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 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