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		Śata		L	s	L/s	Comments
									Chroma-positive	Chroma-negative	Chroma-positive	Chroma-negative
7\10									933.333	400.000	1244.444	533.333	1	1	1.000
					40\57				923.077	392.307	1230.769	523.077	6	5	1.200	Restles↑
						73\104			922.105	391.579	1229.474	522.105	11	9	1.222
							106\151		921.739	391.304	1228.986	521.739	16	13	1.231
				33\47					920.930	390.698	1227.907	520.930	5	4	1.250
						92\131			920.000	390.000	1226.667	520.000	13	11	1.273
					59\84				919.481	389.610	1225.974	519.481	9	7	1.286
						85\121			918.919	389.189	1225.225	518.919	13	10	1.000
							111\158		918.621	388.966	1224.828	518.621	17	13	1.308
			26\37						917.647	388.235	1223.529	517.647	4	3	1.333
						97\138			916.535	387.401	1222.047	516.535	15	11	1.364
					71\101				916.129	387.097	1221.505	516.129	11	8	1.375
				45\64					915.254	386.441	1220.339	515.254	7	5	1.400	Beatles
					64\91				914.286	385.714	1219.048	514.286	10	7	1.428
		19\27							912.000	384.000	1196.296	912.000	3	2	1.500	L/s = 3/2, suhajira/ringo
								126\179	910.843	383.133	1196.648	910.843	20	13	1.538
							107\152		910.638	382.988	1196.711	910.638	17	11	1.545
						88\125			910.345	382.759	1196.800	910.345	14	9	1.556
					69\98				909.890	382.418	1196.938	909.890	11	7	1.571
				50\71					909.091	381.818	1197.183	909.091	8	5	1.600
					81\115				908.411	381.308	1197.391	908.411	13	8	1.625	Golden suhajira
			31\44						907.317	380.489	1197.727	907.317	5	3	1.667
					74\105				906.122	379.592	1198.095	906.122	12	7	1.714
				43\61					905.263	378.947	1198.361	905.263	7	4	1.750
					55\78				904.110	378.082	1198.561	904.110	9	5	1.800
						67\95			903.371	377.528	1198.947	903.371	11	6	1.833
							79\112		902.857	377.143	1199.107	902.857	13	7	1.857
								91\129	902.479	376.860	1199.224	902.479	15	8	1.875
	12\17								900.000	375.000	1200.000	900.000	2	1	2.000	Basic Greater dicoid (Generators smaller than this are proper)
								89\126	897.479	373.109	1200.793	897.479	15	7	2.143
							77\109		897.087	372.816	1200.917	897.087	13	6	2.167
						65\92			896.552	372.414	1201.087	896.552	11	5	2.200
					53\75				895.775	371.831	1201.333	895.775	9	4	2.250
				41\58					894.545	370.909	1201.724	894.545	7	3	2.333
					70\99				893.617	370.213	1202.020	893.617	12	5	2.400	Hemif/hemififths
			29\41						892.307	369.231	1202.439	892.307	5	2	2.500	Mohaha/neutrominant
					75\106				891.089	368.317	1202.830	891.089	13	5	2.600	Hemif/salsa/karadeniz
				46\65					890.322	367.742	1203.077	890.322	8	3	2.667	Mohaha/mohamaq
					63\89				889.412	367.059	1203.371	889.412	11	4	2.750
						80\113			888.889	366.667	1203.540	888.889	14	5	2.800
		17\24							886.957	365.213	1204.167	886.957	3	1	3.000	L/s = 3/1
							90\127		885.246	363.934	1204.724	885.246	16	5	3.200
						73\103			884.848	363.636	1204.854	884.848	13	4	3.250
					56\79				884.210	363.158	1205.063	884.210	10	3	3.333
				39\55					883.018	362.264	1205.455	883.018	7	2	3.500
					61\86				881.928	361.446	1205.814	881.928	11	3	3.667
			22\31						880.000	360.000	1206.452	880.000	4	1	4.000	Mohaha/migration/mohajira
							93\131		878.740	359.055	1206.870	878.740	17	4	4.250
						71\100			878.351	358.762	1207.000	878.351	13	3	4.333
					49\69				877.612	358.209	1207.246	877.612	9	2	4.500
						76\107			876.923	357.692	1207.477	876.923	14	3	4.667
				27\38					875.676	356.757	1207.895	875.676	5	1	5.000
						59\83			874.074	355.556	1208.434	874.074	11	2	5.500
					32\45				872.727	354.545	1208.889	872.727	6	1	6.000	Mohaha/ptolemy
5\7									857.143	342.857	1214.286	485.714	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 (21/10-equivalent) - septimal chromatic minor ninth tuning

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