7-limit

Prime limit

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7-limit tuning

The 7-limit or 7-prime-limit consists of rational intervals where 7 is the highest allowable prime factor, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 7. The 7-limit is the fourth prime limit and is a superset of the 5-limit and a subset of the 11-limit. Some examples of 7-limit intervals include 7/4, 7/5, 7/6, 9/7, 15/14, 21/16, 21/20, 35/27, 49/36, and so on.

These things are contained by the 7-limit, but not the 5-limit:

The 7- and 9-odd-limit;
Mode 4 and 5 of the harmonic or subharmonic series.

The 7-odd-limit is a constraint on the selection of just intervals for a scale or composition such that 7 is the highest allowable odd number, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 7 odd-limit intervals within the octave is 1/1, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4, 2/1, which is known as the 7-odd-limit tonality diamond.

The phrase "7-limit just intonation" usually refers to the 7-prime-limit and includes primes 2, 3, 5, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 5, and 7, which can be represented in 3-dimensional lattice diagrams, each prime represented by a different dimension. Lattices describing scales beyond the 7-limit require more than three dimensions, and in the 7-limit, such lattices have unique features which simplify the relations between 7-limit chords.

For a variety of reasons, common-practice music has been somewhat stuck at the 5-limit for centuries, though 7-limit intervals have a characteristic jazzy sound which is at least partially familiar. Music in the 7-limit thus represents a large step forward, although not as much as 11- or 13-limit, which usually sound much more exotic.

Edo approximation

Here is a list of edos which tunes the 7-limit with more accuracy (decreasing TE error): 10, 12, 19, 27, 31, 41, 53, 72, 99, 171, 441, 612, ….

Here is a list of edos which tunes the 7-limit well relative to their size (TE relative error < 5%): 12, 19, 31, 41, 53, 72, 99, 118, 130, 140, 152, 171, 183, 202, 212, 217, 224, 229, 239, 243, 251, 270, 282, 289, 301, 311, 323, 354, 369, 373, 383, 388, 395, 400, 410, 414, 422, 441, 453, 460, 472, 482, 494, 525, 544, 566, 571, 581, 593, 612, ….

Intervals

Here is a table of intervals in the 7-prime-limit and 81-odd-limit.

