Xenharmonic Wiki

Catalog of 3.5.7 subgroup rank two temperaments

Revision as of 21:17, 31 January 2017 by Wikispaces>spt3125 (**Imported revision 605206821 - Original comment: **)

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[[toc|flat]]
=Table= 

||~ name ||~ comma |> ||~ comma n/d ||~ comma c ||~ comma name ||~ a&b (3) ||~ a&b (2) ||~ complexity ||~ damage ||~ mapping ||~ TE gen1 ||~ TE gen2 ||~ per/3 ||~ MOS ||
|| *Izar || | -11 -7 12 > ||= 13841287201/13839609375 || 0.210 ||   || b271 & b1071 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=171p_676cd&limit=3_5_7|171p & 676cd]] || 11.675 || 0.003 || [< 271 397 480 ], < 1071 1569 1897 ]> || 1901.9584 || 877.2825 || 1 || 2*,3,5,7,9,11*,13*,24,37,50,63,76,...258*,271* ||
|| [[Canopus]] || | 3 4 -5 > ||= 16875/16807 || 6.990 || small BP diesis, [[Mirkwai clan|mirkwai]] || b13 & b101 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_64cd&limit=3_5_7|8d & 64cd]] || 4.866 || 0.249 || [< 13 19 23 ], < 101 148 179 ]> || 1901.7828 || 583.9055 || 1 || 2,3*,4,7,10*,13*,23,36,49,62,75,88,101,114* ||
|| *Procyon || | -8 -3 7 > ||= 823543/820125 || 7.200 ||   || b13 & b144 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_91p&limit=3_5_7|8d & 91p]] || 6.810 || 0.183 || [< 13 19 23 ], < 144 211 255 ]> || 1902.198 || 145.4124 || 1 || ...13*,14,27,40,53,66,79,92,105,118,131,144,157* ||
|| [[Bohlen-Pierce|BP diatonic]] || | -5 1 2 > ||= 245/243 || 14.191 || minor BP diesis, [[Sensamagic clan|sensamagic]] || b13 & b17 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_11cd&limit=3_5_7|8d & 11cd]] || 2.749 || 0.895 || [< 13 19 23 ], < 17 25 30 ]> || 1903.7424 || 440.902 || 1 || 2,3,4*,5,9,13*,17,30,43,56,69,82*,95* ||
|| [[Sirius]] || | -2 5 -3 > ||= 3125/3087 || 21.181 || major BP diesis, [[Gariboh family|gariboh]] || b13 & b19 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_12p&limit=3_5_7|8d & 12p]] || 4.021 || 0.913 || [< 13 19 23 ], < 19 28 34 ]> || 1902.4482 || 293.7397 || 1 || 2,3,4,5,6*,7,13*,19,32,45,58,71,84,97,110,123,136* ||
|| *Betelgeuse || | -13 -2 9 > ||= 40353607/39858075 || 21.391 ||   || b13 & b148 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_93bccd&limit=3_5_7|8d & 93bccd]] || 8.754 || 0.423 || [< 13 19 23 ], < 148 217 262 ]> || 1902.5991 || 732.4139 || 1 || 2*,3*,5*,8,13*,...226* ||
|| *Miaplacidus || | 1 9 -8 > ||= 5859375/5764801 || 28.171 ||   || b13 & b94d || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_59ccdd&limit=3_5_7|8d & 59ccdd]] || 7.788 || 0.627 || [< 13 19 23 ], < 94 138 167 ]> || 1901.88 || 585.8431 || 1 || 2,3*,4,7,10,13*,16,29,42,55,68,81,94,...224* ||
|| [[Arcturus]] || | -7 6 -1 > ||= 15625/15309 || 35.371 || great BP diesis || b13 & b2 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_1cdd&limit=3_5_7|8d & 1cdd]] || 4.823 || 1.271 || [< 13 19 23 ], < 2 3 4 ]> || 1903.8685 || 878.9253 || 1 || 2*,3,5,7,9,11,13* ||
|| *Pollux || | -18 -1 11 > ||= 1977326743/1937102445 || 35.581 ||   || b13 & b139 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_88dd&limit=3_5_7|8d & 88dd]] || 10.697 || 0.576 || [< 13 19 23 ], < 139 204 246 ]> || 1902.8625 || 438.6389 || 1 || 2,3,4*,5,9*,13 ||
|| *Castor || | -1 14 -11 > ||= 6103515625/5931980229 || 49.352 ||   || b13 & b113d || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_71ccddd&limit=3_5_7|8d & 71ccddd]] || 11.259 || 0.760 || [< 13 19 23 ], < 113 166 201 ]> || 1902.0166 || 438.5094 || 1 || 2,3,4*,5,9*,13 ||
|| *Muphrid || | -12 7 1 > ||= 546875/531441 || 49.562 ||   || b13 & b2d || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_1c&limit=3_5_7|8d & 1c]] || 6.599 || 1.302 || [< 13 19 23 ], < 2 3 3 ]> || 1904.3123 || 879.0503 || 1 || 2*,3,5,7,9,11,13* ||
|| *Rigel || | -9 11 -4 > ||= 48828125/47258883 || 56.552 ||   || b13 & b21d || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d_5d&limit=3_5_7|8d & 5d]] || 8.844 || 1.108 || [< 13 19 23 ], < 21 31 38 ]> || 1903.2571 || 731.7183 || 1 || 2*,3*,5*,8*,13* ||
|| *Mirzam || | -17 8 3 > ||= 133984375/129140163 || 63.752 ||   || b13 & b32d || [[http://x31eq.com/cgi-bin/rt.cgi?ets=8d+20c&limit=3.5.7|8d & 20c]] || 9.347 || 1.182 || [< 13 19 23 ], < 32 47 56 ]> || 1904.3391 || 293.1889 || 1 || 2,3,4,5,6*,7,13* ||
