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.
Edo approximations
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.
- 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
- 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)