Ratio	Monzo	Size in ¢s	Color name
1/1	[0⟩	0.000	w1	wa unison
81/80	[-4 4 -1⟩	21.506	g1	gu comma
64/63	[6 -2 0 -1⟩	27.264	r1	ru comma
50/49	[1 0 2 -2⟩	34.976	rryy-2	biruyo comma
49/48	[1 0 2 -2⟩	35.697	zz2	zozo comma
36/35	[2 2 -1 -1⟩	48.770	rg1	rugu comma
28/27	[2 -3 0 1⟩	62.961	z2	zo 2nd
25/24	[-3 -1 2⟩	70.672	yy1	yoyo unison
21/20	[-2 1 -1 1⟩	84.467	zg2	zogu 2nd
16/15	[4 -1 -1⟩	111.731	g2	gu 2nd
15/14	[-1 1 1 -1⟩	119.443	ry1	ruyo unison
27/25	[0 3 -2⟩	133.238	gg2	gugu 2nd
49/45	[0 -2 -1 2⟩	147.428	zzg3	zozogu 3rd
35/32	[-5 0 1 1⟩	155.140	zy2	zoyo 2nd
54/49	[1 3 0 -2⟩	168.213	rr1	ruru unison
10/9	[1 0 2 -2⟩	182.404	y2	yo 2nd
28/25	[2 0 -2 1⟩	196.198	zgg3	zogugu 3rd
9/8	[-3 2⟩	203.910	w2	wa 2nd
8/7	[3 0 0 -1⟩	231.174	r2	ru 2nd
81/70	[-1 4 -1 -1⟩	252.680	rg2	rugu 2nd
7/6	[-1 -1 0 1⟩	266.871	z3	zo 3rd
75/64	[-6 1 2⟩	274.582	yy2	yoyo 2nd
32/27	[5 -3⟩	294.135	w3	wa 3rd
25/21	[0 -1 2 -1⟩	301.847	ryy2	ruyoyo 2nd
6/5	[1 1 -1⟩	315.641	g3	gu 3rd
98/81	[1 -4 0 2⟩	329.832	zz4	zozo 4th
60/49	[2 1 1 -2⟩	350.617	rry2	ruruyo 2nd
49/40	[-3 0 -1 2⟩	351.338	zzg4	zozogu 4th
100/81	[2 -4 2⟩	364.807	yy3	yoyo 3rd
56/45	[3 -2 -1 1⟩	378.602	zg4	zogu 4th
5/4	[-2 0 1⟩	386.314	y3	yo 3rd
63/50	[-1 2 -2 1⟩	400.108	zgg4	zogugu 4th
81/64	[-6 4⟩	407.820	Lw3	lawa 3rd
80/63	[4 -2 1 -1⟩	413.578	ry3	ruyo 3rd
32/25	[5 0 -2⟩	427.373	gg4	gugu 4th
9/7	[0 2 0 -1⟩	435.084	r3	ru 3rd
35/27	[0 -3 1 1⟩	449.275	zy4	zoyo 4th
64/49	[6 0 0 -2⟩	462.348	rr3	ruru 3rd
98/75	[1 -1 -2 2⟩	463.069	zzgg5	bizogu 5th
21/16	[-4 1 0 1⟩	470.781	z4	zo 4th
4/3	[2 -1⟩	498.045	w4	wa 4th
75/56	[-3 1 2 -1⟩	505.757	ryy3	ruyoyo 3rd
27/20	[-2 3 -1⟩	519.551	g4	gu 4th
49/36	[-2 -2 0 2⟩	533.742	zz5	zozo 5th
48/35	[4 1 -1 -1⟩	546.815	rg4	rugu 4th
112/81	[4 -4 0 1⟩	561.006	z5	zo 5th
25/18	[-1 -2 2⟩	568.717	yy4	yoyo 4th
7/5	[0 0 -1 1⟩	582.512	zg5	zogu 5th
45/32	[-5 2 1⟩	590.224	y4	yo 4th
64/45	[6 -2 -1⟩	609.776	g5	gu 5th
10/7	[1 0 1 -1⟩	617.488	ry4	ruyo 4th
36/25	[2 2 -2⟩	631.283	gg5	gugu 5th
81/56	[-3 4 0 -1⟩	638.994	r4	ru 4th
35/24	[-3 -1 1 1⟩	653.185	zy5	zoyo 5th
72/49	[3 2 0 -2⟩	666.258	rr4	ruru 4th
40/27	[3 -3 1⟩	680.449	y5	yo 5th
112/75	[4 -1 -2 1⟩	694.243	zgg6	zogugu 6th
3/2	[-1 1⟩	701.955	w5	wa 5th
32/21	[5 -1 0 -1⟩	729.219	r5	ru 5th
75/49	[0 1 2 -2⟩	736.931	rryy4	biruyo 4th
49/32	[-5 0 0 2⟩	737.652	zz6	zozo 6th
54/35	[1 3 -1 -1⟩	750.725	rg5	rugu 5th
14/9	[1 -2 0 1⟩	764.916	z6	zo 6th
25/16	[-4 0 2⟩	772.627	yy5	yoyo 5th
63/40	[-3 2 -1 1⟩	786.422	zg6	zogu 6th
128/81	[7 -4⟩	792.180	sw6	sawa 6th
100/63	[2 -2 2 -1⟩	799.892	ryy5	ruyoyo 5th
8/5	[3 0 -1⟩	813.686	g6	gu 6th
45/28	[-2 2 1 -1⟩	821.398	ry5	ruyo 5th
81/50	[-1 4 -2⟩	835.193	gg6	gugu 6th
80/49	[4 0 1 -2⟩	848.662	rry5	ruruyo 5th
49/30	[-1 -1 -1 2⟩	849.383	zzg7	zozogu 7th
81/49	[0 4 0 -2⟩	870.168	rr5	ruru 5th
5/3	[0 -1 1⟩	884.359	y6	yo 6th
42/25	[1 1 -2 1⟩	898.153	zgg7	zogugu 7th
27/16	[-4 3⟩	905.865	w6	wa 6th
128/75	[7 -1 -2⟩	925.418	gg7	gugu 7th
12/7	[2 1 0 -1⟩	933.129	r6	ru 6th
140/81	[2 -4 1 1⟩	947.320	zy7	zoyo 7th
7/4	[-2 0 0 1⟩	968.826	z7	zo 7th
16/9	[4 -2⟩	996.090	w7	wa 7th
25/14	[-1 0 2 -1⟩	1003.802	ryy6	ruyoyo 6th
9/5	[0 2 -1⟩	1017.596	g7	gu 7th
49/27	[0 -3 0 2⟩	1031.787	zz8	zozo 8ve
64/35	[6 0 -1 -1⟩	1044.860	rg7	rugu 7th
90/49	[1 2 1 -2⟩	1052.572	rry6	ruruyo 6th
50/27	[1 -3 2⟩	1066.762	yy7	yoyo 7th
28/15	[2 -1 -1 1⟩	1080.557	zg8	zogu 8ve
15/8	[-3 1 1⟩	1088.269	y7	yo 7th
40/21	[3 -1 1 -1⟩	1115.533	ry7	ruyo 7th
48/25	[4 1 -2⟩	1129.328	gg8	gugu 8ve
27/14	[-1 3 0 -1⟩	1137.039	r7	ru 7th
35/18	[-1 -2 1 1⟩	1151.230	zy8	zoyo 8ve
96/49	[5 1 0 -2⟩	1164.303	rr7	ruru 7th
49/25	[0 0 -2 2⟩	1165.024	zzgg9	bizogu 9th
63/32	[-5 2 0 1⟩	1172.736	z8	zo 8ve
160/81	[5 -4 1⟩	1178.494	y8	yo 8ve
2/1	[1⟩	1200.000	w8	wa 8ve