||   || | 15 -9 -1 > ||= 14348907/13671875 || 83.676 ||   || b17 & b32 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=11cd_20cd&limit=3_5_7|11cd & 20cd]] || 8.228 || 1.762 || [< 17 25 30 ], < 32 47 57 ]> || 1898.8326 || 891.5018 || 1 || 2*,3,5,7,9,11,13,15*,17*,32,49*,66,115,164* ||
|| *Capella || | 10 -8 1 > ||= 413343/390625 || 97.866 || small link || b17 & b19 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=11cd_12p&limit=3_5_7|12p & 11cd]] || 6.452 || 2.629 || [< 17 25 30 ], < 19 28 34 ]> || 1897.8856 || 895.415 || 1 || 2*,3,5,7,9,11,13,15,17*,19,36*,53*,89,142*,195* ||
||   || | 13 -4 -4 > ||= 1594323/1500625 || 104.857 ||   || b4 & b32 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=3bcdd_20cd&limit=3_5_7|3bcdd & 20cd]] || 7.126 || 2.550 || [< 4 6 7 ], < 32 47 57 ]> || 474.1446 || 55.0036 || 4 || ...32*,36*,68*,104*,172*,276* ||
||   || | 16 0 -9 > ||= 43046721/40353607 || 111.847 ||   || b9 & b36 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=6bcd_23d&limit=3_5_7|6bcd & 23d]] || 8.773 || 2.209 || [< 9 13 16 ], < 36 53 64 ]> || 210.9385 || 44.1126 || 9 || 18,27,36*,45*,81,126,171*,216*,387* ||
|| Toliman || | 5 -7 3 > ||= 83349/78125 || 112.057 || minor link || b6 & b17 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=4p_11cd&limit=3_5_7|4p & 11cd]] || 5.649 || 3.437 || [< 6 9 11 ], < 17 25 30 ]> || 1898.4178 || 333.298 || 1 || 2,3,4,5*,6*,11,17*,23,40,57*,74,131* ||
|| Vega || | 8 -3 -2 > ||= 6561/6125 || 119.047 || major link || b4 & b15 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=3bcdd_10bccdd&limit=3_5_7|3bcdd & 10bccdd]] || 4.377 || 4.713 || [< 4 6 7 ], < 15 22 27 ]> || 1892.0204 || 494.8658 || 1 || 2,3*,4*,7,11,15,19*,23*,42*,65* ||
|| *Achernar || | 0 -6 5 > ||= 16807/15625 || 126.247 ||   || b7 & b17 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=4bcd_11cd&limit=3_5_7|4bcd & 11cd]] || 4.847 || 4.514 || [< 7 10 12 ], < 17 25 30 ]> || 1901.955 || 559.351 || 1 || 2,3*,4,7*,10,17* ||
|| % || | 3 -2 0 > ||= [[27_25|27/25]] || 133.238 || large limma, BP small semitone || b4 & b6 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=b4+b6&limit=3_5_7|3bcdd & 4p]] || 1.629 || 14.176 || [< 4 6 7 ], < 6 9 11 ]> || 939.6121 || 549.9897 || 2 || 4*,6,10*,14,24*,34,58,82* ||
||   || | 6 2 -5 > ||= 18225/16807 || 140.228 || minimal BP chroma || b9 & b10 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=6bcd_6ccdd&limit=3_5_7|6bcd & 6ccdd]] || 4.885 || 4.974 || [< 9 13 16 ], < 10 15 18 ]> || 1895.1156 || 201.762 || 1 || ...9*,10,19*,28*,47* ||
|| % || | -2 -1 2 > ||= [[49_45|49/45]] || 147.428 || BP minor semitone, [[Swetismic temperaments|swetismic]] neutral second || b4 & b7 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=b4+b7&limit=3_5_7|3bcdd & 4bcd]] || 1.925 || 13.274 || [< 4 6 7 ], < 7 10 12 ]> || 1917.7266 || 516.1077 || 1 || 2,3*,4*,7,11*,15*,26* ||
||   || | 1 3 -3 > ||= 375/343 || 154.418 || BP major semitone, minor BP chroma || b3 & b6 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=2cd_4p&limit=3_5_7|2cd & 4p]] || 2.941 || 9.098 || [< 3 4 5 ], < 6 9 11 ]> || 632.9248 || 234.1367 || 3 || 6*,9*,15,24*,33,57*,81* ||
||   || | -7 0 4 > ||= 2401/2187 || 161.619 ||   || b4 & b40cd || [[http://x31eq.com/cgi-bin/rt.cgi?ets=b4+b25&limit=3_5_7|3bcdd & 25p]] || 3.868 || 7.240 || [< 4 6 7 ], < 40 58 70 ]> || 478.34 || 41.8631 || 4 || ...44*,48,92*,136,228*,320* ||
||   || | -4 4 -1 > ||= 625/567 || 168.609 || BP great semitone, major BP chroma || b2 & b7 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=1cdd_4bcd&limit=3_5_7|1cdd & 4bcd]] || 3.195 || 9.146 || [< 2 3 4 ], < 7 10 12 ]> || 1914.567 || 1069.8876 || 1 || 2*,3,5,7*,9*,16,25,34* ||
||   || | -9 5 1 > ||= 21875/19683 || 182.799 || maximal BP chroma || b9 & b11 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=6bcd_7d&limit=3_5_7|6bcd & 7d]] || 4.970 || 6.373 || [< 9 13 16 ], < 11 16 19 ]> || 1913.8948 || 858.1831 || 1 || 2*,3,5,7,9*,11,20*,29* ||
|| % || | 9 0 -5 > ||= 19683/16807 || 273.465 ||   || b5 & b10 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=3d_6ccdd&limit=3_5_7|3d & 6ccdd]] || 4.904 || 9.663 || [< 5 7 9 ], < 10 15 18 ]> || 377.3036 || 145.1888 || 5 || 10*,15*,25*,40,65* ||

