41edo: Difference between revisions

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<span style="color: #004d25; font-family: 'Times New Roman',Times,serif; font-size: 20px;">**41 Tone Equal Temperament**</span>
[[toc|flat]]
=Introduction= 
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 //[[Schismatic family#Garibaldi|Garibaldi temperament]]// <ref>[[http://x31eq.com/schismic.htm|"Schismic Temperaments"]] at x31eq.com the websize of [[Graham Breed]]</ref> , <ref>[[http://x31eq.com/decimal_lattice.htm|"Lattices with Decimal Notation"]] at x31eq.com</ref> , <ref>[[http://en.wikipedia.org/wiki/Schismatic_temperament|Schismatic temperament]]</ref> the //[[Magic family|Magic temperament]]// <ref>[[http://en.wikipedia.org/wiki/Magic_temperament|Magic temperament]]</ref> and the superkleismic (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%20EDO%20lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|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. Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.

41-ET forms the foundation of the [[http://www.h-pi.com/theory/huntsystem1.html|H-System]], which uses the scale degrees of 41-ET as the basic [[13-limit]] intervals requiring fine tuning +/- 1 [[http://www.h-pi.com/theory/huntsystem2.html|average JND]] from the 41-ET circle in [[205edo]].

41edo is the 13th [[prime numbers|prime]] edo, following [[37edo]] and coming before [[43edo]].

=Temperaments=
[[List of edo-distinct 41et rank two temperaments]]

=Intervals= 
||~ degrees of 41edo ||~ cents value ||~ Approximate Ratios in the [[11-limit]] ||~ Andrew's solfege syllable ||~ generator for ||~ some MOS and MODMOS Scales available ||
|| 0 || 0.00 || [[1_1|1/1]] || do ||   ||   ||
|| 1 || 29.27 || [[81_80|81/80]] || di ||   ||   ||
|| 2 || 58.54 || [[25_24|25/24]], [[28_27|28/27]], [[33_32|33/32]] || ro ||   ||   ||
|| 3 || 87.80 || [[21_20|21/20]], [[22_21|22/21]] || rih || 88cET (approx) / [[octacot]] ||   ||
|| 4 || 117.07 || [[16_15|16/15]], [[15_14|15/14]] || ra || [[Miracle]] ||   ||
|| 5 || 146.34 || [[12_11|12/11]] || ru || [[Bohlen-Pierce]]/[[bohpier]] ||   ||
|| 6 || 175.61 || [[10_9|10/9]], [[11_10|11/10]] || reh || [[Tetracot]]/[[bunya]]/[[monkey]] || 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5 ||
|| 7 || 204.88 || [[9_8|9/8]] || re || [[Baldy]] || 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1 ||
|| 8 || 234.15 || [[8_7|8/7]] || ri || [[Rodan]]/[[guiron]] || 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1 ||
|| 9 || 263.41 || [[7_6|7/6]], [[32_25|32/25]] || ma || [[Septimin]] || 9-tone MOS: 5 4 5 5 4 5 4 5 4 ||
|| 10 || 292.68 || [[32_27|32/27]] || meh ||   ||   ||
|| 11 || 321.95 || [[6_5|6/5]] || me || [[Superkleismic]] || 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3 ||
|| 12 || 351.22 || [[11_9|11/9]],[[27_22|27/22]] || mu || [[Hemififths]]/[[karadeniz]] || 10-tone MOS: 5 2 5 5 2 5 5 5 2 5 ||
|| 13 || 380.49 || [[5_4|5/4]] || mi || [[Magic]]/[[witchcraft]] || 10-tone MOS: 2 9 2 2 9 2 2 9 2 2 ||
|| 14 || 409.76 || [[14_11|14/11]], [[81_64|81/64]] || maa || [[Hocus]] ||   ||
|| 15 || 439.02 || [[9_7|9/7]] || mo ||   || 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4 ||
|| 16 || 468.29 || [[21_16|21/16]] || fe ||   ||   ||
|| 17 || 497.56 || [[4_3|4/3]] || fa || [[Schismatic]] ([[helmholtz]], [[Garibaldi temperament|garibaldi]], [[cassandra]]) ||   ||
|| 18 || 526.83 || [[15_11|15/11]], [[27_20|27/20]] || fih ||   || 9-tone MOS: 5 5 3 5 5 5 5 3 5 ||
|| 19 || 556.10 || [[11_8|11/8]] || fu ||   ||   ||
|| 20 || 585.37 || [[7_5|7/5]] || fi || [[Pluto]] ||   ||
|| 21 || 614.63 || [[10_7|10/7]] || se ||   ||   ||
|| 22 || 643.90 || [[16_11|16/11]] || su ||   ||   ||
|| 23 || 673.17 || [[22_15|22/15]], [[40_27|40/27]] || sih ||   ||   ||
|| 24 || 702.44 || [[3_2|3/2]] || sol ||   ||   ||
|| 25 || 731.71 || [[32_21|32/21]] || si ||   ||   ||
|| 26 || 760.98 || [[14_9|14/9]], [[25_16|25/16]] || lo ||   ||   ||
|| 27 || 790.24 || [[11_7|11/7]], [[128_81|128/81]] || leh ||   ||   ||
|| 28 || 819.51 || [[8_5|8/5]] || le ||   ||   ||
|| 29 || 848.78 || [[18_11|18/11]], [[44_27|44/27]] || lu ||   ||   ||
|| 30 || 878.05 || [[5_3|5/3]] || la ||   ||   ||
|| 31 || 907.32 || [[27_16|27/16]] || laa ||   ||   ||
|| 32 || 936.59 || [[12_7|12/7]] || li ||   ||   ||
|| 33 || 965.85 || [[7_4|7/4]] || ta ||   ||   ||
|| 34 || 995.12 || [[16_9|16/9]] || teh ||   ||   ||
|| 35 || 1024.39 || [[9_5|9/5]], [[20_11|20/11]] || te ||   ||   ||
|| 36 || 1053.66 || [[11_6|11/6]] || tu ||   ||   ||
|| 37 || 1082.93 || [[15_8|15/8]] || ti ||   ||   ||
|| 38 || 1112.20 || [[40_21|40/21]], [[21_11|21/11]] || taa ||   ||   ||
|| 39 || 1141.46 || [[48_25|48/25]], [[27_14|27/14]], [[64_33|64/33]] || to ||   ||   ||
|| 40 || 1170.73 || [[160_81|160/81]] || da ||   ||   ||

