13/10
In 13-limit just intonation, 13/10, the tridecimal semisixth is an interseptimal interval measuring about 454.2 cents. It falls in an ambiguous zone between a wide major third such as 9/7 and a flat perfect fourth such as 21/16. The descriptor "interseptimal" comes from Margo Schulter, and indicates its position between those two septimal (7-based) extremes.
| Interval information |
tridecimal semisixth,
tridecimal third-fourth
[sound info]
In many notation systems based on the diatonic chain-of-fifths notation with commatic alterations (e.g. FJS, HEJI), 13/10 is a fourth, as it is a perfect fourth (4/3) minus an instance of 40/39, which is a Pythagorean apotome minus a stack consisting of an syntonic comma (81/80) and a tridecimal quartertone (1053/1024), none of which changes the scale degree. It functions as such in the harmonic thirteenth chord, 4:5:6:7:9:11:13.
However, 13/10 also appears in the relatively simple 10:13:15 triad, which consists of 13/10 and 15/13 that stack to make a 3/2 perfect fifth. This makes 13/10 function as an ultramajor third (if the chord is not taken as a suspension).
Approximation
13/10 is well-approximated in 8edo (as well as all multiples of it such as 16edo, 24edo and 32edo), 21edo, 29edo, 37edo, and of course, infinitely many other edo systems.
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 8 | 3\8 | 450.00 | -4.21 | -2.81 |
| 13 | 5\13 | 461.54 | +7.32 | +7.93 |
| 16 | 6\16 | 450.00 | -4.21 | -5.62 |
| 21 | 8\21 | 457.14 | +2.93 | +5.13 |
| 24 | 9\24 | 450.00 | -4.21 | -8.43 |
| 29 | 11\29 | 455.17 | +0.96 | +2.32 |
| 37 | 14\37 | 454.05 | -0.16 | -0.49 |
| 45 | 17\45 | 453.33 | -0.88 | -3.30 |
| 50 | 19\50 | 456.00 | +1.79 | +7.44 |
| 53 | 20\53 | 452.83 | -1.38 | -6.11 |
| 58 | 22\58 | 455.17 | +0.96 | +4.63 |
| 61 | 23\61 | 452.46 | -1.75 | -8.92 |
| 66 | 25\66 | 454.55 | +0.33 | +1.82 |
| 74 | 28\74 | 454.05 | -0.16 | -0.99 |
| 79 | 30\79 | 455.70 | +1.48 | +9.76 |
Interval chain
Because 13/10 is an interseptimal interval, stacking it four times will result in a good approximation of a septimal interval. In this case, (13/10)4 approximates 20/7 (compound 10/7) remarkably well, with less than 1 ¢ error.
Additionally, while it may seem as though (13/10)2 does not approximate 17/10 very well at first glance, it allows for an elegant interpretation of the tetrad formed by stacking 13/10 three times on top of itself: ~10:13:17:22.
| # | Cents | Approximated ratios | Associated commas |
|---|---|---|---|
| 1 | 454.2 | 13/10 17/13 (+10.2 ¢) |
170/169 (major naiadma) |
| 2 | 908.4 | 27/16 (-2.6 ¢) 22/13 (+2.4 ¢) 17/10 (+10.2 ¢) |
676/675 (island comma) 2200/2197 (petrma) 170/169 (major naiadma) |
| 3 | 1362.6 | 11/5 (+2.4 ¢) | 2200/2197 (petrma) |
| 4 | 1816.9 | 20/7 (+0.6 ¢) | 200000/199927 |
| 5 | 2271.1 | 26/7 (+0.6 ¢) | 200000/199927 |