13-limit

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The 13-prime-limit refers to a constraint on selecting just intonation intervals such that the highest prime number in all ratios is 13. Thus, 40/39 would be allowable, since 40 is 2*2*2*5 and 39 is 3*13, but 34/33 would not be allowable, since 34 is 2*17, and 17 is a prime number higher than 13. An interval doesn't need to contain a 13 to be considered within the 13-limit. For instance, 3/2 is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a 13 in it is not necessarily within the 13-limit. 23/13 is not within the 13-limit, since 23 is a prime number higher than 13).

The 13-prime-limit can be modeled in a 5-dimensional lattice, with the primes 3, 5, 7, 11, and 13 represented by each dimension. The prime 2 does not appear in the typical 13-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a sixth dimension is need.

Edos good for 13-limit are 5, 6, 7, 9, 10, 15, 16, 17, 19, 20, 22, 24, 26, 31, 37, 46, 50, 53, 63, 77, 84, 87, 130, 140, 161, 183, 207, 217, 224, 270, 494, 851, 1075, 1282, 1578, 2159, 2190, 2684, 3265, 3535, 4573, 5004, 5585, 6079, 8269, 8539, 13854, 14124, 16808, 20203, 22887, 28742, 32007, 37011, 50434, 50928, 51629, 54624, 56202, 59467, 64471, 65052, ... .

Intervals

Here are all the 15-odd-limit intervals of 13:

Ratio Cents Value Interval name
14/13 128 3uz2 thuzo 2nd
13/12 139 3o2 tho 2nd
15/13 248 3uy2 thuyo 2nd
13/11 289 3o1u3 tholu 3rd
16/13 359 3u3 thu 3rd
13/10 454 3og4 thogu 4th
18/13 563 3u4 thu 4th
13/9 637 3o5 tho 5th
20/13 746 3uy5 thuyo 5th
13/8 841 3o6 tho 6th
22/13 911 3u1o6 thulo 6th
26/15 952 3og7 thogu 7th
24/13 1061 3u7 thu 7th
13/7 1072 3or7 thoru 7th

See: Gallery of Just Intervals

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Music

Venusian Cataclysms play by Dave Hill

Chord Progression on the Harmonic Overtone Series play by Dave Hill

See also

Harmonic limit