13-limit

Prime limit

The 13-limit or 13-prime-limit consists of just intonation intervals such that the highest prime factor in all ratios is 13. Thus, 40/39 would be within the 13-limit, since 40 is 2 × 2 × 2 × 5 and 39 is 3 × 13, but 34/33 would not, since 34 is 2 × 17, and 17 is a prime number higher than 13. The 13-limit is the 6th prime limit and is a superset of the 11-limit and a subset of the 17-limit.

The 13-limit is a rank-6 system, and 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.

These things are contained by the 13-limit, but not the 11-limit:

The 13- and 15-odd-limit;
Mode 7 and 8 of the harmonic or subharmonic series.

Edo approximation

Edos which represent 13-limit intervals better (with decreasing TE error): 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 and so on.

Here is a list of edos which tunes the 13-limit well relative to their size (TE 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.

Note: Wart notation is used to specify the val chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

Intervals

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

Ratio	Cents value	Color name		Name
14/13	128.298	3uz2	thuzo 2nd	tridecimal supraminor second
13/12	138.573	3o2	tho 2nd	tridecimal subneutral second
15/13	247.741	3uy2	thuyo 2nd	tridecimal semifourth
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 naiadic
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 cocytic
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 semitwelfth
24/13	1061.427	3u7	thu 7th	tridecimal supraneutral seventh
13/7	1071.702	3or7	thoru 7th	tridecimal submajor seventh

Music

E8 Heterotic

Justification (2022)

Francium

Bicycle Wheels (2023)

Dave Hill

Venusian Cataclysms⁠ ⁠^[dead link] play⁠ ⁠^[dead link]
Chord Progression on the Harmonic Overtone Series⁠ ⁠^[dead link] play⁠ ⁠^[dead link]

Ben Johnston

String Quartet No. 5 (1979) – Bandcamp | YouTube – performed by Kepler Quartet
String Quartet No. 7 (1984)
- "Movt. 1" – Bandcamp | YouTube
- "Movt. 2" – Bandcamp | YouTube
- "Movt. 3" – Bandcamp | YouTube

performed by Kepler Quartet

Kaiveran Lugheidh

Unlicensed Copy (2017) – mostly 7-limit with some erstwhile 13-based chromaticisms

Claudi Meneghin

Canon on a ground – in 2.11.13 subgroup

Gene Ward Smith

Threnody for the Victims of Wolfgang Amadeus Mozart (archived 2010) – 13-limit JI in 6079edo tuning
Rough Diamond (archived 2010) a.k.a. Diamond in the Rough^[1] – symphonic con brio using the Partch 13-odd-limit tonality diamond as a scale.