names: * = proposed
names: % = not shown on PTS plots
MOS: * = maximally even

complexity = ln(n*d)/4
damage = 1200 *log(n/d) / log(n*d)
(complexity and damage are calculated as in the "Middle Path" tables)


=Graphs= 

[[image:357plot_cplx_damage.png caption="Complexity vs. damage plot.  z<1 corresponds to the "Middle Path" inclusion criterion."]]


=Projective tuning space diagrams= 

[[image:tn_357ptslines1n.png caption="Temperaments with smaller commas, labeled by name" link="file:357ptslines1n.png"]]

[[image:tn_357ptslines1c.png caption="Temperaments with smaller commas, labeled by comma" link="file:357ptslines1c.png"]]

[[image:tn_357ptslines2n.png caption="Temperaments with larger commas, labeled by name" link="file:357ptslines2n.png"]]

[[image:tn_357ptslines2c.png caption="Temperaments with larger commas, labeled by comma" link="file:357ptslines2c.png"]]

[[image:tn_357ptslines12n.png caption="Both sets, labeled by name" link="file:357ptslines12n.png"]]

[[image:tn_357ptslines12c.png caption="Both sets, labeled by comma" link="file:357ptslines12c.png"]]
<span style="display: block; height: 1px; left: 0px; overflow: hidden; position: absolute; top: 2151px; width: 1px;">
|| 11.675 ||
</span>

Original HTML content:

<html><head><title>Catalog of 3.5.7 subgroup rank two temperaments</title></head><body><!-- ws:start:WikiTextTocRule:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Table">Table</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Graphs">Graphs</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Projective tuning space diagrams">Projective tuning space diagrams</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
<!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Table"></a><!-- ws:end:WikiTextHeadingRule:0 -->Table</h1>
 <br />


