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
[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). It 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.
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 |
Approximation
| 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 |