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27edt: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = 27edt
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-02-24 06:39:09 UTC</tt>.<br>
| de = 27-EDT
: The original revision id was <tt>204608830</tt>.<br>
}}
: The revision comment was: <tt></tt><br>
{{Infobox ET}}
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
{{ED intro}}
<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">=[[#Division of the tritave (3/1) into 12 equal parts]]Division of the tritave (3/1) into 27 equal parts=


Dividing the interval of 3/1 into 27 equal parts gives a scale with a basic step of 70.44 cents, which is nearly identical to one step of [[17edo]] (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a prime number.
== Theory ==
27edt corresponds to 17.035…edo, which is nearly identical to one step of [[17edo]] (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a [[prime number]]. In fact, the [[prime edo]]s that approximate the 3-limit well often correspond to composite edts: e.g. [[19edo]] → [[30edt]], [[29edo]] → [[46edt]] and [[31edo]] → [[49edt]].


27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for Klingon music (since the tradtional Klingon number system is also based on 3). See, e.g., [[http://launch.dir.groups.yahoo.com/group/tuning/message/86909]] and [[http://www.klingon.org/smboard/index.php?topic=1810.0]].
Compared to 17edo, 27edt approximates the [[prime interval|primes]] [[7/1|7]], [[11/1|11]], and [[13/1|13]] better; it approximates prime [[5/1|5]] equally poorly, but maps it to 40 steps rather than 39 in the [[patent val]], corresponding to the 17c [[val]], often considered the better mapping as it equates [[5/4]] and [[6/5]] to major and minor thirds rather than to a neutral third, and 5 has the same sharp tendency as 7 and 11.  


==Intervals==
From a purely tritave-based perspective, it [[support]]s the [[minalzidar]] temperament, but otherwise it can be used as a retuning of 17edo with closer-to-just harmonic properties in the no-fives 2.3.7.11.13 subgroup.
||~ degrees of 27edt ||~ cents value ||~ approximation in 17edo ||
|| 0 || 0.00 || 0.00 ||
|| 1 || 70.44 || 70.59 ||
|| 2 || 140.89 || 141.18 ||
|| 3 || 211.33 || 211.76 ||
|| 4 || 281.77 || 282.35 ||
|| 5 || 352.21 || 352.94 ||
|| 6 || 422.66 || 423.53 ||
|| 7 || 493.10 || 494.12 ||
|| 8 || 563.54 || 564.71 ||
|| 9 || 633.99 || 635.29 ||
|| 10 || 704.43 || 705.88 ||
|| 11 || 774.87 || 776.47 ||
|| 12 || 845.31 || 847.06 ||
|| 13 || 915.76 || 917.65 ||
|| 14 || 986.20 || 988.24 ||
|| 15 || 1056.64 || 1058.82 ||
|| 16 || 1127.08 || 1129.41 ||
|| 17 || 1197.53 || 1200.00 ||
|| 18 || 1267.97 || 1270.59 ||
|| 19 || 1338.41 || 1341.18 ||
|| 20 || 1408.86 || 1411.76 ||
|| 21 || 1479.30 || 1482.35 ||
|| 22 || 1549.74 || 1551.94 ||
|| 23 || 1620.18 || 1623.53 ||
|| 24 || 1690.63 || 1694.12 ||
|| 25 || 1761.07 || 1764.71 ||
|| 26 || 1831.51 || 1835.29 ||
|| 27 || 1901.96 || 1905.88 ||</pre></div>
<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;27edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Division of the tritave (3/1) into 27 equal parts"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;!-- ws:start:WikiTextAnchorRule:4:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@Division of the tritave (3/1) into 12 equal parts&amp;quot; title=&amp;quot;Anchor: Division of the tritave (3/1) into 12 equal parts&amp;quot;/&amp;gt; --&gt;&lt;a name="Division of the tritave (3/1) into 12 equal parts"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:4 --&gt;Division of the tritave (3/1) into 27 equal parts&lt;/h1&gt;
&lt;br /&gt;
Dividing the interval of 3/1 into 27 equal parts gives a scale with a basic step of 70.44 cents, which is nearly identical to one step of &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt; (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a prime number.&lt;br /&gt;
&lt;br /&gt;
27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for Klingon music (since the tradtional Klingon number system is also based on 3). See, e.g., &lt;a class="wiki_link_ext" href="http://launch.dir.groups.yahoo.com/group/tuning/message/86909" rel="nofollow"&gt;http://launch.dir.groups.yahoo.com/group/tuning/message/86909&lt;/a&gt; and &lt;a class="wiki_link_ext" href="http://www.klingon.org/smboard/index.php?topic=1810.0" rel="nofollow"&gt;http://www.klingon.org/smboard/index.php?topic=1810.0&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Division of the tritave (3/1) into 27 equal parts-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals&lt;/h2&gt;


&lt;table class="wiki_table"&gt;
=== Harmonics ===
    &lt;tr&gt;
{{Harmonics in equal|27|3|1|intervals=integer|columns=11}}
        &lt;th&gt;degrees of 27edt&lt;br /&gt;
{{Harmonics in equal|27|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 27edt (continued)}}
&lt;/th&gt;
        &lt;th&gt;cents value&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;approximation in 17edo&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.00&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.00&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;70.44&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;70.59&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;140.89&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;141.18&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;211.33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;211.76&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;281.77&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;282.35&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;352.21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;352.94&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;422.66&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;423.53&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;493.10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;494.12&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;563.54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;564.71&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;633.99&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;635.29&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;704.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;705.88&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;774.87&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;776.47&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;845.31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;847.06&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;915.76&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;917.65&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;986.20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;988.24&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1056.64&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1058.82&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1127.08&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1129.41&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1197.53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200.00&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1267.97&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1270.59&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1338.41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1341.18&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1408.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1411.76&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1479.30&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1482.35&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1549.74&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1551.94&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;23&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1620.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1623.53&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1690.63&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1694.12&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1761.07&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1764.71&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;26&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1831.51&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1835.29&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1901.96&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1905.88&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 27 factors into primes as 3<sup>3</sup>, 27edt contains [[3edt]] and [[9edt]] as subset edts.
 
=== Miscellany ===
27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for {{w|Klingon}} music since the tradtional Klingon number system is also based on 3. The rather harsh harmonic character of 27edt would suit very well, too<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_86909.html Yahoo! Tuning Group | ''the evil 27 equal temp scale from outer space'']</ref><ref>[https://web.archive.org/web/20100624113458/https://www.klingon.org/smboard/index.php?topic=1810.0 Klingon Imperial Forums | ''klingon music theory'']</ref>.
 
This being said, such a proposal is rather short-sighted from a general cultural perspective, since any kind of living creature would most likely gravitate towards some form of [[low-complexity JI]], and while 27edt will gain appreciation in base-3 cultures at some point, it may not be the first temperament they discover. That would be like aliens assuming dominant tuning in human music is [[100ed10]] (or 1000ed10 or variation thereof) just because we count in base 10.
 
== Intervals ==
{{Interval table}}
 
== See also ==
* [[10edf]] – relative edf
* [[17edo]] – relative edo
* [[44ed6]] – relative ed6
 
== Notes ==
Retrieved from "https://en.xen.wiki/w/27edt"