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)
173\245	898.701	374.026	29	14	2.071
161\228	898.605	373.953	27	13	2.077
149\211	898.492	373.869	25	12	2.083
137\194	898.361	373.770	23	11	2.091
125\177	898.204	373.653	21	10	2.100
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
165\233	887.892	365.919	29	10	2.900
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