13/10: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = Barbados third, tridecimal semisixth
| Ratio = 13/10
| Monzo = -1 0 -1 0 0 1
| Cents = 454.21395
| Name = Barbados third, <br>tridecimal 9/4 tone, <br>tridecimal semidiminished fourth, <br>tridecimal ultramajor third
| Color name = 3og4, thogu 4th
| Color name = 3og4, thogu 4th
| FJS name = d4<sup>13</sup><sub>5</sub>
| Sound = jid_13_10_pluck_adu_dr220.mp3
| Sound = jid_13_10_pluck_adu_dr220.mp3
}}
}}


In [[13-limit]] [[Just Intonation]], '''13/10''' 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.
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 ==
Line 19: Line 52:
* [[The Archipelago]]
* [[The Archipelago]]


[[Category:13-limit]]
[[Category:Interseptimal intervals]]
[[Category:Interval]]
[[Category:Naiadic]]
[[Category:Just interval]]
[[Category:Ratio]]
[[Category:Fourth]]
[[Category:Fourth]]
[[Category:Subfourth]]
[[Category:Subfourth]]
[[Category:Third]]
[[Category:Third]]
[[Category:Supermajor third]]
[[Category:Supermajor third]]
[[Category:Interseptimal]]
[[Category:Over-5 intervals]]
[[Category:Naiadic]]
[[Category:Over-5]]

Latest revision as of 11:37, 1 August 2025

Interval information
Ratio 13/10
Factorization 2-1 × 5-1 × 13
Monzo [-1 0 -1 0 0 1⟩
Size in cents 454.2139¢
Names Barbados third,
tridecimal semisixth
Color name 3og4, thogu 4th
FJS name [math]\displaystyle{ \text{d4}^{13}_{5} }[/math]
Special properties reduced
Tenney height (log2 nd) 7.02237
Weil height (log2 max(n, d)) 7.40088
Wilson height (sopfr(nd)) 20

[sound info]
Open this interval in xen-calc

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.

Interval chain generated by 13/10
# 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

See also

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