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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 a pitch class representing that interval in every possible octave. 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
1/1
|0>
0.000
81/80
|-4 4 -1>
21.506
64/63
|6 -2 0 -1>
27.264
50/49
|1 0 2 -2>
34.976
36/35
|2 2 -1 -1>
48.770
28/27
|2 -3 0 1>
62.961
25/24
|-3 -1 2>
70.672
21/20
|-2 1 -1 1>
84.467
16/15
|4 -1 -1>
111.731
15/14
|-1 1 1 -1>
119.443
27/25
|0 3 -2>
133.238
49/45
|0 -2 -1 2>
147.428
35/32
|-5 0 1 1>
155.140
54/49
|1 3 0 -2>
168.213
28/25
|2 0 -2 1>
196.198
9/8
|-3 2>
203.910
8/7
|3 0 0 -1>
231.174
81/70
|-1 4 -1 -1>
252.68
7/6
|-1 -1 0 1>
266.871
75/64
|-6 1 2>
274.582
32/27
|5 -3>
294.135
25/21
|0 -1 2 -1>
301.847
6/5
|1 1 -1>
315.641
98/81
|1 -4 0 2>
329.832
60/49
|2 1 1 -2>
350.617
49/40
|-3 0 -1 2>
351.338
100/81
|2 -4 2>
364.807
56/45
|3 -2 -1 1>
378.602
63/50
|-1 2 -2 1>
400.108
81/64
|-6 4>
407.820
80/63
|4 -2 1 -1>
413.578
32/25
|5 0 -2>
427.373
9/7
|0 2 0 -1>
435.084
35/27
|0 -3 1 1>
449.275
64/49
|6 0 0 -2>
462.348
98/75
|1 -1 -2 2>
463.069
21/16
|-4 1 0 1>
470.781
4/3
|2 -1>
498.045
75/56
|-3 1 2 -1>
505.757
27/20
|-2 3 -1>
519.551
49/36
|-2 -2 0 2>
533.742
48/35
|4 1 -1 -1>
546.815
112/81
|4 -4 0 1>
561.006
7/5
|0 0 -1 1>
582.512
45/32
|-5 2 1>
590.224
64/45
|6 -2 -1>
609.776
10/7
|1 0 1 -1>
617.488
81/56
|-3 4 0 -1>
638.994
35/24
|-3 -1 1 1>
653.185
72/49
|3 2 0 -2>
666.258
40/27
|3 -3 1>
680.449
112/75
|4 -1 -2 1>
694.243
3/2
|-1 1>
701.955
32/21
|5 -1 0 -1>
729.219
75/49
|0 1 2 -2>
736.931
49/32
|-5 0 0 2>
737.652
54/35
|1 3 -1 -1>
750.725
14/9
|1 -2 0 1>
764.916
25/16
|-4 0 2>
772.627
63/40
|-3 2 -1 1>
786.422
128/81
|7 -4>
792.180
100/63
|2 -2 2 -1>
799.892
45/28
|-2 2 1 -1>
821.398
81/50
|-1 4 -2>
835.193
80/49
|4 0 1 -2>
848.662
49/30
|-1 -1 -1 2>
849.383
81/49
|0 4 0 -2>
870.168
5/3
|0 -1 1>
884.359
42/25
|1 1 -2 1>
898.153
27/16
|-4 3>
905.865
128/75
|7 -1 -2>
925.418
12/7
|2 1 0 -1>
933.129
140/81
|2 -4 1 1>
947.320
7/4
|-2 0 0 1>
968.826
16/9
|4 -2>
996.090
25/14
|-1 0 2 -1>
1003.802
49/27
|0 -3 0 2>
1031.787
64/35
|6 0 -1 -1>
1044.860
90/49
|1 2 1 -2>
1052.572
50/27
|1 -3 2>
1066.762
28/15
|2 -1 -1 1>
1080.557
15/8
|-3 1 1>
1088.269
40/21
|3 -1 1 -1>
1115.533
48/25
|4 1 -2>
1129.328
27/14
|-1 3 0 -1>
1137.039
35/18
|-1 -2 1 1>
1151.230
49/25
|0 0 -2 2>
1165.024
63/32
|-5 2 0 1>
1172.736
160/81
|5 -4 1>
1178.494
2/1
|1>
1200.000


Music

Ruckus From the Quiet Zone by Ralph Lewis
Excluded by Peers by Chris Vaisvil
Prelude for Centaur Tuned Piano by Chris Vaisvil
Prelude #1 in 7-limit JI by Ivor Darreg <-- are there any notations for it?
Clinton Variations play by Gene Ward Smith
Pachelbel's Canon in D in 7-limit JI play
Mars in 7-Limit JI from The Planets the orchestral suite by Gustav Holst arranged by Chris Vaisvil (Blog entry: Gustav Holst’s Mars arranged for 7-limit JI Orchestra « Music & Techniques by Chris Vaisvil)
Liszt Consolation #3 Ken Stillwell performance, retuned by Kite Giedraitis to the kite33 7-limit JI scale

see also