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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 = Pythagorean minor third |
| : This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-08 15:12:56 UTC</tt>.<br>
| | | Color name = w3, wa 3rd |
| : The original revision id was <tt>513254024</tt>.<br>
| | | Sound = jid_32_27_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>
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| <h4>Original Wikitext content:</h4>
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| <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">**32/27**
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| |5 -3> | |
| 294.135 cents
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| [[media type="file" key="jid_32_27_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_32_27_pluck_adu_dr220.mp3|sound sample]]
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| The Pythagorean minor third of 32/27 is the interval between [[9_8|9/8]] and [[4_3|4/3]] which arises naturally in 3-limit just intonation. It is 352/351 sharp of [[13_11|13/11]], and tempering 352/351 out equates it with 13/11 and leads to [[minthmic chords]].</pre></div> | | The '''Pythagorean minor third''' of '''32/27''' is the interval between [[9/8]] and [[4/3]] which arises naturally in [[3-limit]] [[just intonation]]. Compared to the more typical [[6/5]]- with which it is conflated in [[meantone]]- this interval is more dissonant, with a [[harmonic entropy]] level roughly on par with that of 9/8. |
| <h4>Original HTML content:</h4>
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| <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"><html><head><title>32_27</title></head><body><strong>32/27</strong><br />
| | It is 352/351 sharp of [[13/11]], and tempering 352/351 out equates it with 13/11 and leads to [[minthmic chords]]. |
| |5 -3&gt;<br /> | | |
| 294.135 cents<br />
| | == Temperaments == |
| <!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_32_27_pluck_adu_dr220.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;jid_32_27_pluck_adu_dr220.mp3&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><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_32_27_pluck_adu_dr220.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:0 --> <a href="http://xenharmonic.wikispaces.com/file/view/jid_32_27_pluck_adu_dr220.mp3/513250248/jid_32_27_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_32_27_pluck_adu_dr220.mp3/513250248/jid_32_27_pluck_adu_dr220.mp3');">sound sample</a><br />
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| <br />
| | 32/27 is treated as a comma in edos 3 & 6, where the best approximation of a perfect 5th is the 800 cent interval that wraps around to the octave again after only three iterations, producing [[alteraugment]]. Temperaments it can be interpreted as if used as a generator include [[Kleismic_family#Kleiboh|Kleiboh]] or [[Gariberttet]]. |
| The Pythagorean minor third of 32/27 is the interval between <a class="wiki_link" href="/9_8">9/8</a> and <a class="wiki_link" href="/4_3">4/3</a> which arises naturally in 3-limit just intonation. It is 352/351 sharp of <a class="wiki_link" href="/13_11">13/11</a>, and tempering 352/351 out equates it with 13/11 and leads to <a class="wiki_link" href="/minthmic%20chords">minthmic chords</a>.</body></html></pre></div>
| | == Approximation == |
| | {{Interval edo approximation|32/27}} |
| | == See also == |
| | * [[27/16]] – its [[octave complement]] |
| | * [[81/64]] – its [[fifth complement]] |
| | * [[9/8]] – its [[fourth complement]] |
| | * [[Gallery of just intervals]] |
| | * [[Pythagorean tuning]] |
| | |
| | [[Category:Third]] |
| | [[Category:Minor third]] |
The Pythagorean minor third of 32/27 is the interval between 9/8 and 4/3 which arises naturally in 3-limit just intonation. Compared to the more typical 6/5- with which it is conflated in meantone- this interval is more dissonant, with a harmonic entropy level roughly on par with that of 9/8.
It is 352/351 sharp of 13/11, and tempering 352/351 out equates it with 13/11 and leads to minthmic chords.
Temperaments
32/27 is treated as a comma in edos 3 & 6, where the best approximation of a perfect 5th is the 800 cent interval that wraps around to the octave again after only three iterations, producing alteraugment. Temperaments it can be interpreted as if used as a generator include Kleiboh or Gariberttet.
Approximation
Edo approximations for 32/27 (294.13 ¢)
≤ 80edo, relative error ≤ 10%
| Edo |
Step size |
Cents (¢) |
Absolute error (¢) |
Relative error (%)
|
| 4 |
1\4 |
300.00 |
+5.87 |
+1.96
|
| 8 |
2\8 |
300.00 |
+5.87 |
+3.91
|
| 12 |
3\12 |
300.00 |
+5.87 |
+5.87
|
| 16 |
4\16 |
300.00 |
+5.87 |
+7.82
|
| 20 |
5\20 |
300.00 |
+5.87 |
+9.78
|
| 33 |
8\33 |
290.91 |
-3.23 |
-8.87
|
| 37 |
9\37 |
291.89 |
-2.24 |
-6.92
|
| 41 |
10\41 |
292.68 |
-1.45 |
-4.96
|
| 45 |
11\45 |
293.33 |
-0.80 |
-3.01
|
| 49 |
12\49 |
293.88 |
-0.26 |
-1.05
|
| 53 |
13\53 |
294.34 |
+0.20 |
+0.90
|
| 57 |
14\57 |
294.74 |
+0.60 |
+2.86
|
| 61 |
15\61 |
295.08 |
+0.95 |
+4.81
|
| 65 |
16\65 |
295.38 |
+1.25 |
+6.77
|
| 69 |
17\69 |
295.65 |
+1.52 |
+8.72
|
See also