=Instruments= 
[[image:Ron_Sword_with_a_41ET_Guitar.jpg]]
//41-EDO Classical guitar, by Ron Sword.//

A possible system to tune keyboards in 41EDO is discussed in [[http://launch.groups.yahoo.com/group/tuning/message/74155]].

=Harmonic Scale= 
41edo is the first edo to do some justice to Mode 8 of the [[OverToneSeries|harmonic series]], which Dante Rosati calls the "[[overtone scales|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).

|| Overtones in "Mode 8": || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 ||
|| ...as JI Ratio from 1/1: || 1/1 || 9/8 || 5/4 || 11/8 || 3/2 || 13/8 || 7/4 || 15/8 || 2/1 ||
|| ...in cents: || 0 || 203.9 || 386.3 || 551.3 || 702.0 || 840.5 || 968.8 || 1088.3 || 1200.0 ||
|| Nearest degree of 41edo: || 0 || 7 || 13 || 19 || 24 || 29 || 33 || 37 || 41 ||
|| ...in cents: || 0 || 204.9 || 380.5 || 556.1 || 702.4 || 848.8 || 965.9 || 1082.9 || 1200.0 ||

While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)

7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).

The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.

=Nonoctave Temperaments= 
Taking every third degree of 41edo produces a scale extremely close to [[88cET]] 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">[[BP|Bohlen-Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See chart:

||||||= 3 degrees of 41edo (near 88cET) ||= overlap ||||||= 5 degrees of 41edo (near BP) ||
||~ deg of 41edo ||~ deg of 88cET ||~ cents ||~ cents ||~ cents ||~ deg of BP ||~ deg of 41edo ||
||= 0 ||= 0 ||=   ||= 0 ||=   ||= 0 ||= 0 ||
||= 3 ||= 1 ||= 87.8 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 146.3 ||= 1 ||= 5 ||
||= 6 ||= 2 ||= 175.6 ||=   ||=   ||=   ||=   ||
||= 9 ||= 3 ||= 263.4 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 292.7 ||= 2 ||= 10 ||
||= 12 ||= 4 ||= 351.2 ||=   ||=   ||=   ||=   ||
||= 15 ||= 5 ||=   ||= 439.0 ||=   ||= 3 ||= 15 ||
||= 18 ||= 6 ||= 526.8 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 585.4 ||= 4 ||= 20 ||
||= 21 ||= 7 ||= 614.6 ||=   ||=   ||=   ||=   ||
||= 24 ||= 8 ||= 702.4 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 731.7 ||= 5 ||= 25 ||
||= 27 ||= 9 ||= 790.2 ||=   ||=   ||=   ||=   ||
||= 30 ||= 10 ||=   ||= 878.0 ||=   ||= 6 ||= 30 ||
||= 33 ||= 11 ||= 965.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1024.4 ||= 7 ||= 35 ||
||= 36 ||= 12 ||= 1053.7 ||=   ||=   ||=   ||=   ||
||= 39 ||= 13 ||= 1141.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1170.7 ||= 8 ||= 40 ||
||||||||||||||~ [ second octave ] ||
||= 1 ||= 14 ||= 29.2 ||=   ||=   ||=   ||=   ||
||= 4 ||= 15 ||=   ||= 117.1 ||=   ||= 9 ||= 4 ||
||= 7 ||= 16 ||= 204.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 263.4 ||= 10 ||= 9 ||
||= 10 ||= 17 ||= 292.7 ||=   ||=   ||=   ||=   ||
||= 13 ||= 18 ||= 380.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 409.8 ||= 11 ||= 14 ||
||= 16 ||= 19 ||= 468.3 ||=   ||=   ||=   ||=   ||
||= 19 ||= 20 ||=   ||= 556.1 ||=   ||= 12 ||= 19 ||
||= 22 ||= 21 ||= 643.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 702.4 ||= 13 ||= 24 ||
||= 25 ||= 22 ||= 731.7 ||=   ||=   ||=   ||=   ||
||= 28 ||= 23 ||= 819.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 848.8 ||= 14 ||= 29 ||
||= 31 ||= 24 ||= 907.3 ||=   ||=   ||=   ||=   ||
||= 34 ||= 25 ||=   ||= 995.1 ||=   ||= 15 ||= 34 ||
||= 37 ||= 26 ||= 1082.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1141.5 ||= 16 ||= 39 ||
||= 40 ||= 27 ||= 1170.7 ||=   ||=   ||=   ||=   ||
||||||||||||||~ [ third octave ] ||
||= 2 ||= 28 ||= 58.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 87.8 ||= 17 ||= 3 ||
||= 5 ||= 29 ||= 146.3 ||=   ||=   ||=   ||=   ||
||= 8 ||= 30 ||=   ||= 234.1 ||=   ||= 18 ||= 8 ||
||= 11 ||= 31 ||= 322.0 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 380.5 ||= 19 ||= 13 ||
||= 14 ||= 32 ||= 409.8 ||=   ||=   ||=   ||=   ||
||= 17 ||= 33 ||= 497.6 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 526.8 ||= 20 ||= 18 ||
||= 20 ||= 34 ||= 585.3 ||=   ||=   ||=   ||=   ||
||= 23 ||= 35 ||=   ||= 673.2 ||=   ||= 21 ||= 23 ||
||= 26 ||= 36 ||= 761.0 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 819.5 ||= 22 ||= 28 ||
||= 29 ||= 37 ||= 848.8 ||=   ||=   ||=   ||=   ||
||= 32 ||= 38 ||= 936.6 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 965.9 ||= 23 ||= 33 ||
||= 35 ||= 39 ||= 1024.4 ||=   ||=   ||=   ||=   ||
||= 38 ||= 40 ||=   ||= 1112.2 ||=   ||= 24 ||= 38 ||


=Music= 
[[http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro|EveningHorizon]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3|play]] by Cameron Bobro

=Links= 
* [[http://en.wikipedia.org/wiki/41_equal_temperament|Wikipedia article on 41edo]]
* [[Magic22 as srutis#magic22assrutis]] describes a possible use of 41edo for [[indian]] music.
* see also [[Magic family]]
* Sword, Ron.[[@http://www.ronsword.com| "Tetracontamonophonic Scales for Guitar"]]

Original HTML content:

<html><head><title>41edo</title></head><body><span style="color: #004d25; font-family: 'Times New Roman',Times,serif; font-size: 20px;"><strong>41 Tone Equal Temperament</strong></span><br />
<!-- ws:start:WikiTextTocRule:25:&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:25 --><!-- ws:start:WikiTextTocRule:26: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Temperaments">Temperaments</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Instruments">Instruments</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Nonoctave Temperaments">Nonoctave Temperaments</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: -->
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:9 -->Introduction</h1>
 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 <em><a class="wiki_link" href="/Schismatic%20family#Garibaldi">Garibaldi temperament</a></em> <!-- ws:start:WikiTextRefRule:2:&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; at x31eq.com the websize of &lt;a class=&quot;wiki_link&quot; href=&quot;/Graham%20Breed&quot;&gt;Graham Breed&lt;/a&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><!-- ws:end:WikiTextRefRule:2 --> , <!-- ws:start:WikiTextRefRule:4:&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; at x31eq.com&amp;lt;/ref&amp;gt; --><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><!-- ws:end:WikiTextRefRule:4 --> , <!-- ws:start:WikiTextRefRule:6:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://en.wikipedia.org/wiki/Schismatic_temperament&quot; rel=&quot;nofollow&quot;&gt;Schismatic temperament&lt;/a&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup><!-- ws:end:WikiTextRefRule:6 --> the <em><a class="wiki_link" href="/Magic%20family">Magic temperament</a></em> <!-- ws:start:WikiTextRefRule:8:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://en.wikipedia.org/wiki/Magic_temperament&quot; rel=&quot;nofollow&quot;&gt;Magic temperament&lt;/a&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup><!-- ws:end:WikiTextRefRule:8 --> and the superkleismic (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%20EDO%20lists">zeta integral edo</a> after 31; it is not, however, a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">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. Various 13-limit <a class="wiki_link" href="/magic%20extensions">magic extensions</a> are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.<br />
<br />
41-ET forms the foundation of the <a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem1.html" rel="nofollow">H-System</a>, which uses the scale degrees of 41-ET as the basic <a class="wiki_link" href="/13-limit">13-limit</a> intervals requiring fine tuning +/- 1 <a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem2.html" rel="nofollow">average JND</a> from the 41-ET circle in <a class="wiki_link" href="/205edo">205edo</a>.<br />
<br />
41edo is the 13th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/37edo">37edo</a> and coming before <a class="wiki_link" href="/43edo">43edo</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc1"><a name="Temperaments"></a><!-- ws:end:WikiTextHeadingRule:11 -->Temperaments</h1>
<a class="wiki_link" href="/List%20of%20edo-distinct%2041et%20rank%20two%20temperaments">List of edo-distinct 41et rank two temperaments</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:13 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <th>degrees of 41edo<br />
</th>
        <th>cents value<br />
</th>
        <th>Approximate Ratios in the <a class="wiki_link" href="/11-limit">11-limit</a><br />
</th>
        <th>Andrew's solfege syllable<br />
</th>
        <th>generator for<br />
</th>
        <th>some MOS and MODMOS Scales available<br />
</th>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0.00<br />
</td>
        <td><a class="wiki_link" href="/1_1">1/1</a><br />
</td>
        <td>do<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>29.27<br />
</td>
        <td><a class="wiki_link" href="/81_80">81/80</a><br />
</td>
        <td>di<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>58.54<br />
</td>
        <td><a class="wiki_link" href="/25_24">25/24</a>, <a class="wiki_link" href="/28_27">28/27</a>, <a class="wiki_link" href="/33_32">33/32</a><br />
</td>
        <td>ro<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>87.80<br />
</td>
        <td><a class="wiki_link" href="/21_20">21/20</a>, <a class="wiki_link" href="/22_21">22/21</a><br />
</td>
        <td>rih<br />
</td>
        <td>88cET (approx) / <a class="wiki_link" href="/octacot">octacot</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>117.07<br />
</td>
        <td><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_14">15/14</a><br />
</td>
        <td>ra<br />
</td>
        <td><a class="wiki_link" href="/Miracle">Miracle</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>146.34<br />
</td>
        <td><a class="wiki_link" href="/12_11">12/11</a><br />
</td>
        <td>ru<br />
</td>
        <td><a class="wiki_link" href="/Bohlen-Pierce">Bohlen-Pierce</a>/<a class="wiki_link" href="/bohpier">bohpier</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>175.61<br />
</td>
        <td><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/11_10">11/10</a><br />
</td>
        <td>reh<br />
</td>
        <td><a class="wiki_link" href="/Tetracot">Tetracot</a>/<a class="wiki_link" href="/bunya">bunya</a>/<a class="wiki_link" href="/monkey">monkey</a><br />
</td>
        <td>13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>204.88<br />
</td>
        <td><a class="wiki_link" href="/9_8">9/8</a><br />
</td>
        <td>re<br />
</td>
        <td><a class="wiki_link" href="/Baldy">Baldy</a><br />
</td>
        <td>11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>234.15<br />
</td>
        <td><a class="wiki_link" href="/8_7">8/7</a><br />
</td>
        <td>ri<br />
</td>
        <td><a class="wiki_link" href="/Rodan">Rodan</a>/<a class="wiki_link" href="/guiron">guiron</a><br />
</td>
        <td>11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>263.41<br />
</td>
        <td><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/32_25">32/25</a><br />
</td>
        <td>ma<br />
</td>
        <td><a class="wiki_link" href="/Septimin">Septimin</a><br />
</td>
        <td>9-tone MOS: 5 4 5 5 4 5 4 5 4<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>292.68<br />
</td>
        <td><a class="wiki_link" href="/32_27">32/27</a><br />
</td>
        <td>meh<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>321.95<br />
</td>
        <td><a class="wiki_link" href="/6_5">6/5</a><br />
