13-limit

Prime limit

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← 11-limit  13-limit  17-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.

The 13-limit intervals of the 2.3.13 subgroup, such as 13/12 and 16/13, are close to neutral intervals, but are further from true (hemipythagorean) neutral intervals than 2.3.11 subgroup intervals, and may thus be termed "subneutral" and "superneutral".

The 13- or 15-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 13 or 15, respectively. As prime limits increase, ratios containing different primes over 3 in the numerator and denominator become more and more numerous. Such ratios include 15/14, 14/13, 11/10, 15/13, 13/11, 14/11, 13/10, 15/11, 7/5, and their octave complements. In a 13-limit system, all the ratios of the 13- or 15-odd-limit can be treated as potential consonances. An example of a way to use these intervals is to build tertian triads such as 10:13:15, which consists of a 13/10 ultramajor third and a 15/13 inframinor third. Another example is the neogothic major and minor triads of 22:28:33 and 22:26:33, which can be tempered to the 13-odd-limit via vanishing of 364/363, but can also be used as they are.

Edo approximation

Edos which represent 13-limit intervals better (monotonicity limit ≥ 13 and decreasing TE error): 15, 17c, 19, 26, 27e, 29, 31, 41, 46, 53, 58, 72, 87, 103, 111, 121, 130, 183, 190, 198, 224, 270, 494 and so on. For a more comprehensive list, see Sequence of equal temperaments by error.

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

Tristan Bay

• Junp – in a 2.3.11/7.13/7 subgroup JI scale

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

Thomas Leroy Meier

• WIP cover of Sheik's Theme by Koji Kondo (2025; original was 1996) - tuning adapted from Ibn Sina

Claudi Meneghin

• Canon on a ground – in 2.11.13 subgroup

Hojo Minori

• P`rismatic fut`URE (2025)

Claire Rose

• fretless harp guitar study (2026)

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 – symphonic con brio using the Partch 13-odd-limit tonality diamond as a scale.

Randy Wells

• Dying Visions of a Lonesome Machine (2021)
• Avenoir (2022)
• I Was a Teenage Boltzmann Brain (2022)
• Atlas Apassionata (2022)

See also

• Gallery of just intervals
• Tridecimal neutral seventh chord
• Augmented chords in just intonation, some (they are 13-limit)