<table class="wiki_table">
    <tr>
        <th>name<br />
</th>
        <th>comma |&gt;<br />
</th>
        <th>comma n/d<br />
</th>
        <th>comma c<br />
</th>
        <th>comma name<br />
</th>
        <th>a&amp;b (3)<br />
</th>
        <th>a&amp;b (2)<br />
</th>
        <th>complexity<br />
</th>
        <th>damage<br />
</th>
        <th>mapping<br />
</th>
        <th>TE gen1<br />
</th>
        <th>TE gen2<br />
</th>
        <th>per/3<br />
</th>
        <th>MOS<br />
</th>
    </tr>
    <tr>
        <td>*Izar<br />
</td>
        <td>| -11 -7 12 &gt;<br />
</td>
        <td style="text-align: center;">13841287201/13839609375<br />
</td>
        <td>0.210<br />
</td>
        <td><br />
</td>
        <td>b271 &amp; b1071<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=171p_676cd&amp;limit=3_5_7" rel="nofollow">171p &amp; 676cd</a><br />
</td>
        <td>11.675<br />
</td>
        <td>0.003<br />
</td>
        <td>[&lt; 271 397 480 ], &lt; 1071 1569 1897 ]&gt;<br />
</td>
        <td>1901.9584<br />
</td>
        <td>877.2825<br />
</td>
        <td>1<br />
</td>
        <td>2*,3,5,7,9,11*,13*,24,37,50,63,76,...258*,271*<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/Canopus">Canopus</a><br />
</td>
        <td>| 3 4 -5 &gt;<br />
</td>
        <td style="text-align: center;">16875/16807<br />
</td>
        <td>6.990<br />
</td>
        <td>small BP diesis, <a class="wiki_link" href="/Mirkwai%20clan">mirkwai</a><br />
</td>
        <td>b13 &amp; b101<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_64cd&amp;limit=3_5_7" rel="nofollow">8d &amp; 64cd</a><br />
</td>
        <td>4.866<br />
</td>
        <td>0.249<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 101 148 179 ]&gt;<br />
</td>
        <td>1901.7828<br />
</td>
        <td>583.9055<br />
</td>
        <td>1<br />
</td>
        <td>2,3*,4,7,10*,13*,23,36,49,62,75,88,101,114*<br />
</td>
    </tr>
    <tr>
        <td>*Procyon<br />
</td>
        <td>| -8 -3 7 &gt;<br />
</td>
        <td style="text-align: center;">823543/820125<br />
</td>
        <td>7.200<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b144<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_91p&amp;limit=3_5_7" rel="nofollow">8d &amp; 91p</a><br />
</td>
        <td>6.810<br />
</td>
        <td>0.183<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 144 211 255 ]&gt;<br />
</td>
        <td>1902.198<br />
</td>
        <td>145.4124<br />
</td>
        <td>1<br />
</td>
        <td>...13*,14,27,40,53,66,79,92,105,118,131,144,157*<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/Bohlen-Pierce">BP diatonic</a><br />
</td>
        <td>| -5 1 2 &gt;<br />
</td>
        <td style="text-align: center;">245/243<br />
</td>
        <td>14.191<br />
</td>
        <td>minor BP diesis, <a class="wiki_link" href="/Sensamagic%20clan">sensamagic</a><br />
</td>
        <td>b13 &amp; b17<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_11cd&amp;limit=3_5_7" rel="nofollow">8d &amp; 11cd</a><br />
</td>
        <td>2.749<br />
</td>
        <td>0.895<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 17 25 30 ]&gt;<br />
</td>
        <td>1903.7424<br />
</td>
        <td>440.902<br />
</td>
        <td>1<br />
</td>
        <td>2,3,4*,5,9,13*,17,30,43,56,69,82*,95*<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/Sirius">Sirius</a><br />
</td>
        <td>| -2 5 -3 &gt;<br />
</td>
        <td style="text-align: center;">3125/3087<br />
</td>
        <td>21.181<br />
</td>
        <td>major BP diesis, <a class="wiki_link" href="/Gariboh%20family">gariboh</a><br />
</td>
        <td>b13 &amp; b19<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_12p&amp;limit=3_5_7" rel="nofollow">8d &amp; 12p</a><br />
</td>
        <td>4.021<br />
</td>
        <td>0.913<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 19 28 34 ]&gt;<br />
</td>
        <td>1902.4482<br />
</td>
        <td>293.7397<br />
</td>
        <td>1<br />
</td>
        <td>2,3,4,5,6*,7,13*,19,32,45,58,71,84,97,110,123,136*<br />
</td>
    </tr>
    <tr>
        <td>*Betelgeuse<br />
</td>
        <td>| -13 -2 9 &gt;<br />
</td>
        <td style="text-align: center;">40353607/39858075<br />
</td>
        <td>21.391<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b148<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_93bccd&amp;limit=3_5_7" rel="nofollow">8d &amp; 93bccd</a><br />
</td>
        <td>8.754<br />
</td>
        <td>0.423<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 148 217 262 ]&gt;<br />
</td>
        <td>1902.5991<br />
</td>
        <td>732.4139<br />
</td>
        <td>1<br />
</td>
