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 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
- Venusian Cataclysms[dead link] play[dead link] (404 error as of 2/5/2020)
- Chord Progression on the Harmonic Overtone Series[dead link] play[dead link] (404 error as of 2/5/2020)