15/8: 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 = just major seventh, classic(al) major seventh, ptolemaic major seventh
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-08 15:45:34 UTC</tt>.<br>
| Color name = y7, yo 7th
: The original revision id was <tt>513256132</tt>.<br>
| Sound = jid_15_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>
{{Wikipedia|Major seventh}}
<h4>Original Wikitext content:</h4>
In [[5-limit]] [[just intonation]], '''15/8''' is the '''just major seventh''', '''classic(al) major seventh''', or '''ptolemaic major seventh'''<ref>For reference, see [[5-limit]]. </ref> of about 1088.3¢. It is also the [[octave-reduced]] 15th [[harmonic]], and appears as a complex consonance in chords such as [[8:10:12:15]], a just version of a major seventh chord. Since 15/8 = [[3/2]] × [[5/4]], it can be seen as a perfect fifth above a major third or vice versa, and this understanding works in [[12edo]], as the sum of [[~]]3/2 and ~5/4 is 700{{c}} + 400{{c}} = 1100{{c}}, which 15/8 is mapped to.
<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">**15/8**
|-3 1 1&gt;
1088.2687 cents
[[media type="file" key="jid_15_8_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_15_8_pluck_adu_dr220.mp3|sound sample]]


In [[5-limit]] [[Just Intonation]], 15/8 is a major seventh of about 1088.3¢. It is also the 15th overtone (octave-reduced), and appears as a complex consonance in chords such as 8:10:12:15, a just version of a major seventh chord. Since 15 is 3*5, it can be seen as a perfect fifth above a major third or vice versa, and this understanding is compatible with the 1100¢ interval of [[12edo]].
Since 15 is a perfect fifth above 10 (15/10 = [[3/2]]), seventh chords can be formed with the 10th harmonic as major third and 15th harmonic as major seventh. The simplest and most familiar example is the classical major seventh chord 8:10:12:15 with steps 5/4, 6/5 and 5/4. Another example replaces the 12 with 13, which leads to [[8:10:13:15]] with steps 5/4, 13/10 and 15/13, and contains the [[10:13:15]] barbados triad. A particularly uncommon but mentionable example is the [[23-limit]] seventh chord [[16:20:23:30]].
== Approximation ==
{{Interval edo approximation|15/8}}


//&lt;range type="comment" id="513214416_1"&gt;Since 15 is a perfect fifth above 10 (15/10 = [[3_2|3/2]]), [[List of root-3rd-P5 triads in JI|root-3rd-P5 triads]] can be formed with the 10th harmonic as root and 15th harmonic as perfect fifth. The simplest and most familiar example is the classic minor triad 10:12:15 -- a [[6_5|6/5]] with a [[5_4|5/4]] stacked on top of it. Another is the Barbados triad, 10:13:15 -- a [[13_10|13/10]] on bottom and a [[15_13|15/13]] on top. And a particularly uncommon but mentionable example is the [[23-limit]] inframinor triad 20:23:30.&lt;/range id="513214416_1"&gt;//
== See also ==
* [[16/15]] – its [[octave complement]]
* [[8/5]] – its [[twelfth complement]]
* [[Ed15/8]]
* [[Gallery of just intervals]]


See: [[Gallery of Just Intervals]]</pre></div>
== Notes ==
<h4>Original HTML content:</h4>
<references/>
<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;15_8&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;15/8&lt;/strong&gt;&lt;br /&gt;
 