        <td>2*,3*,5*,8,13*,...226*<br />
</td>
    </tr>
    <tr>
        <td>*Miaplacidus<br />
</td>
        <td>| 1 9 -8 &gt;<br />
</td>
        <td style="text-align: center;">5859375/5764801<br />
</td>
        <td>28.171<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b94d<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_59ccdd&amp;limit=3_5_7" rel="nofollow">8d &amp; 59ccdd</a><br />
</td>
        <td>7.788<br />
</td>
        <td>0.627<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 94 138 167 ]&gt;<br />
</td>
        <td>1901.88<br />
</td>
        <td>585.8431<br />
</td>
        <td>1<br />
</td>
        <td>2,3*,4,7,10,13*,16,29,42,55,68,81,94,...224*<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/Arcturus">Arcturus</a><br />
</td>
        <td>| -7 6 -1 &gt;<br />
</td>
        <td style="text-align: center;">15625/15309<br />
</td>
        <td>35.371<br />
</td>
        <td>great BP diesis<br />
</td>
        <td>b13 &amp; b2<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_1cdd&amp;limit=3_5_7" rel="nofollow">8d &amp; 1cdd</a><br />
</td>
        <td>4.823<br />
</td>
        <td>1.271<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 2 3 4 ]&gt;<br />
</td>
        <td>1903.8685<br />
</td>
        <td>878.9253<br />
</td>
        <td>1<br />
</td>
        <td>2*,3,5,7,9,11,13*<br />
</td>
    </tr>
    <tr>
        <td>*Pollux<br />
</td>
        <td>| -18 -1 11 &gt;<br />
</td>
        <td style="text-align: center;">1977326743/1937102445<br />
</td>
        <td>35.581<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b139<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_88dd&amp;limit=3_5_7" rel="nofollow">8d &amp; 88dd</a><br />
</td>
        <td>10.697<br />
</td>
        <td>0.576<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 139 204 246 ]&gt;<br />
</td>
        <td>1902.8625<br />
</td>
        <td>438.6389<br />
</td>
        <td>1<br />
</td>
        <td>2,3,4*,5,9*,13<br />
</td>
    </tr>
    <tr>
        <td>*Castor<br />
</td>
        <td>| -1 14 -11 &gt;<br />
</td>
        <td style="text-align: center;">6103515625/5931980229<br />
</td>
        <td>49.352<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b113d<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_71ccddd&amp;limit=3_5_7" rel="nofollow">8d &amp; 71ccddd</a><br />
</td>
        <td>11.259<br />
</td>
        <td>0.760<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 113 166 201 ]&gt;<br />
</td>
        <td>1902.0166<br />
</td>
        <td>438.5094<br />
</td>
        <td>1<br />
</td>
        <td>2,3,4*,5,9*,13<br />
</td>
    </tr>
    <tr>
        <td>*Muphrid<br />
</td>
        <td>| -12 7 1 &gt;<br />
</td>
        <td style="text-align: center;">546875/531441<br />
</td>
        <td>49.562<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b2d<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_1c&amp;limit=3_5_7" rel="nofollow">8d &amp; 1c</a><br />
</td>
        <td>6.599<br />
</td>
        <td>1.302<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 2 3 3 ]&gt;<br />
</td>
        <td>1904.3123<br />
</td>
        <td>879.0503<br />
</td>
        <td>1<br />
</td>
        <td>2*,3,5,7,9,11,13*<br />
</td>
    </tr>
    <tr>
        <td>*Rigel<br />
</td>
        <td>| -9 11 -4 &gt;<br />
</td>
        <td style="text-align: center;">48828125/47258883<br />
</td>
        <td>56.552<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b21d<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d_5d&amp;limit=3_5_7" rel="nofollow">8d &amp; 5d</a><br />
</td>
        <td>8.844<br />
</td>
        <td>1.108<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 21 31 38 ]&gt;<br />
</td>
        <td>1903.2571<br />
</td>
        <td>731.7183<br />
</td>
        <td>1<br />
</td>
        <td>2*,3*,5*,8*,13*<br />
</td>
    </tr>
    <tr>
        <td>*Mirzam<br />
</td>
        <td>| -17 8 3 &gt;<br />
</td>
        <td style="text-align: center;">133984375/129140163<br />
</td>
        <td>63.752<br />
</td>
        <td><br />
</td>
        <td>b13 &amp; b32d<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=8d+20c&amp;limit=3.5.7" rel="nofollow">8d &amp; 20c</a><br />
</td>
        <td>9.347<br />
</td>
        <td>1.182<br />
</td>
        <td>[&lt; 13 19 23 ], &lt; 32 47 56 ]&gt;<br />
</td>
        <td>1904.3391<br />
</td>
        <td>293.1889<br />
</td>
        <td>1<br />
</td>
        <td>2,3,4,5,6*,7,13*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| 15 -9 -1 &gt;<br />
</td>
        <td style="text-align: center;">14348907/13671875<br />
</td>
        <td>83.676<br />
</td>
        <td><br />
</td>
