13/10
| Interval information |
tridecimal semisixth
[sound info]
In 13-limit just intonation, 13/10, the tridecimal semisixth is an interseptimal interval measuring about 454.2¢. 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. 13/10 appears between the 10th and 13th overtones of the harmonic series and appears in such chords as 8:10:13, a quasi-augmented triad. 13/10 also appears in the relatively-simple 10:13:15 triad, which consists of an interseptimal ultramajor third (13/10) and an interseptimal inframinor third (15/13) which stack to make a 3/2 perfect fifth. It is well-approximated in 16edo, 21edo, 24edo, 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 doesn't 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 |