7-limit
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[edit]
Warning: No 49/48. Recalculate and expand! And don't blindly add this interval because this warning will just change to no 10/9!
7-limit-81-odd-limit odd numbers: 1 3 5 7 9 15 21 25 27 35 45 49 63 75 81
{{#rreplace:
1/1 3/2 5/4 7/4 9/8 15/8 21/16 25/16 27/16 35/32 45/32 49/32 63/32 75/64 81/64 4/3 5/3 7/6 25/24 35/24 49/48 8/5 6/5 7/5 9/5 21/20 27/20 49/40 63/40 81/80 8/7 12/7 10/7 9/7 15/14 25/14 27/14 45/28 75/56 81/56 16/9 10/9 14/9 25/18 35/18 49/36 16/15 28/15 49/30 32/21 40/21 25/21 32/25 48/25 28/25 36/25 42/25 27/25 49/25 63/50 81/50 32/27 40/27 28/27 50/27 35/27 49/27 64/35 48/35 36/35 54/35 81/70 64/45 56/45 49/45 64/49 96/49 80/49 72/49 60/49 50/49 54/49 90/49 75/49 81/49 64/63 80/63 100/63 128/75 112/75 98/75 128/81 160/81 112/81 100/81 140/81 98/81
|/(\d+)(\D+)?\/(\d+)(\D+)?/|\1/\3\2\4}}
| 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[edit]
- 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