</td>
        <td>me<br />
</td>
        <td><a class="wiki_link" href="/Superkleismic">Superkleismic</a><br />
</td>
        <td>11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>351.22<br />
</td>
        <td><a class="wiki_link" href="/11_9">11/9</a>,<a class="wiki_link" href="/27_22">27/22</a><br />
</td>
        <td>mu<br />
</td>
        <td><a class="wiki_link" href="/Hemififths">Hemififths</a>/<a class="wiki_link" href="/karadeniz">karadeniz</a><br />
</td>
        <td>10-tone MOS: 5 2 5 5 2 5 5 5 2 5<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>380.49<br />
</td>
        <td><a class="wiki_link" href="/5_4">5/4</a><br />
</td>
        <td>mi<br />
</td>
        <td><a class="wiki_link" href="/Magic">Magic</a>/<a class="wiki_link" href="/witchcraft">witchcraft</a><br />
</td>
        <td>10-tone MOS: 2 9 2 2 9 2 2 9 2 2<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>409.76<br />
</td>
        <td><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/81_64">81/64</a><br />
</td>
        <td>maa<br />
</td>
        <td><a class="wiki_link" href="/Hocus">Hocus</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>439.02<br />
</td>
        <td><a class="wiki_link" href="/9_7">9/7</a><br />
</td>
        <td>mo<br />
</td>
        <td><br />
</td>
        <td>11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>468.29<br />
</td>
        <td><a class="wiki_link" href="/21_16">21/16</a><br />
</td>
        <td>fe<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>497.56<br />
</td>
        <td><a class="wiki_link" href="/4_3">4/3</a><br />
</td>
        <td>fa<br />
</td>
        <td><a class="wiki_link" href="/Schismatic">Schismatic</a> (<a class="wiki_link" href="/helmholtz">helmholtz</a>, <a class="wiki_link" href="/Garibaldi%20temperament">garibaldi</a>, <a class="wiki_link" href="/cassandra">cassandra</a>)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>526.83<br />
</td>
        <td><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/27_20">27/20</a><br />
</td>
        <td>fih<br />
</td>
        <td><br />
</td>
        <td>9-tone MOS: 5 5 3 5 5 5 5 3 5<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>556.10<br />
</td>
        <td><a class="wiki_link" href="/11_8">11/8</a><br />
</td>
        <td>fu<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>585.37<br />
</td>
        <td><a class="wiki_link" href="/7_5">7/5</a><br />
</td>
        <td>fi<br />
</td>
        <td><a class="wiki_link" href="/Pluto">Pluto</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>614.63<br />
</td>
        <td><a class="wiki_link" href="/10_7">10/7</a><br />
</td>
        <td>se<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>643.90<br />
</td>
        <td><a class="wiki_link" href="/16_11">16/11</a><br />
</td>
        <td>su<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>673.17<br />
</td>
        <td><a class="wiki_link" href="/22_15">22/15</a>, <a class="wiki_link" href="/40_27">40/27</a><br />
</td>
        <td>sih<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>702.44<br />
</td>
        <td><a class="wiki_link" href="/3_2">3/2</a><br />
</td>
        <td>sol<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>731.71<br />
</td>
        <td><a class="wiki_link" href="/32_21">32/21</a><br />
</td>
        <td>si<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>760.98<br />
</td>
        <td><a class="wiki_link" href="/14_9">14/9</a>, <a class="wiki_link" href="/25_16">25/16</a><br />
</td>
        <td>lo<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>790.24<br />
</td>
        <td><a class="wiki_link" href="/11_7">11/7</a>, <a class="wiki_link" href="/128_81">128/81</a><br />
</td>
        <td>leh<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>819.51<br />
</td>
        <td><a class="wiki_link" href="/8_5">8/5</a><br />
</td>
        <td>le<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>848.78<br />
</td>
        <td><a class="wiki_link" href="/18_11">18/11</a>, <a class="wiki_link" href="/44_27">44/27</a><br />
</td>
        <td>lu<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>878.05<br />
</td>
        <td><a class="wiki_link" href="/5_3">5/3</a><br />
</td>
        <td>la<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>907.32<br />
</td>
        <td><a class="wiki_link" href="/27_16">27/16</a><br />
</td>
        <td>laa<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>936.59<br />
</td>
        <td><a class="wiki_link" href="/12_7">12/7</a><br />
</td>
        <td>li<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>965.85<br />
</td>
        <td><a class="wiki_link" href="/7_4">7/4</a><br />
</td>
        <td>ta<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>995.12<br />
</td>
        <td><a class="wiki_link" href="/16_9">16/9</a><br />
</td>
        <td>teh<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>1024.39<br />
</td>
        <td><a class="wiki_link" href="/9_5">9/5</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
        <td>te<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>1053.66<br />
</td>
        <td><a class="wiki_link" href="/11_6">11/6</a><br />
</td>
        <td>tu<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>1082.93<br />
</td>
        <td><a class="wiki_link" href="/15_8">15/8</a><br />
</td>
        <td>ti<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>1112.20<br />
</td>
        <td><a class="wiki_link" href="/40_21">40/21</a>, <a class="wiki_link" href="/21_11">21/11</a><br />
</td>
        <td>taa<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>39<br />
</td>
        <td>1141.46<br />
</td>
        <td><a class="wiki_link" href="/48_25">48/25</a>, <a class="wiki_link" href="/27_14">27/14</a>, <a class="wiki_link" href="/64_33">64/33</a><br />
</td>
        <td>to<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>40<br />
</td>
        <td>1170.73<br />
</td>
        <td><a class="wiki_link" href="/160_81">160/81</a><br />
</td>
        <td>da<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc3"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:15 -->Instruments</h1>
 <!-- ws:start:WikiTextLocalImageRule:1693:&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:1693 --><br />
<em>41-EDO Classical guitar, by Ron Sword.</em><br />
<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 />
<br />
<!-- ws:start:WikiTextHeadingRule:17:&lt;h1&gt; --><h1 id="toc4"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:17 -->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 />
<br />


