308
			26\37								917.647	388.235	4	3	1.333
						97\138					916.535	387.401	15	11	1.364
					71\101						916.129	387.097	11	8	1.375
				45\64							915.254	386.441	7	5	1.400	Beatles
					64\91						914.286	385.714	10	7	1.428
						83\118					913.716	385.321	13	9	1.444
		19\27									912.000	384.000	3	2	1.500	L/s = 3/2, suhajira/ringo
										164\233	911.111	383.333	26	17	1.529
									145\206		910.995	383.246	23	15	1.533
								126\179			910.843	383.133	20	13	1.538
							107\152				910.638	382.988	17	11	1.545
						88\125					910.345	382.759	14	9	1.556
					69\98						909.890	382.418	11	7	1.571
				50\71							909.091	381.818	8	5	1.600
					81\115						908.411	381.308	13	8	1.625	Golden suhajira
			31\44								907.317	380.489	5	3	1.667
					74\105						906.122	379.592	12	7	1.714
				43\61							905.263	378.947	7	4	1.750
					55\78						904.110	378.082	9	5	1.800
						67\95					903.371	377.528	11	6	1.833
							79\112				902.857	377.143	13	7	1.857
								91\129			902.479	376.860	15	8	1.875
									103\146		902.190	376.642	17	9	1.889
										115\163	901.961	376.471	19	10	1.900
	12\17										900.000	375.000	2	1	2.000	Basic Greater dicoid (Generators smaller than this are proper)
										113\160	898.013	373.510	19	9	2.111
									101\143		897.778	373.333	17	8	2.125
								89\126			897.479	373.109	15	7	2.143
							77\109				897.087	372.816	13	6	2.167
						65\92					896.552	372.414	11	5	2.200
					53\75						895.775	371.831	9	4	2.250
				41\58							894.545	370.909	7	3	2.333
					70\99						893.617	370.213	12	5	2.400	Hemif/hemififths
			29\41								892.307	369.231	5	2	2.500	Mohaha/neutrominant
					75\106						891.089	368.317	13	5	2.600	Hemif/salsa/karadeniz
				46\65							890.322	367.742	8	3	2.667	Mohaha/mohamaq
					63\89						889.412	367.059	11	4	2.750
						80\113					888.889	366.667	14	5	2.800
							97\137				888.550	366.412	17	6	2.833
								114\161			888.312	366.234	20	7	2.857
									131\185		888.136	366.102	23	8	2.875
										148\209	888.000	366.000	26	9	2.889
		17\24									886.957	365.213	3	1	3.000	L/s = 3/1
							90\127				885.246	363.934	16	5	3.200
						73\103					884.848	363.636	13	4	3.250
					56\79						884.210	363.158	10	3	3.333
				39\55							883.018	362.264	7	2	3.500
					61\86						881.928	361.446	11	3	3.667
			22\31								880.000	360.000	4	1	4.000	Mohaha/migration/mohajira
									137\193		879.144	359.358	25	6	4.167
								115\162			878.980	359.236	21	5	4.200
							93\131				878.740	359.055	17	4	4.250
						71\100					878.351	358.762	13	3	4.333
					49\69						877.612	358.209	9	2	4.500
						76\107					876.923	357.692	14	3	4.667
				27\38							875.676	356.757	5	1	5.000
						59\83					874.074	355.556	11	2	5.500
							91\128				873.600	355.200	17	3	5.667
					32\45						872.727	354.545	6	1	6.000	Mohaha/ptolemy
5\7											857.143	342.857	1	0	→ inf

The scale produced by stacks of 5\17 is the 17edIX neutral scale. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a chromatic pair.

Other compatible edIXs include: 37edIX, 27edIX, 44edIX, 41edIX, 24edIX, 31edIX.

You can also build this scale by stacking neutral thirds that are not members of edIXs – for instance, frequency ratios 11:9, 5:4, 21:17, 16:13 – or the square root of 3:2 or 11:7 (a bisected just perfect fifth or undecimal subminor sixth).

7-note subsets

If you stop the chain at 7 tones, you have a heptatonic scale of the form 3L 4s:

L s s L s L s

The large steps here consist of t+s of the 10-tone system, and the small step is the same as t. Graham proposes calling the large step here T for "tone," uppercase because it is a wider tone than t. Thus, we have:

T t t T t T t

This scale (and its rotations) is not the only possible heptatonic scale. Graham also gives us:

T t t T t t T

which is not a complete moment of symmetry scale in itself, but a subset of one.

Tetrachordal structure

Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a tetrachordal scale. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.

I ( - Andrew Heathwaite) offer "a" to refer to a step of 2t (for "augmented second")

Thus, the possible tetrachords are:

T t t

t T t

t t T

a q t

a t q

t a q

t q a

q a t

q t a

See also

7L 3s (33/16-equivalent) - harmonic subminor ninth tuning

7L 3s (44/21-equivalent) - Neogothic minor ninth tuning

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

7L 3s (15/7-equivalent) - septimal diatonic minor ninth tuning