13/10: Difference between revisions
No edit summary |
mNo edit summary |
||
| (10 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox Interval | {{Infobox Interval | ||
| Name = Barbados third, tridecimal semisixth | |||
| Name = Barbados third, | |||
| Color name = 3og4, thogu 4th | | Color name = 3og4, thogu 4th | ||
| Sound = jid_13_10_pluck_adu_dr220.mp3 | | Sound = jid_13_10_pluck_adu_dr220.mp3 | ||
}} | }} | ||
In [[13-limit]] [[ | 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)<sup>4</sup> approximates 20/7 (compound [[10/7]]) remarkably well, with less than 1{{cent}} error. | |||
Additionally, while it may seem as though (13/10)<sup>2</sup> 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. | |||
{| class="wikitable" | |||
|+ [[Interval chain]] generated by 13/10 | |||
! # | |||
! [[Cent]]s | |||
! Approximated [[ratio]]s | |||
! Associated [[comma]]s | |||
|- | |||
| 1 | |||
| 454.2 | |||
| 13/10<br>[[17/13]] (+10.2{{cent}}) | |||
| <br>[[170/169]] (major naiadma) | |||
|- | |||
| 2 | |||
| 908.4 | |||
| [[27/16]] (-2.6{{cent}})<br>[[22/13]] (+2.4{{cent}})<br>[[17/10]] (+10.2{{cent}}) | |||
| [[676/675]] (island comma)<br>[[2200/2197]] (petrma)<br>[[170/169]] (major naiadma) | |||
|- | |||
| 3 | |||
| 1362.6 | |||
| [[11/5]] (+2.4{{cent}}) | |||
| [[2200/2197]] (petrma) | |||
|- | |||
| 4 | |||
| 1816.9 | |||
| [[20/7]] (+0.6{{cent}}) | |||
| [[200000/199927]] | |||
|- | |||
| 5 | |||
| 2271.1 | |||
| [[26/7]] (+0.6{{cent}}) | |||
| [[200000/199927]] | |||
|} | |||
== See also == | == See also == | ||
* [[20/13]] – its [[octave complement]] | |||
* [[15/13]] – its [[fifth complement]] | |||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[List of root-3rd-P5 triads in JI]] | * [[List of root-3rd-P5 triads in JI]] | ||
* [[ | * [[The Archipelago]] | ||
[[Category: | [[Category:Interseptimal intervals]] | ||
[[Category:Naiadic]] | |||
[[Category:Fourth]] | [[Category:Fourth]] | ||
[[Category: | [[Category:Subfourth]] | ||
[[Category:Third]] | [[Category:Third]] | ||
[[Category: | [[Category:Supermajor third]] | ||
[[Category:Over-5 intervals]] | |||
[[Category:Over-5]] | |||
Latest revision as of 11:37, 1 August 2025
| 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 |