41edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 239335091 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-29 11: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-29 11:59:18 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>239335565</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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<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">**<span style="color: #004d25; font-size: 20px;">41 Tone Equal Temperament</span>** | <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">**<span style="color: #004d25; font-size: 20px;">41 Tone Equal Temperament</span>** | ||
The 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29,268 [[cent]]s, an [[interval]] close in size to [[64_63|64/63]], the [[Septimal comma|septimal comma]]. 41-ET can be seen as a tuning of the [[http://en.wikipedia.org/wiki/Schismatic_temperament|Garibaldi temperament]] <ref>[http://x31eq.com/schismic.htm "Schismic Temperaments "], | The 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29,268 [[cent]]s, an [[interval]] close in size to [[64_63|64/63]], the [[Septimal comma|septimal comma]]. 41-ET can be seen as a tuning of the [[http://en.wikipedia.org/wiki/Schismatic_temperament|Garibaldi temperament]] <ref>[[http://x31eq.com/schismic.htm|"Schismic Temperaments"]], //Intonation Information//</ref> , the [[http://en.wikipedia.org/wiki/Schismatic_temperament|miracle temperament]], <ref>[[http://x31eq.com/decimal_lattice.htm|"Lattices with Decimal Notation"]], //Intonation Information//</ref> the [[http://en.wikipedia.org/wiki/Magic_temperament|magic temperament]] and the valentine (41&26) temperament. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], and is the seventh [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit]] fairly well, and the [[13-limit]] perhaps close enough for government work, though its [[13_10|13/10]] is 14 cents sharp. | ||
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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>41edo</title></head><body><strong><span style="color: #004d25; font-size: 20px;">41 Tone Equal Temperament</span></strong><br /> | <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>41edo</title></head><body><strong><span style="color: #004d25; font-size: 20px;">41 Tone Equal Temperament</span></strong><br /> | ||
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The 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29,268 <a class="wiki_link" href="/cent">cent</a>s, an <a class="wiki_link" href="/interval">interval</a> close in size to <a class="wiki_link" href="/64_63">64/63</a>, the <a class="wiki_link" href="/Septimal%20comma">septimal comma</a>. 41-ET can be seen as a tuning of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">Garibaldi temperament</a> <!-- ws:start:WikiTextRefRule: | The 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29,268 <a class="wiki_link" href="/cent">cent</a>s, an <a class="wiki_link" href="/interval">interval</a> close in size to <a class="wiki_link" href="/64_63">64/63</a>, the <a class="wiki_link" href="/Septimal%20comma">septimal comma</a>. 41-ET can be seen as a tuning of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">Garibaldi temperament</a> <!-- ws:start:WikiTextRefRule:3:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://x31eq.com/schismic.htm&quot; rel=&quot;nofollow&quot;&gt;&amp;quot;Schismic Temperaments&amp;quot;&lt;/a&gt;, &lt;em&gt;Intonation Information&lt;/em&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><!-- ws:end:WikiTextRefRule:3 --> , the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">miracle temperament</a>, <!-- ws:start:WikiTextRefRule:7:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://x31eq.com/decimal_lattice.htm&quot; rel=&quot;nofollow&quot;&gt;&amp;quot;Lattices with Decimal Notation&amp;quot;&lt;/a&gt;, &lt;em&gt;Intonation Information&lt;/em&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><!-- ws:end:WikiTextRefRule:7 --> the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Magic_temperament" rel="nofollow">magic temperament</a> and the valentine (41&amp;26) temperament. It is the second smallest equal temperament (after <a class="wiki_link" href="/29edo">29edo</a>) whose perfect fifth is closer to just intonation than that of <a class="wiki_link" href="/12edo">12-ET</a>, and is the seventh <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">zeta integral edo</a> after 31; it is not, however, a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">zeta gap edo</a>. This has to do with the fact that it can deal with the <a class="wiki_link" href="/11-limit">11-limit</a> fairly well, and the <a class="wiki_link" href="/13-limit">13-limit</a> perhaps close enough for government work, though its <a class="wiki_link" href="/13_10">13/10</a> is 14 cents sharp.<br /> | ||