        <td>b17 &amp; b32<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=11cd_20cd&amp;limit=3_5_7" rel="nofollow">11cd &amp; 20cd</a><br />
</td>
        <td>8.228<br />
</td>
        <td>1.762<br />
</td>
        <td>[&lt; 17 25 30 ], &lt; 32 47 57 ]&gt;<br />
</td>
        <td>1898.8326<br />
</td>
        <td>891.5018<br />
</td>
        <td>1<br />
</td>
        <td>2*,3,5,7,9,11,13,15*,17*,32,49*,66,115,164*<br />
</td>
    </tr>
    <tr>
        <td>*Capella<br />
</td>
        <td>| 10 -8 1 &gt;<br />
</td>
        <td style="text-align: center;">413343/390625<br />
</td>
        <td>97.866<br />
</td>
        <td>small link<br />
</td>
        <td>b17 &amp; b19<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=11cd_12p&amp;limit=3_5_7" rel="nofollow">12p &amp; 11cd</a><br />
</td>
        <td>6.452<br />
</td>
        <td>2.629<br />
</td>
        <td>[&lt; 17 25 30 ], &lt; 19 28 34 ]&gt;<br />
</td>
        <td>1897.8856<br />
</td>
        <td>895.415<br />
</td>
        <td>1<br />
</td>
        <td>2*,3,5,7,9,11,13,15,17*,19,36*,53*,89,142*,195*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| 13 -4 -4 &gt;<br />
</td>
        <td style="text-align: center;">1594323/1500625<br />
</td>
        <td>104.857<br />
</td>
        <td><br />
</td>
        <td>b4 &amp; b32<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=3bcdd_20cd&amp;limit=3_5_7" rel="nofollow">3bcdd &amp; 20cd</a><br />
</td>
        <td>7.126<br />
</td>
        <td>2.550<br />
</td>
        <td>[&lt; 4 6 7 ], &lt; 32 47 57 ]&gt;<br />
</td>
        <td>474.1446<br />
</td>
        <td>55.0036<br />
</td>
        <td>4<br />
</td>
        <td>...32*,36*,68*,104*,172*,276*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| 16 0 -9 &gt;<br />
</td>
        <td style="text-align: center;">43046721/40353607<br />
</td>
        <td>111.847<br />
</td>
        <td><br />
</td>
        <td>b9 &amp; b36<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=6bcd_23d&amp;limit=3_5_7" rel="nofollow">6bcd &amp; 23d</a><br />
</td>
        <td>8.773<br />
</td>
        <td>2.209<br />
</td>
        <td>[&lt; 9 13 16 ], &lt; 36 53 64 ]&gt;<br />
</td>
        <td>210.9385<br />
</td>
        <td>44.1126<br />
</td>
        <td>9<br />
</td>
        <td>18,27,36*,45*,81,126,171*,216*,387*<br />
</td>
    </tr>
    <tr>
        <td>Toliman<br />
</td>
        <td>| 5 -7 3 &gt;<br />
</td>
        <td style="text-align: center;">83349/78125<br />
</td>
        <td>112.057<br />
</td>
        <td>minor link<br />
</td>
        <td>b6 &amp; b17<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=4p_11cd&amp;limit=3_5_7" rel="nofollow">4p &amp; 11cd</a><br />
</td>
        <td>5.649<br />
</td>
        <td>3.437<br />
</td>
        <td>[&lt; 6 9 11 ], &lt; 17 25 30 ]&gt;<br />
</td>
        <td>1898.4178<br />
</td>
        <td>333.298<br />
</td>
        <td>1<br />
</td>
        <td>2,3,4,5*,6*,11,17*,23,40,57*,74,131*<br />
</td>
    </tr>
    <tr>
        <td>Vega<br />
</td>
        <td>| 8 -3 -2 &gt;<br />
</td>
        <td style="text-align: center;">6561/6125<br />
</td>
        <td>119.047<br />
</td>
        <td>major link<br />
</td>
        <td>b4 &amp; b15<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=3bcdd_10bccdd&amp;limit=3_5_7" rel="nofollow">3bcdd &amp; 10bccdd</a><br />
</td>
        <td>4.377<br />
</td>
        <td>4.713<br />
</td>
        <td>[&lt; 4 6 7 ], &lt; 15 22 27 ]&gt;<br />
</td>
        <td>1892.0204<br />
</td>
        <td>494.8658<br />
</td>
        <td>1<br />
</td>
        <td>2,3*,4*,7,11,15,19*,23*,42*,65*<br />
</td>
    </tr>
    <tr>
        <td>*Achernar<br />
</td>
        <td>| 0 -6 5 &gt;<br />
</td>
        <td style="text-align: center;">16807/15625<br />
</td>
        <td>126.247<br />
</td>
        <td><br />
</td>
        <td>b7 &amp; b17<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=4bcd_11cd&amp;limit=3_5_7" rel="nofollow">4bcd &amp; 11cd</a><br />
</td>
        <td>4.847<br />
</td>
        <td>4.514<br />
</td>
        <td>[&lt; 7 10 12 ], &lt; 17 25 30 ]&gt;<br />
</td>
        <td>1901.955<br />
</td>
        <td>559.351<br />
</td>
        <td>1<br />
</td>
        <td>2,3*,4,7*,10,17*<br />
</td>
    </tr>
    <tr>
        <td>%<br />
</td>
        <td>| 3 -2 0 &gt;<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/27_25">27/25</a><br />
</td>
        <td>133.238<br />
</td>
        <td>large limma, BP small semitone<br />
</td>
        <td>b4 &amp; b6<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=b4+b6&amp;limit=3_5_7" rel="nofollow">3bcdd &amp; 4p</a><br />
</td>
        <td>1.629<br />
</td>
        <td>14.176<br />
