166edo: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-28 20:45:28 UTC</tt>.<br>~~

~~: The original revision id was <tt>269750516</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 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank three temperament [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[Marvel family#Hecate|hecate]], and the rank two temperament wizard, which also tempers out 4000/3993, 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 [[Marvel temperaments|gizzard]], the 72&94 temperament, for which 166 is an excellent tuning through the [[19-limit|19 limit]].

Its ~~prime factorization is 166 =~~ [[~~2edo~~|2]] * [[~~83edo~~|83]].

== Theory ==

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]].

166edo (as 83edo) contains a very good approximation of the [[~~7_4~~|harmonic 7th]]. It's 0.~~15121~~ [[~~cent~~]] ~~close~~ to the ~~just interval 7:4.~~

166edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.

=== Prime harmonics ===

=== 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~~]]~~</pre></div>~~

* [[Prisun]]

~~<h4>Original HTML content:</h4>~~

* [[Hecatehex]]

<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>166edo</title></head><body>The 166 equal temperament (in short 166-<a class="wiki_link" href="/EDO">EDO</a>) divides the <a class="wiki_link" href="/octave">octave</a> into 166 equal steps of size 7.229 <a class="wiki_link" href="/cent">cent</a>s each. Its principle interest lies in the usefulness of its approximations; it tempers out 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank three temperament <a class="wiki_link" href="/marvel">marvel</a>, in both the <a class="wiki_link" href="/11-limit">11-limit</a> and in the 13-limit extension <a class="wiki_link" href="/Marvel%20family#Hecate">hecate</a>, and the rank two temperament wizard, which also tempers out 4000/3993, giving the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for all of these. In the <a class="wiki_link" href="/13-limit">13-limit</a> it tempers out 325/324, leading to hecate, and 1573/1568, leading to marvell, and tempering out both gives <a class="wiki_link" href="/Marvel%20temperaments">gizzard</a>, the 72&amp;94 temperament, for which 166 is an excellent tuning through the <a class="wiki_link" href="/19-limit">19 limit</a>.<br />

* [[Marvel1]]

~~<br />~~

* [[Marvel2]]

~~Its prime factorization is 166 = <a class="wiki_link" href="/2edo">2</a>~~ * ~~<a class="wiki_link" href="/83edo">83</a>.<br />~~

* [[Marvel3]]

~~<br />~~

* [[Marvel11max7a]]

166edo (as 83edo) contains a very good approximation of the <a class="wiki_link" href="/7_4">harmonic 7th</a>. It's 0.15121 <a class="wiki_link" href="/cent">cent</a> close to the just interval 7:4.<br />

* [[Marvel11max7b]]

~~<br />~~

* [[Marveldene]]

~~<h2 id="toc0"><a name="x-Scales"></a> Scales </h2>~~

* [[Fifteentofourteen]]

~~<ul><li><a class="wiki_link" href="/prisun">prisun</a></li></ul></body></html></pre></div>~~

[[Category:Wizard]]

[[Category:Houborizic]]

[[Category:Marvel]]

@@ Line 1: / Line 1: @@
-<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:genewardsmith|genewardsmith]] and made on <tt>2011-10-28 20:45:28 UTC</tt>.<br>
-: The original revision id was <tt>269750516</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 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank three temperament [[marvel]], in both the [[11-limit]] and in the 13-limit extension [[Marvel family#Hecate|hecate]], and the rank two temperament wizard, which also tempers out 4000/3993, 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 [[Marvel temperaments|gizzard]], the 72&amp;94 temperament, for which 166 is an excellent tuning through the [[19-limit|19 limit]].
-Its prime factorization is 166 = [[2edo|2]] * [[83edo|83]].
+== Theory ==
+edo 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]].
-edo (as 83edo) contains a very good approximation of the [[7_4|harmonic 7th]]. It's 0.15121 [[cent]] close to the just interval 7:4.
+edo (as 83edo) contains a very good approximation of the [[7/4|harmonic 7th]], of which it is only flat by 0.15121 cent.
+=== Prime harmonics ===
+{{Harmonics in equal|166|intervals=prime}}
+=== Octave stretch ===
+edo'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]]</pre></div>
+* [[Prisun]]
-<h4>Original HTML content:</h4>
+* [[Hecatehex]]
-<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 1600000/1594323, 225/224, 385/384, 540/539, 4000/3993, 325/324 and 729/728. It is an excellent tuning for the rank three temperament &lt;a class="wiki_link" href="/marvel"&gt;marvel&lt;/a&gt;, in both the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; and in the 13-limit extension &lt;a class="wiki_link" href="/Marvel%20family#Hecate"&gt;hecate&lt;/a&gt;, 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 all of these. In the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; it tempers out 325/324, leading to hecate, and 1573/1568, leading to marvell, and  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;
+* [[Marvel1]]
-&lt;br /&gt;
+* [[Marvel2]]
-Its prime factorization is 166 = &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt; * &lt;a class="wiki_link" href="/83edo"&gt;83&lt;/a&gt;.&lt;br /&gt;
+* [[Marvel3]]
-&lt;br /&gt;
+* [[Marvel11max7a]]
-edo (as 83edo) contains a very good approximation of the &lt;a class="wiki_link" href="/7_4"&gt;harmonic 7th&lt;/a&gt;. It's 0.15121 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt; close to the just interval 7:4.&lt;br /&gt;
+* [[Marvel11max7b]]
-&lt;br /&gt;
+* [[Marveldene]]
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; Scales &lt;/h2&gt;
+* [[Fifteentofourteen]]
-&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/prisun"&gt;prisun&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+[[Category:Wizard]]
+[[Category:Houborizic]]
+[[Category:Marvel]]