13-limit

The 13-limit or 13-prime-limit consists of 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.

A list of edos which represent 13-limit intervals better: 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 and so on. Another list of edos which tunes the 13-limit well relative to their size (relative error < 5.5%): 31, 41, 46, 53, 58, 72, 87, 94, 103, 111, 121, 130, 140, 152f, 159, 183, 190, 198, 212, 217, 224, 270, 282, 296, 301, 311, 320, 328, 342f, 354, 364, 369f, 373, 383, 400, 414, 422, 431, 441, 460, 472, 494 and so on.

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

Music

• Venusian Cataclysms play by Dave Hill ^{[dead link]} (404 error as of 2/5/2020)
• Chord Progression on the Harmonic Overtone Series play by Dave Hill ^{[dead link]} (404 error as of 2/5/2020)

See also

• Harmonic limit
• 13-odd-limit
• Gallery of just intervals