|-3 1 1&amp;gt;&lt;br /&gt;
[[Category:Seventh]]
1088.2687 cents&lt;br /&gt;
[[Category:Major seventh]]
&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_15_8_pluck_adu_dr220.mp3?h=20&amp;amp;w=240&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;jid_15_8_pluck_adu_dr220.mp3&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot; height=&amp;amp;quot;20&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;20&amp;quot; width=&amp;quot;240&amp;quot;/&amp;gt; --&gt;&lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_15_8_pluck_adu_dr220.mp3?file_extension=mp3&amp;autostart=false&amp;repeat=false&amp;showdigits=true&amp;showfsbutton=false&amp;width=240&amp;height=20"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt; &lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_15_8_pluck_adu_dr220.mp3/513214372/jid_15_8_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_15_8_pluck_adu_dr220.mp3/513214372/jid_15_8_pluck_adu_dr220.mp3');"&gt;sound sample&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
In &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt;, 15/8 is a major seventh of about 1088.3¢. It is also the 15th overtone (octave-reduced), and appears as a complex consonance in chords such as 8:10:12:15, a just version of a major seventh chord. Since 15 is 3*5, it can be seen as a perfect fifth above a major third or vice versa, and this understanding is compatible with the 1100¢ interval of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;&lt;a name="comment-513214416_1-open" class="range"&gt;&lt;/a&gt;Since 15 is a perfect fifth above 10 (15/10 = &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;), &lt;a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI"&gt;root-3rd-P5 triads&lt;/a&gt; can be formed with the 10th harmonic as root and 15th harmonic as perfect fifth. The simplest and most familiar example is the classic minor triad 10:12:15 -- a &lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt; with a &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt; stacked on top of it. Another is the Barbados triad, 10:13:15 -- a &lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt; on bottom and a &lt;a class="wiki_link" href="/15_13"&gt;15/13&lt;/a&gt; on top. And a particularly uncommon but mentionable example is the &lt;a class="wiki_link" href="/23-limit"&gt;23-limit&lt;/a&gt; inframinor triad 20:23:30.&lt;a name="comment-513214416_1-close" class="range"&gt;&lt;/a&gt;&lt;/em&gt; &lt;br /&gt;
&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>

Latest revision as of 21:53, 22 December 2025

Interval information
Ratio 15/8
Factorization 2-3 × 3 × 5
Monzo [-3 1 1⟩
Size in cents 1088.269¢
Names just major seventh,
classic(al) major seventh,
ptolemaic major seventh
Color name y7, yo 7th
FJS name [math]\displaystyle{ \text{M7}^{5} }[/math]
Special properties reduced,
reduced harmonic
Tenney norm (log2 nd) 6.90689
Weil norm (log2 max(n, d)) 7.81378
Wilson norm (sopfr(nd)) 14

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

In 5-limit just intonation, 15/8 is the just major seventh, classic(al) major seventh, or ptolemaic major seventh[1] of about 1088.3¢. It is also the octave-reduced 15th harmonic, and appears as a complex consonance in chords such as 8:10:12:15, a just version of a major seventh chord. Since 15/8 = 3/2 × 5/4, it can be seen as a perfect fifth above a major third or vice versa, and this understanding works in 12edo, as the sum of ~3/2 and ~5/4 is 700 ¢ + 400 ¢ = 1100 ¢, which 15/8 is mapped to.

Since 15 is a perfect fifth above 10 (15/10 = 3/2), seventh chords can be formed with the 10th harmonic as major third and 15th harmonic as major seventh. The simplest and most familiar example is the classical major seventh chord 8:10:12:15 with steps 5/4, 6/5 and 5/4. Another example replaces the 12 with 13, which leads to 8:10:13:15 with steps 5/4, 13/10 and 15/13, and contains the 10:13:15 barbados triad. A particularly uncommon but mentionable example is the 23-limit seventh chord 16:20:23:30.

Approximation

Edo approximations for 15/8 (1088.27 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
10 9\10 1080.00 -8.27 -6.89
11 10\11 1090.91 +2.64 +2.42
21 19\21 1085.71 -2.55 -4.47
22 20\22 1090.91 +2.64 +4.84
32 29\32 1087.50 -0.77 -2.05
33 30\33 1090.91 +2.64 +7.26
42 38\42 1085.71 -2.55 -8.94
43 39\43 1088.37 +0.10 +0.37
44 40\44 1090.91 +2.64 +9.68
53 48\53 1086.79 -1.48 -6.52
54 49\54 1088.89 +0.62 +2.79
64 58\64 1087.50 -0.77 -4.10
65 59\65 1089.23 +0.96 +5.21
75 68\75 1088.00 -0.27 -1.68
76 69\76 1089.47 +1.20 +7.63

See also

Notes

  1. ↑ For reference, see 5-limit.
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