<table class="wiki_table">
    <tr>
        <td>Overtones in &quot;Mode 8&quot;:<br />
</td>
        <td>8<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
        <td>13<br />
</td>
        <td>14<br />
</td>
        <td>15<br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>...as JI Ratio from 1/1:<br />
</td>
        <td>1/1<br />
</td>
        <td>9/8<br />
</td>
        <td>5/4<br />
</td>
        <td>11/8<br />
</td>
        <td>3/2<br />
</td>
        <td>13/8<br />
</td>
        <td>7/4<br />
</td>
        <td>15/8<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>203.9<br />
</td>
        <td>386.3<br />
</td>
        <td>551.3<br />
</td>
        <td>702.0<br />
</td>
        <td>840.5<br />
</td>
        <td>968.8<br />
</td>
        <td>1088.3<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 41edo:<br />
</td>
        <td>0<br />
</td>
        <td>7<br />
</td>
        <td>13<br />
</td>
        <td>19<br />
</td>
        <td>24<br />
</td>
        <td>29<br />
</td>
        <td>33<br />
</td>
        <td>37<br />
</td>
        <td>41<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>204.9<br />
</td>
        <td>380.5<br />
</td>
        <td>556.1<br />
</td>
        <td>702.4<br />
</td>
        <td>848.8<br />
</td>
        <td>965.9<br />
</td>
        <td>1082.9<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
</table>

