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18edf: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on <tt>2011-12-26 16:22:16 UTC</tt>.<br>
: The original revision id was <tt>288516678</tt>.<br>
: 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">=18edf=


0: 1/1 0.000 unison, perfect prime
== Theory ==
1: 38.998 cents 38.998
18edf corresponds to [[31edo]] with an [[octave stretching]] of about 9 [[cent]]s. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor [[7/6]] (+6{{cent}}), minor [[6/5]] (-3{{cent}}), neutral [[11/9]] (+4{{cent}}), major [[5/4]] (+4{{cent}}), and supermajor [[9/7]] (-6{{cent}}). These intervals may be used to form a variety of [[triad]]s and [[tetrad]]s in close harmony along with the tuning's pure fifth.
2: 77.995 cents 77.995
3: 116.993 cents 116.993
4: 155.990 cents 155.990
5: 194.988 cents 194.988
6: 233.985 cents 233.985
7: 272.983 cents 272.983
8: 311.980 cents 311.980
9: 350.978 cents 350.978
10: 389.975 cents 389.975
11: 428.973 cents 428.973
12: 467.970 cents 467.970
13: 506.968 cents 506.968
14: 545.965 cents 545.965
15: 584.963 cents 584.963
16: 623.960 cents 623.960
17: 662.958 cents 662.958
18: 3/2 701.955 perfect fifth


Lookalikes: [[31edo]]</pre></div>
In comparison, [[20edf]] (and [[Carlos Gamma]]) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds.
<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;18edf&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="x18edf"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;18edf&lt;/h1&gt;
=== Regular temperaments ===
&lt;br /&gt;
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].
0: 1/1 0.000 unison, perfect prime&lt;br /&gt;
 
1: 38.998 cents 38.998&lt;br /&gt;
=== Harmonics ===
2: 77.995 cents 77.995&lt;br /&gt;
{{Harmonics in equal|18|3|2|intervals=integer|columns=11}}
3: 116.993 cents 116.993&lt;br /&gt;
{{Harmonics in equal|18|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edf (continued)}}
4: 155.990 cents 155.990&lt;br /&gt;
 
5: 194.988 cents 194.988&lt;br /&gt;
=== Subsets and supersets ===
6: 233.985 cents 233.985&lt;br /&gt;
Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18edf has subset edfs {{EDs|equave=f| 2, 3, 6, and 9 }}.
7: 272.983 cents 272.983&lt;br /&gt;
 
8: 311.980 cents 311.980&lt;br /&gt;
== Intervals ==
9: 350.978 cents 350.978&lt;br /&gt;
{| class="wikitable center-1 right-2 mw-collapsible"
10: 389.975 cents 389.975&lt;br /&gt;
|-
11: 428.973 cents 428.973&lt;br /&gt;
! #
12: 467.970 cents 467.970&lt;br /&gt;
! Cents
13: 506.968 cents 506.968&lt;br /&gt;
! Approximate ratios
14: 545.965 cents 545.965&lt;br /&gt;
|-
15: 584.963 cents 584.963&lt;br /&gt;
| 0
16: 623.960 cents 623.960&lt;br /&gt;
| 0.0
17: 662.958 cents 662.958&lt;br /&gt;
| [[1/1]]
18: 3/2 701.955 perfect fifth&lt;br /&gt;
|-
&lt;br /&gt;
| 1
Lookalikes: &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
| 39.0
| [[33/32]], [[36/35]], [[49/48]], [[50/49]], [[64/63]]
|-
| 2
| 78.0
| [[21/20]], [[22/21]], [[25/24]], [[28/27]]
|-
| 3
| 117.0
| [[15/14]], [[16/15]]
|-
| 4
| 156.0
| [[11/10]], [[12/11]]
|-
| 5
| 195.0
| [[9/8]], [[10/9]]
|-
| 6
| 234.0
| [[8/7]]
|-
| 7
| 273.0
| [[7/6]]
|-
| 8
| 312.0
| [[6/5]]
|-
| 9
| 351.0
| [[11/9]], [[16/13]]
|-
| 10
| 390.0
| [[5/4]]
|-
| 11
| 429.0
| [[9/7]], [[14/11]]
|-
| 12
| 468.0
| [[13/10]], [[21/16]]
|-
| 13
| 507.0
| [[4/3]]
|-
| 14
| 546.0
| [[11/8]], [[15/11]]
|-
| 15
| 585.0
| [[7/5]]
|-
| 16
| 624.0
| [[10/7]]
|-
| 17
| 663.0
| [[16/11]], [[22/15]]
|-
| 18
| 702.0
| [[3/2]]
|-
| 19
| 741.0
| [[20/13]], [[32/21]]
|-
| 20
| 780.0
| [[11/7]], [[14/9]]
|-
| 21
| 818.9
| [[8/5]]
|-
| 22
| 857.9
| [[18/11]]
|-
| 23
| 896.9
| [[5/3]]
|-
| 24
| 935.9
| [[12/7]]
|-
| 25
| 974.9
| [[7/4]]
|-
| 26
| 1013.9
| [[9/5]]
|-
| 27
| 1052.9
| [[11/6]]
|-
| 28
| 1091.9
| [[15/8]]
|-
| 29
| 1130.9
| [[27/14]]
|-
| 30
| 1169.9
| [[35/18]], [[49/25]], [[63/32]]
|-
| 31
| 1208.9
| [[2/1]]
|-
| 32
| 1247.9
| [[33/16]], [[45/22]], [[49/24]], [[55/27]]
|-
| 33
| 1286.9
| [[21/10]], [[25/12]]
|-
| 34
| 1325.9
| [[15/7]]
|-
| 35
| 1364.9
| [[11/5]]
|-
| 36
| 1403.9
| [[9/4]]
|}
 
== Related regular temperaments ==
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.
 
=== 7-limit 31 &amp; 369 ===
Commas: 2401/2400, 8589934592/8544921875
 
POTE generator: ~5/4 = 386.997
 
Mapping: [{{map| 1 19 2 7 }}, {{map| 0 -54 1 -13 }}]
 
EDOs: {{EDOs|31, 369, 400, 431, 462}}
 
=== 11-limit 31 &amp; 369 ===
Commas: 2401/2400, 5632/5625, 46656/46585
 
POTE generator: ~5/4 = 386.999
 
Mapping: [{{map| 1 19 2 7 37 }}, {{map| 0 -54 1 -13 -104 }}]
 
EDOs: 31, 369, 400, 431, 462
 
=== 13-limit 31 &amp; 369 ===
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585
 
POTE generator: ~5/4 = 387.003
 
Mapping: [{{map| 1 19 2 7 37 -35 }}, {{map| 0 -54 1 -13 -104 120 }}]
 
EDOs: 31, 369, 400, 431, 462
 
{{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}}
 
== See also ==
* [[31edo]] – relative edo
* [[49edt]] – relative edt
* [[72ed5]] – relative ed5
* [[80ed6]] – relative ed6
* [[87ed7]] – relative ed7
* [[107ed11]] – relative ed11
* [[111ed12]] – relative ed12
* [[138ed22]] – relative ed22
* [[204ed96]] – close to the zeta-optimized tuning for 31edo
* [[39cET]]
 
[[Category:31edo]]
Retrieved from "https://en.xen.wiki/w/18edf"