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:

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