<br />
While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)<br />
<br />
7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.<br />
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).<br />
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).<br />
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).<br />
<br />
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:19:&lt;h1&gt; --><h1 id="toc5"><a name="Nonoctave Temperaments"></a><!-- ws:end:WikiTextHeadingRule:19 -->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 />
<br />


<table class="wiki_table">
    <tr>
        <td colspan="3" style="text-align: center;">3 degrees of 41edo (near 88cET)<br />
</td>
        <td style="text-align: center;">overlap<br />
</td>
        <td colspan="3" style="text-align: center;">5 degrees of 41edo (near BP)<br />
</td>
    </tr>
    <tr>
        <th>deg of 41edo<br />
</th>
        <th>deg of 88cET<br />
</th>
        <th>cents<br />
</th>
        <th>cents<br />
</th>
        <th>cents<br />
</th>
        <th>deg of BP<br />
</th>
        <th>deg of 41edo<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">87.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">146.3<br />
</td>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">5<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">175.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">263.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">292.7<br />
</td>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">10<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">351.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">439.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">15<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">526.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">585.4<br />
</td>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">20<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">614.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">702.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">731.7<br />
</td>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">25<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">790.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">878.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">30<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">965.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1024.4<br />
</td>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">35<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">1053.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">1141.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1170.7<br />
</td>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">40<br />
</td>
    </tr>
    <tr>
        <th colspan="7">[ second octave ]<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">29.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">117.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">4<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">204.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">263.4<br />
</td>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">9<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">292.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">380.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">409.8<br />
</td>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">14<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">468.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">556.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">19<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">643.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">702.4<br />
</td>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">24<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">731.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">819.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">848.8<br />
</td>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">29<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">907.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">995.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">34<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">1082.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1141.5<br />
</td>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">39<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">1170.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <th colspan="7">[ third octave ]<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">58.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">87.8<br />
</td>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">3<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">146.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">234.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">8<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">322.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">380.5<br />
</td>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">13<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">409.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">497.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">526.8<br />
</td>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">18<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">585.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">673.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">23<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">761.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">819.5<br />
</td>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">28<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">848.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">936.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">965.9<br />
</td>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">33<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">1024.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1112.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">38<br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:21:&lt;h1&gt; --><h1 id="toc6"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:21 -->Music</h1>
 <a class="wiki_link_ext" href="http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro" rel="nofollow">EveningHorizon</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3" rel="nofollow">play</a> by Cameron Bobro<br />
<br />
<!-- ws:start:WikiTextHeadingRule:23:&lt;h1&gt; --><h1 id="toc7"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:23 -->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:2568: --><hr class="references" /><ol class="references">
<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> at x31eq.com the websize of <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a></li>
<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> at x31eq.com</li>
<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">Schismatic temperament</a></li>
<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Magic_temperament" rel="nofollow">Magic temperament</a></li>
</ol><!-- ws:end:WikiTextReferencesRule:2568 --></body></html>