</td>
        <td>[&lt; 4 6 7 ], &lt; 6 9 11 ]&gt;<br />
</td>
        <td>939.6121<br />
</td>
        <td>549.9897<br />
</td>
        <td>2<br />
</td>
        <td>4*,6,10*,14,24*,34,58,82*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| 6 2 -5 &gt;<br />
</td>
        <td style="text-align: center;">18225/16807<br />
</td>
        <td>140.228<br />
</td>
        <td>minimal BP chroma<br />
</td>
        <td>b9 &amp; b10<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=6bcd_6ccdd&amp;limit=3_5_7" rel="nofollow">6bcd &amp; 6ccdd</a><br />
</td>
        <td>4.885<br />
</td>
        <td>4.974<br />
</td>
        <td>[&lt; 9 13 16 ], &lt; 10 15 18 ]&gt;<br />
</td>
        <td>1895.1156<br />
</td>
        <td>201.762<br />
</td>
        <td>1<br />
</td>
        <td>...9*,10,19*,28*,47*<br />
</td>
    </tr>
    <tr>
        <td>%<br />
</td>
        <td>| -2 -1 2 &gt;<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/49_45">49/45</a><br />
</td>
        <td>147.428<br />
</td>
        <td>BP minor semitone, <a class="wiki_link" href="/Swetismic%20temperaments">swetismic</a> neutral second<br />
</td>
        <td>b4 &amp; b7<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=b4+b7&amp;limit=3_5_7" rel="nofollow">3bcdd &amp; 4bcd</a><br />
</td>
        <td>1.925<br />
</td>
        <td>13.274<br />
</td>
        <td>[&lt; 4 6 7 ], &lt; 7 10 12 ]&gt;<br />
</td>
        <td>1917.7266<br />
</td>
        <td>516.1077<br />
</td>
        <td>1<br />
</td>
        <td>2,3*,4*,7,11*,15*,26*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| 1 3 -3 &gt;<br />
</td>
        <td style="text-align: center;">375/343<br />
</td>
        <td>154.418<br />
</td>
        <td>BP major semitone, minor BP chroma<br />
</td>
        <td>b3 &amp; b6<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=2cd_4p&amp;limit=3_5_7" rel="nofollow">2cd &amp; 4p</a><br />
</td>
        <td>2.941<br />
</td>
        <td>9.098<br />
</td>
        <td>[&lt; 3 4 5 ], &lt; 6 9 11 ]&gt;<br />
</td>
        <td>632.9248<br />
</td>
        <td>234.1367<br />
</td>
        <td>3<br />
</td>
        <td>6*,9*,15,24*,33,57*,81*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| -7 0 4 &gt;<br />
</td>
        <td style="text-align: center;">2401/2187<br />
</td>
        <td>161.619<br />
</td>
        <td><br />
</td>
        <td>b4 &amp; b40cd<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=b4+b25&amp;limit=3_5_7" rel="nofollow">3bcdd &amp; 25p</a><br />
</td>
        <td>3.868<br />
</td>
        <td>7.240<br />
</td>
        <td>[&lt; 4 6 7 ], &lt; 40 58 70 ]&gt;<br />
</td>
        <td>478.34<br />
</td>
        <td>41.8631<br />
</td>
        <td>4<br />
</td>
        <td>...44*,48,92*,136,228*,320*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| -4 4 -1 &gt;<br />
</td>
        <td style="text-align: center;">625/567<br />
</td>
        <td>168.609<br />
</td>
        <td>BP great semitone, major BP chroma<br />
</td>
        <td>b2 &amp; b7<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=1cdd_4bcd&amp;limit=3_5_7" rel="nofollow">1cdd &amp; 4bcd</a><br />
</td>
        <td>3.195<br />
</td>
        <td>9.146<br />
</td>
        <td>[&lt; 2 3 4 ], &lt; 7 10 12 ]&gt;<br />
</td>
        <td>1914.567<br />
</td>
        <td>1069.8876<br />
</td>
        <td>1<br />
</td>
        <td>2*,3,5,7*,9*,16,25,34*<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>| -9 5 1 &gt;<br />
</td>
        <td style="text-align: center;">21875/19683<br />
</td>
        <td>182.799<br />
</td>
        <td>maximal BP chroma<br />
</td>
        <td>b9 &amp; b11<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=6bcd_7d&amp;limit=3_5_7" rel="nofollow">6bcd &amp; 7d</a><br />
</td>
        <td>4.970<br />
</td>
        <td>6.373<br />
</td>
        <td>[&lt; 9 13 16 ], &lt; 11 16 19 ]&gt;<br />
</td>
        <td>1913.8948<br />
</td>
        <td>858.1831<br />
</td>
        <td>1<br />
</td>
        <td>2*,3,5,7,9*,11,20*,29*<br />
</td>
    </tr>
    <tr>
        <td>%<br />
</td>
        <td>| 9 0 -5 &gt;<br />
</td>
        <td style="text-align: center;">19683/16807<br />
</td>
        <td>273.465<br />
</td>
        <td><br />
</td>
        <td>b5 &amp; b10<br />
</td>
        <td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=3d_6ccdd&amp;limit=3_5_7" rel="nofollow">3d &amp; 6ccdd</a><br />
</td>
        <td>4.904<br />
</td>
        <td>9.663<br />
</td>
        <td>[&lt; 5 7 9 ], &lt; 10 15 18 ]&gt;<br />
</td>
        <td>377.3036<br />
</td>
        <td>145.1888<br />
</td>
        <td>5<br />
</td>
        <td>10*,15*,25*,40,65*<br />
</td>
    </tr>
</table>