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<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:18:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Instruments">Instruments</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Nonoctave Temperaments">Nonoctave Temperaments</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc0"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:8 -->Intervals</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc1"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:10 -->Instruments</h1> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1516:&lt;img src=&quot;/file/view/Ron_Sword_with_a_41ET_Guitar.jpg/221056094/Ron_Sword_with_a_41ET_Guitar.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/Ron_Sword_with_a_41ET_Guitar.jpg/221056094/Ron_Sword_with_a_41ET_Guitar.jpg" alt="Ron_Sword_with_a_41ET_Guitar.jpg" title="Ron_Sword_with_a_41ET_Guitar.jpg" /><!-- ws:end:WikiTextLocalImageRule:1516 --><br /> | ||
<em>41-EDO Classical guitar, by Ron Sword.</em><br /> | <em>41-EDO Classical guitar, by Ron Sword.</em><br /> | ||
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A possible system to tune keyboards in 41EDO is discussed in <a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/74155" rel="nofollow">http://launch.groups.yahoo.com/group/tuning/message/74155</a>.<br /> | A possible system to tune keyboards in 41EDO is discussed in <a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/74155" rel="nofollow">http://launch.groups.yahoo.com/group/tuning/message/74155</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc2"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:12 -->Harmonic Scale</h1> | ||
41edo is the first edo to do some justice to Mode 8 of the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>, which Dante Rosati calls the &quot;<a class="wiki_link" href="/overtone%20scales">Diatonic Harmonic Series Scale</a>,&quot; consisting of overtones 8 through 16 (sometimes made to repeat at the octave).<br /> | 41edo is the first edo to do some justice to Mode 8 of the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>, which Dante Rosati calls the &quot;<a class="wiki_link" href="/overtone%20scales">Diatonic Harmonic Series Scale</a>,&quot; consisting of overtones 8 through 16 (sometimes made to repeat at the octave).<br /> | ||
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The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.<br /> | The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc3"><a name="Nonoctave Temperaments"></a><!-- ws:end:WikiTextHeadingRule:14 -->Nonoctave Temperaments</h1> | ||
Taking every third degree of 41edo produces a scale extremely close to <a class="wiki_link" href="/88cET">88cET</a> or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <span class="wiki_link_new"><a class="wiki_link" href="/BP">Bohlen-Pierce</a></span><a class="wiki_link" href="/BP"> Scale</a> (or the 13th root of 3). See chart:<br /> | Taking every third degree of 41edo produces a scale extremely close to <a class="wiki_link" href="/88cET">88cET</a> or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <span class="wiki_link_new"><a class="wiki_link" href="/BP">Bohlen-Pierce</a></span><a class="wiki_link" href="/BP"> Scale</a> (or the 13th root of 3). See chart:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc4"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:16 -->Links</h1> | ||
<ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/41_equal_temperament" rel="nofollow">Wikipedia article on 41edo</a></li><li><a class="wiki_link" href="/Magic22%20as%20srutis#magic22assrutis">Magic22 as srutis</a> describes a possible use of 41edo for <a class="wiki_link" href="/indian">indian</a> music.</li><li>see also <a class="wiki_link" href="/Magic%20family">Magic family</a></li><li>Sword, Ron.<a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow" target="_blank"> &quot;Tetracontamonophonic Scales for Guitar&quot;</a></li></ul><!-- ws:start:WikiTextReferencesRule: | <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/41_equal_temperament" rel="nofollow">Wikipedia article on 41edo</a></li><li><a class="wiki_link" href="/Magic22%20as%20srutis#magic22assrutis">Magic22 as srutis</a> describes a possible use of 41edo for <a class="wiki_link" href="/indian">indian</a> music.</li><li>see also <a class="wiki_link" href="/Magic%20family">Magic family</a></li><li>Sword, Ron.<a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow" target="_blank"> &quot;Tetracontamonophonic Scales for Guitar&quot;</a></li></ul><!-- ws:start:WikiTextReferencesRule:2206: --><hr class="references" /><ol class="references"> | ||
<li id="cite_note-1"><a href="#cite_ref-1">^</a> | <li id="cite_note-1"><a href="#cite_ref-1">^</a> <a class="wiki_link_ext" href="http://x31eq.com/schismic.htm" rel="nofollow">&quot;Schismic Temperaments&quot;</a>, <em>Intonation Information</em></li> | ||
<li id="cite_note-2"><a href="#cite_ref-2">^</a> | <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a class="wiki_link_ext" href="http://x31eq.com/decimal_lattice.htm" rel="nofollow">&quot;Lattices with Decimal Notation&quot;</a>, <em>Intonation Information</em></li> | ||
</ol><!-- ws:end:WikiTextReferencesRule: | </ol><!-- ws:end:WikiTextReferencesRule:2206 --></body></html></pre></div> | ||