13/10: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox Interval
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Name = Barbados third, tridecimal semisixth
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-10-07 19:22:18 UTC</tt>.<br>
| Color name = 3og4, thogu 4th
: The original revision id was <tt>262737198</tt>.<br>
| Sound = jid_13_10_pluck_adu_dr220.mp3
: The revision comment was: <tt></tt><br>
}}
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
In [[13-limit]] [[just intonation]], '''13/10''', the '''tridecimal semisixth''' is an [[interseptimal]] interval measuring about 454.2 [[cent]]s. 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.  
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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|9/7]] and a flat perfect fourth such as [[21_16|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 [[OverToneSeries|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|15/13]]) which stack to make a [[3_2|3/2]] perfect fifth. It is well-approximated in [[24edo]], [[29edo]], [[37edo]], and of course, infinitely many other [[EDO]] systems.


See: [[Gallery of Just Intervals]], [[List of root-3rd-P5 triads in JI]]</pre></div>
In many notation systems based on the [[5L 2s|diatonic]] [[chain-of-fifths notation]] with commatic alterations (e.g. [[FJS]], [[HEJI]]), 13/10 is a fourth, as it is a [[4/3|perfect fourth (4/3)]] minus an instance of [[40/39]], which is a [[2187/2048|Pythagorean apotome]] minus a stack consisting of an [[81/80|syntonic comma (81/80)]] and a [[1053/1024|tridecimal quartertone (1053/1024)]], none of which changes the [[scale|scale degree]]. It functions as such in the harmonic thirteenth chord, [[4:5:6:7:9:11:13]].
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;13_10&lt;/title&gt;&lt;/head&gt;&lt;body&gt;In &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt;, 13/10 is an &lt;a class="wiki_link" href="/interseptimal"&gt;interseptimal&lt;/a&gt; interval measuring about 454.2¢. It falls in an ambiguous zone between a wide major third such as &lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt; and a flat perfect fourth such as &lt;a class="wiki_link" href="/21_16"&gt;21/16&lt;/a&gt;. The descriptor &amp;quot;interseptimal&amp;quot; comes from &lt;a class="wiki_link" href="/Margo%20Schulter"&gt;Margo Schulter&lt;/a&gt;, and indicates its position between those two septimal (7-based) extremes. 13/10 appears between the 10th and 13th overtones of the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt; 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 (&lt;a class="wiki_link" href="/15_13"&gt;15/13&lt;/a&gt;) which stack to make a &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt; perfect fifth. It is well-approximated in &lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt;, &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37edo&lt;/a&gt;, and of course, infinitely many other &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt; systems.&lt;br /&gt;
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 [[16edo]], [[21edo]], [[24edo]], [[29edo]], [[37edo]], and of course, infinitely many other [[edo]] systems.
&lt;br /&gt;
 
See: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;, &lt;a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI"&gt;List of root-3rd-P5 triads in JI&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
== 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> 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.
 
{| 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]]
|}
 
== Approximation ==
{{Interval edo approximation|13/10}}
 
== See also ==
* [[20/13]] – its [[octave complement]]
* [[15/13]] – its [[fifth complement]]
* [[Gallery of just intervals]]
* [[List of root-3rd-P5 triads in JI]]
* [[The Archipelago]]
 
[[Category:Interseptimal intervals]]
[[Category:Naiadic]]
[[Category:Fourth]]
[[Category:Subfourth]]
[[Category:Third]]
[[Category:Supermajor third]]
[[Category:Over-5 intervals]]

Latest revision as of 16:29, 22 January 2026

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 norm (log2 nd) 7.02237
Weil norm (log2 max(n, d)) 7.40088
Wilson norm (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 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.

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 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 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.

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

Approximation

Edo approximations for 13/10 (454.21 ¢)
≤ 80edo, relative error ≤ 10%
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

See also

Retrieved from "https://en.xen.wiki/index.php?title=13/10&oldid=222201"