7-limit

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The 7-limit or "7 prime-limit" refers to a constraint on rational intervals such that 7 is the highest allowable prime number, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the octave include 9/7, 14/9, 15/14, 28/15, 21/16, 32/21, 25/14, 28/25, 25/21, 42/25, 28/27, 27/14, 35/27, 54/35, 45/28, 56/45, 49/32, 64/49, 49/36, 72/49, 49/30, 60/49, 49/25, 50/49, 49/27, 54/49, 49/35, 70/49, 49/45, 90/49.

"7 odd-limit" refers to 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-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.

Relative to their size, the equal divisions 1edo, 2edo, 3edo, 4edo, 5edo, 7edo, 9edo, 10edo, 12edo, 15edo, 19edo, 21edo, 22edo, 31edo, 53edo, 84edo, 87edo, 94edo, 99edo, 118edo, 130edo, 140edo, 171edo, 270edo, 410edo, 441edo and 612edo provide good approximations to the 7-limit.

List of Intervals in the 7-Prime Limit and 81-Odd Limit

Ratio Monzo Cents Value 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 double ruyo 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> 252.68 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 | 0 > 386.314 y3 yo 3rd
63/50 |-1 2 -2 1> 400.108 zgg4 zogugu 4th
81/64 |-6 4> 407.820 Lw3 large wa 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 double zogu 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 double ruyo 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 small wa 6th
100/63 |2 -2 2 -1> 799.892 ryy5 ruyoyo 5th
8/5 | 0 > 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 double zogu 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

Music

See also