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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>2012-11-17 09:49:27 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-11-17 09:49:57 UTC</tt>.<br>
-: The original revision id was <tt>383534932</tt>.<br>
+: The original revision id was <tt>383534998</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>
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 [[toc|flat]]
 =Introduction=
-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 //[[Schismatic family#Garibaldi|Garibaldi temperament]]// &lt;ref&gt;[[http://x31eq.com/schismic.htm|"Schismic Temperaments"]] at x31eq.com the websize of [[Graham Breed]]&lt;/ref&gt; , &lt;ref&gt;[[http://x31eq.com/decimal_lattice.htm|"Lattices with Decimal Notation"]] at x31eq.com&lt;/ref&gt; , &lt;ref&gt;[[http://en.wikipedia.org/wiki/Schismatic_temperament|Schismatic temperament]]&lt;/ref&gt; the //[[Magic family|Magic temperament]]// &lt;ref&gt;[[http://en.wikipedia.org/wiki/Magic_temperament|Magic temperament]]&lt;/ref&gt; and the superkleismic (41&amp;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%20EDO%20lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|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. Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy at witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.
+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 //[[Schismatic family#Garibaldi|Garibaldi temperament]]// &lt;ref&gt;[[http://x31eq.com/schismic.htm|"Schismic Temperaments"]] at x31eq.com the websize of [[Graham Breed]]&lt;/ref&gt; , &lt;ref&gt;[[http://x31eq.com/decimal_lattice.htm|"Lattices with Decimal Notation"]] at x31eq.com&lt;/ref&gt; , &lt;ref&gt;[[http://en.wikipedia.org/wiki/Schismatic_temperament|Schismatic temperament]]&lt;/ref&gt; the //[[Magic family|Magic temperament]]// &lt;ref&gt;[[http://en.wikipedia.org/wiki/Magic_temperament|Magic temperament]]&lt;/ref&gt; and the superkleismic (41&amp;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%20EDO%20lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|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. Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.
 -ET forms the foundation of the [[http://www.h-pi.com/theory/huntsystem1.html|H-System]], which uses the scale degrees of 41-ET as the basic [[13-limit]] intervals requiring fine tuning +/- 1 [[http://www.h-pi.com/theory/huntsystem2.html|average JND]] from the 41-ET circle in [[205edo]].
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 &lt;!-- ws:start:WikiTextTocRule:25:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;a href="#Introduction"&gt;Introduction&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Temperaments"&gt;Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Instruments"&gt;Instruments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#Harmonic Scale"&gt;Harmonic Scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Nonoctave Temperaments"&gt;Nonoctave Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Music"&gt;Music&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Links"&gt;Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Introduction&lt;/h1&gt;
-  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 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s, an &lt;a class="wiki_link" href="/interval"&gt;interval&lt;/a&gt; close in size to &lt;a class="wiki_link" href="/64_63"&gt;64/63&lt;/a&gt;, the &lt;a class="wiki_link" href="/Septimal%20comma"&gt;septimal comma&lt;/a&gt;. 41-ET can be seen as a tuning of the &lt;em&gt;&lt;a class="wiki_link" href="/Schismatic%20family#Garibaldi"&gt;Garibaldi temperament&lt;/a&gt;&lt;/em&gt; &lt;!-- ws:start:WikiTextRefRule:2:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://x31eq.com/schismic.htm&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;amp;quot;Schismic Temperaments&amp;amp;quot;&amp;lt;/a&amp;gt; at x31eq.com the websize of &amp;lt;a class=&amp;quot;wiki_link&amp;quot; href=&amp;quot;/Graham%20Breed&amp;quot;&amp;gt;Graham Breed&amp;lt;/a&amp;gt;&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-1" class="reference"&gt;&lt;a href="#cite_note-1"&gt;[1]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:2 --&gt; , &lt;!-- ws:start:WikiTextRefRule:4:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://x31eq.com/decimal_lattice.htm&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;amp;quot;Lattices with Decimal Notation&amp;amp;quot;&amp;lt;/a&amp;gt; at x31eq.com&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-2" class="reference"&gt;&lt;a href="#cite_note-2"&gt;[2]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:4 --&gt; , &lt;!-- ws:start:WikiTextRefRule:6:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://en.wikipedia.org/wiki/Schismatic_temperament&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;Schismatic temperament&amp;lt;/a&amp;gt;&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-3" class="reference"&gt;&lt;a href="#cite_note-3"&gt;[3]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:6 --&gt; the &lt;em&gt;&lt;a class="wiki_link" href="/Magic%20family"&gt;Magic temperament&lt;/a&gt;&lt;/em&gt; &lt;!-- ws:start:WikiTextRefRule:8:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://en.wikipedia.org/wiki/Magic_temperament&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;Magic temperament&amp;lt;/a&amp;gt;&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-4" class="reference"&gt;&lt;a href="#cite_note-4"&gt;[4]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:8 --&gt; and the superkleismic (41&amp;amp;26) temperament. It is the second smallest equal temperament (after &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;) whose perfect fifth is closer to just intonation than that of &lt;a class="wiki_link" href="/12edo"&gt;12-ET&lt;/a&gt;, and is the seventh &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt; after 31; it is not, however, a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta gap edo&lt;/a&gt;. This has to do with the fact that it can deal with the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; fairly well, and the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; perhaps close enough for government work, though its &lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt; is 14 cents sharp. Various 13-limit &lt;a class="wiki_link" href="/magic%20extensions"&gt;magic extensions&lt;/a&gt; are supported by 41: 13-limit magic, and less successfully necromancy at witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.&lt;br /&gt;
+  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 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s, an &lt;a class="wiki_link" href="/interval"&gt;interval&lt;/a&gt; close in size to &lt;a class="wiki_link" href="/64_63"&gt;64/63&lt;/a&gt;, the &lt;a class="wiki_link" href="/Septimal%20comma"&gt;septimal comma&lt;/a&gt;. 41-ET can be seen as a tuning of the &lt;em&gt;&lt;a class="wiki_link" href="/Schismatic%20family#Garibaldi"&gt;Garibaldi temperament&lt;/a&gt;&lt;/em&gt; &lt;!-- ws:start:WikiTextRefRule:2:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://x31eq.com/schismic.htm&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;amp;quot;Schismic Temperaments&amp;amp;quot;&amp;lt;/a&amp;gt; at x31eq.com the websize of &amp;lt;a class=&amp;quot;wiki_link&amp;quot; href=&amp;quot;/Graham%20Breed&amp;quot;&amp;gt;Graham Breed&amp;lt;/a&amp;gt;&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-1" class="reference"&gt;&lt;a href="#cite_note-1"&gt;[1]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:2 --&gt; , &lt;!-- ws:start:WikiTextRefRule:4:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://x31eq.com/decimal_lattice.htm&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;&amp;amp;quot;Lattices with Decimal Notation&amp;amp;quot;&amp;lt;/a&amp;gt; at x31eq.com&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-2" class="reference"&gt;&lt;a href="#cite_note-2"&gt;[2]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:4 --&gt; , &lt;!-- ws:start:WikiTextRefRule:6:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://en.wikipedia.org/wiki/Schismatic_temperament&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;Schismatic temperament&amp;lt;/a&amp;gt;&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-3" class="reference"&gt;&lt;a href="#cite_note-3"&gt;[3]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:6 --&gt; the &lt;em&gt;&lt;a class="wiki_link" href="/Magic%20family"&gt;Magic temperament&lt;/a&gt;&lt;/em&gt; &lt;!-- ws:start:WikiTextRefRule:8:&amp;amp;lt;ref&amp;amp;gt;&amp;lt;a class=&amp;quot;wiki_link_ext&amp;quot; href=&amp;quot;http://en.wikipedia.org/wiki/Magic_temperament&amp;quot; rel=&amp;quot;nofollow&amp;quot;&amp;gt;Magic temperament&amp;lt;/a&amp;gt;&amp;amp;lt;/ref&amp;amp;gt; --&gt;&lt;sup id="cite_ref-4" class="reference"&gt;&lt;a href="#cite_note-4"&gt;[4]&lt;/a&gt;&lt;/sup&gt;&lt;!-- ws:end:WikiTextRefRule:8 --&gt; and the superkleismic (41&amp;amp;26) temperament. It is the second smallest equal temperament (after &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;) whose perfect fifth is closer to just intonation than that of &lt;a class="wiki_link" href="/12edo"&gt;12-ET&lt;/a&gt;, and is the seventh &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt; after 31; it is not, however, a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta gap edo&lt;/a&gt;. This has to do with the fact that it can deal with the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; fairly well, and the &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; perhaps close enough for government work, though its &lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt; is 14 cents sharp. Various 13-limit &lt;a class="wiki_link" href="/magic%20extensions"&gt;magic extensions&lt;/a&gt; are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.&lt;br /&gt;
 &lt;br /&gt;
 -ET forms the foundation of the &lt;a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem1.html" rel="nofollow"&gt;H-System&lt;/a&gt;, which uses the scale degrees of 41-ET as the basic &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; intervals requiring fine tuning +/- 1 &lt;a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem2.html" rel="nofollow"&gt;average JND&lt;/a&gt; from the 41-ET circle in &lt;a class="wiki_link" href="/205edo"&gt;205edo&lt;/a&gt;.&lt;br /&gt;

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