# 13-limit

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 needed.

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 Color name Interval name
14/13 128.298 3uz2 thuzo 2nd tridecimal large semitone
tridecimal large limma
13/12 138.573 3o2 tho 2nd tridecimal subneutral second
15/13 247.741 3uy2 thuyo 2nd tridecimal second-third
13/11 289.210 3o1u3 tholu 3rd tridecimal minor third
16/13 359.472 3u3 thu 3rd tridecimal supraneutral third
13/10 454.214 3og4 thogu 4th tridecimal third-fourth
18/13 563.382 3u4 thu 4th tridecimal sub-tritone
13/9 636.618 3o5 tho 5th tridecimal super-tritone
20/13 745.786 3uy5 thuyo 5th tridecimal fifth-sixth
13/8 840.528 3o6 tho 6th tridecimal subneutral sixth
22/13 910.790 3u1o6 thulo 6th tridecimal major sixth
26/15 952.259 3og7 thogu 7th tridecimal sixth-seventh
24/13 1061.427 3u7 thu 7th tridecimal supraneutral seventh
13/7 1071.702 3or7 thoru 7th tridecimal submajor seventh