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166edo: Difference between revisions

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**Imported revision 215739964 - Original comment: **
→Theory: note its flat tuning tendency
 
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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:xenwolf|xenwolf]] and made on <tt>2011-03-31 03:40:48 UTC</tt>.<br>
: The original revision id was <tt>215739964</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">The 166 equal temperament (in short 166-[[EDO]]) divides the [[octave]] into 166 equal steps of size 7.229 [[cent]]s each. Its principle interest lies in the usefulness of its approximations; it tempers out 225/224, 385/384, 540/539 and 4000/3993. It is an excellent tuning for the rank three temperament [[marvel]], especially the [[11-limit]] version tempering out both 225/224 and 385/384, and the rank two temperament wizard, which also tempers out 4000/3993, giving the [[optimal patent val]] for both of these. In the 13-limit it also tempers out 325/324, leading to 13-limit marvel, and 1573/1568, leading to marvell. Tempering out both gives [[Marvel temperaments|gizzard]], the 72&amp;94 temperament, for which 166 is an excellent tuning through the [[19-limit|19 limit]].


166 = 2 * 83.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
166edo is [[consistent]] through the [[13-odd-limit]]. It has a flat tendency, with [[harmonic]]s 3 to 13 all tuned flat. Its principal interest lies in the usefulness of its approximations. In addition to the 5-limit [[amity comma]], it [[tempering out|tempers out]] [[225/224]], [[325/324]], [[385/384]], [[540/539]], and [[729/728]], hence being an excellent tuning for the [[rank-3 temperament]] [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[hecate]], the [[rank-2 temperament]] [[wizard]], which also tempers out [[4000/3993]], and [[houborizic]], which also tempers out [[2200/2197]], giving the [[optimal patent val]] for all of these. In the [[13-limit]] it tempers out 325/324, leading to hecate, and [[1573/1568]], leading to marvell, and tempering out both gives [[gizzard]], the {{nowrap|72 & 94}} temperament, for which 166 is an excellent tuning through the [[19-limit]].
<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;166edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 166 equal temperament (in short 166-&lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;) divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 166 equal steps of size 7.229 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. Its principle interest lies in the usefulness of its approximations; it tempers out 225/224, 385/384, 540/539 and 4000/3993. It is an excellent tuning for the rank three temperament &lt;a class="wiki_link" href="/marvel"&gt;marvel&lt;/a&gt;, especially the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; version tempering out both 225/224 and 385/384, and the rank two temperament wizard, which also tempers out 4000/3993, giving the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for both of these. In the 13-limit it also tempers out 325/324, leading to 13-limit marvel, and 1573/1568, leading to marvell. Tempering out both gives &lt;a class="wiki_link" href="/Marvel%20temperaments"&gt;gizzard&lt;/a&gt;, the 72&amp;amp;94 temperament, for which 166 is an excellent tuning through the &lt;a class="wiki_link" href="/19-limit"&gt;19 limit&lt;/a&gt;.&lt;br /&gt;
 
&lt;br /&gt;
166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.
166 = 2 * 83.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Prime harmonics ===
{{Harmonics in equal|166|intervals=prime}}
 
=== Octave stretch ===
166edo's approximated harmonics 3, 5, 7, 11, and 13 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[263edt]] or [[429ed6]].
 
=== Subsets and supersets ===
Since 166 factors into primes as 2 × 83, 166edo contains [[2edo]] and [[83edo]] as subsets.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -263 166 }}
| {{mapping| 166 263 }}
| +0.237
| 0.237
| 3.27
|-
| 2.3.5
| 1600000/1594323, {{monzo| -31 2 12 }}
| {{mapping| 166 263 385 }}
| +0.615
| 0.568
| 7.86
|-
| 2.3.5.7
| 225/224, 118098/117649, 1250000/1240029
| {{mapping| 166 263 385 466 }}
| +0.474
| 0.549
| 7.59
|-
| 2.3.5.7.11
| 225/224, 385/384, 4000/3993, 322102/321489
| {{mapping| 166 263 385 466 574 }}
| +0.490
| 0.492
| 6.80
|-
| 2.3.5.7.11.13
| 225/224, 325/324, 385/384, 1573/1568, 2200/2197
| {{mapping| 166 263 385 466 574 614 }}
| +0.498
| 0.449
| 6.21
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 33\166
| 238.55
| 147/128
| [[Tokko]]
|-
| 1
| 47\166
| 339.76
| 243/200
| [[Houborizic]]
|-
| 1
| 81\166
| 585.54
| 7/5
| [[Merman]] (7-limit)
|-
| 2
| 30\166
| 216.87
| 17/15
| [[Wizard]] / gizzard
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Scales ==
* [[Prisun]]
* [[Hecatehex]]
* [[Marvel1]]
* [[Marvel2]]
* [[Marvel3]]
* [[Marvel11max7a]]
* [[Marvel11max7b]]
* [[Marveldene]]
* [[Fifteentofourteen]]
 
[[Category:Wizard]]
[[Category:Houborizic]]
[[Category:Marvel]]
Retrieved from "https://en.xen.wiki/w/166edo"