13/8: 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 = (lesser) tridecimal neutral sixth
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-14 20:48:09 UTC</tt>.<br>
| Color name = 3o6, tho 6th
: The original revision id was <tt>254165610</tt>.<br>
| Sound = jid_13_8_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>
<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">13/8 is a neutral sixth measuring about 840.5¢, falling between the categories of minor sixth and major sixth. In [[13-limit]] [[Just Intonation]], 13/8, as an octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of 3/2 and 7/4, separated from each by the [[superparticular]] ratios [[13_12|13/12]] and [[14_13|14/13]], respectively. It differs from the 11-limit neutral sixth [[18_11|18/11]] by 144/143, about 12.1¢.


See: [[Gallery of Just Intervals]]</pre></div>
'''13/8''' is the '''(lesser) tridecimal neutral sixth''', which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In [[13-limit]] [[just intonation]], 13/8, as an octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of [[3/2]] and [[7/4]], separated from each by the [[superparticular]] ratios [[13/12]] and [[14/13]], respectively. The word "lesser" is added when necessary to differentiate it from [[64/39]], another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of [[5/4]] and [[13/10]].
<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_8&lt;/title&gt;&lt;/head&gt;&lt;body&gt;13/8 is a neutral sixth measuring about 840.5¢, falling between the categories of minor sixth and major sixth. 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;, 13/8, as an octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of 3/2 and 7/4, separated from each by the &lt;a class="wiki_link" href="/superparticular"&gt;superparticular&lt;/a&gt; ratios &lt;a class="wiki_link" href="/13_12"&gt;13/12&lt;/a&gt; and &lt;a class="wiki_link" href="/14_13"&gt;14/13&lt;/a&gt;, respectively. It differs from the 11-limit neutral sixth &lt;a class="wiki_link" href="/18_11"&gt;18/11&lt;/a&gt; by 144/143, about 12.1¢.&lt;br /&gt;
13/8 differs from the Pythagorean minor sixth [[128/81]] by [[1053/1024]], about 48¢, from the classic minor sixth [[8/5]] by [[65/64]], about 27¢, from the undecimal neutral sixth [[18/11]] by [[144/143]], about 12¢, and from the rastmic neutral sixth [[44/27]] by [[352/351]], about 4.9¢.  
&lt;br /&gt;
 
See: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
== Approximation ==
13/8 is a fraction of a cent away from the neutral sixth found in the [[10edo|10''n''-edo]] family (7\10).
 
This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the [[golden ratio]]. In this case, 13/8 is ~7.4 [[cent|¢]] sharp of the golden ratio.
 
== See also ==
* [[16/13]] – its [[octave complement]]
* [[64/39]] – the greater tridecimal neutral sixth
* [[Gallery of just intervals]]
 
[[Category:Sixth]]
[[Category:Neutral sixth]]
[[Category:Golden ratio approximations]]

Latest revision as of 11:53, 25 October 2022

Interval information
Ratio 13/8
Subgroup monzo 2.13 [-3 1⟩
Size in cents 840.5277¢
Name (lesser) tridecimal neutral sixth
Color name 3o6, tho 6th
FJS name [math]\displaystyle{ \text{m6}^{13} }[/math]
Special properties reduced,
reduced harmonic
Tenney height (log2 nd) 6.70044
Weil height (log2 max(n, d)) 7.40088
Wilson height (sopfr(nd)) 19

[sound info]
Open this interval in xen-calc

13/8 is the (lesser) tridecimal neutral sixth, which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In 13-limit just intonation, 13/8, as an octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of 3/2 and 7/4, separated from each by the superparticular ratios 13/12 and 14/13, respectively. The word "lesser" is added when necessary to differentiate it from 64/39, another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of 5/4 and 13/10.

13/8 differs from the Pythagorean minor sixth 128/81 by 1053/1024, about 48¢, from the classic minor sixth 8/5 by 65/64, about 27¢, from the undecimal neutral sixth 18/11 by 144/143, about 12¢, and from the rastmic neutral sixth 44/27 by 352/351, about 4.9¢.

Approximation

13/8 is a fraction of a cent away from the neutral sixth found in the 10n-edo family (7\10).

This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the golden ratio. In this case, 13/8 is ~7.4 ¢ sharp of the golden ratio.

See also

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