<br />
names: * = proposed<br />
names: % = not shown on PTS plots<br />
MOS: * = maximally even<br />
<br />
complexity = ln(n*d)/4<br />
damage = 1200 *log(n/d) / log(n*d)<br />
(complexity and damage are calculated as in the &quot;Middle Path&quot; tables)<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Graphs"></a><!-- ws:end:WikiTextHeadingRule:2 -->Graphs</h1>
 <br />
<!-- ws:start:WikiTextLocalImageRule:889:&lt;img src=&quot;/file/view/357plot_cplx_damage.png/605206705/357plot_cplx_damage.png&quot; alt=&quot;Complexity vs. damage plot.  z&amp;lt;1 corresponds to the &amp;quot;Middle Path&amp;quot; inclusion criterion.&quot; title=&quot;Complexity vs. damage plot.  z&amp;lt;1 corresponds to the &amp;quot;Middle Path&amp;quot; inclusion criterion.&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/357plot_cplx_damage.png/605206705/357plot_cplx_damage.png" alt="357plot_cplx_damage.png" title="357plot_cplx_damage.png" /></td></tr><tr><td class="imageCaption">Complexity vs. damage plot.  z&lt;1 corresponds to the &quot;Middle Path&quot; inclusion criterion.</td></tr></table><!-- ws:end:WikiTextLocalImageRule:889 --><br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Projective tuning space diagrams"></a><!-- ws:end:WikiTextHeadingRule:4 -->Projective tuning space diagrams</h1>
 <br />
<!-- ws:start:WikiTextLocalImageRule:891:&lt;a href=&quot;/file/view/357ptslines1n.png&quot;&gt;&lt;img src=&quot;/file/view/tn_357ptslines1n.png/604981805/tn_357ptslines1n.png&quot; alt=&quot;Temperaments with smaller commas, labeled by name&quot; title=&quot;Temperaments with smaller commas, labeled by name&quot; /&gt;&lt;/a&gt; --><table class="captionBox"><tr><td class="captionedImage"><a href="/file/view/357ptslines1n.png"><img src="/file/view/tn_357ptslines1n.png/604981805/tn_357ptslines1n.png" alt="tn_357ptslines1n.png" title="tn_357ptslines1n.png" /></a></td></tr><tr><td class="imageCaption">Temperaments with smaller commas, labeled by name</td></tr></table><!-- ws:end:WikiTextLocalImageRule:891 --><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:893:&lt;a href=&quot;/file/view/357ptslines1c.png&quot;&gt;&lt;img src=&quot;/file/view/tn_357ptslines1c.png/604981803/tn_357ptslines1c.png&quot; alt=&quot;Temperaments with smaller commas, labeled by comma&quot; title=&quot;Temperaments with smaller commas, labeled by comma&quot; /&gt;&lt;/a&gt; --><table class="captionBox"><tr><td class="captionedImage"><a href="/file/view/357ptslines1c.png"><img src="/file/view/tn_357ptslines1c.png/604981803/tn_357ptslines1c.png" alt="tn_357ptslines1c.png" title="tn_357ptslines1c.png" /></a></td></tr><tr><td class="imageCaption">Temperaments with smaller commas, labeled by comma</td></tr></table><!-- ws:end:WikiTextLocalImageRule:893 --><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:895:&lt;a href=&quot;/file/view/357ptslines2n.png&quot;&gt;&lt;img src=&quot;/file/view/tn_357ptslines2n.png/604981813/tn_357ptslines2n.png&quot; alt=&quot;Temperaments with larger commas, labeled by name&quot; title=&quot;Temperaments with larger commas, labeled by name&quot; /&gt;&lt;/a&gt; --><table class="captionBox"><tr><td class="captionedImage"><a href="/file/view/357ptslines2n.png"><img src="/file/view/tn_357ptslines2n.png/604981813/tn_357ptslines2n.png" alt="tn_357ptslines2n.png" title="tn_357ptslines2n.png" /></a></td></tr><tr><td class="imageCaption">Temperaments with larger commas, labeled by name</td></tr></table><!-- ws:end:WikiTextLocalImageRule:895 --><br />
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<!-- ws:start:WikiTextLocalImageRule:897:&lt;a href=&quot;/file/view/357ptslines2c.png&quot;&gt;&lt;img src=&quot;/file/view/tn_357ptslines2c.png/604981811/tn_357ptslines2c.png&quot; alt=&quot;Temperaments with larger commas, labeled by comma&quot; title=&quot;Temperaments with larger commas, labeled by comma&quot; /&gt;&lt;/a&gt; --><table class="captionBox"><tr><td class="captionedImage"><a href="/file/view/357ptslines2c.png"><img src="/file/view/tn_357ptslines2c.png/604981811/tn_357ptslines2c.png" alt="tn_357ptslines2c.png" title="tn_357ptslines2c.png" /></a></td></tr><tr><td class="imageCaption">Temperaments with larger commas, labeled by comma</td></tr></table><!-- ws:end:WikiTextLocalImageRule:897 --><br />
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<!-- ws:start:WikiTextLocalImageRule:899:&lt;a href=&quot;/file/view/357ptslines12n.png&quot;&gt;&lt;img src=&quot;/file/view/tn_357ptslines12n.png/604981817/tn_357ptslines12n.png&quot; alt=&quot;Both sets, labeled by name&quot; title=&quot;Both sets, labeled by name&quot; /&gt;&lt;/a&gt; --><table class="captionBox"><tr><td class="captionedImage"><a href="/file/view/357ptslines12n.png"><img src="/file/view/tn_357ptslines12n.png/604981817/tn_357ptslines12n.png" alt="tn_357ptslines12n.png" title="tn_357ptslines12n.png" /></a></td></tr><tr><td class="imageCaption">Both sets, labeled by name</td></tr></table><!-- ws:end:WikiTextLocalImageRule:899 --><br />
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<!-- ws:start:WikiTextLocalImageRule:901:&lt;a href=&quot;/file/view/357ptslines12c.png&quot;&gt;&lt;img src=&quot;/file/view/tn_357ptslines12c.png/604981815/tn_357ptslines12c.png&quot; alt=&quot;Both sets, labeled by comma&quot; title=&quot;Both sets, labeled by comma&quot; /&gt;&lt;/a&gt; --><table class="captionBox"><tr><td class="captionedImage"><a href="/file/view/357ptslines12c.png"><img src="/file/view/tn_357ptslines12c.png/604981815/tn_357ptslines12c.png" alt="tn_357ptslines12c.png" title="tn_357ptslines12c.png" /></a></td></tr><tr><td class="imageCaption">Both sets, labeled by comma</td></tr></table><!-- ws:end:WikiTextLocalImageRule:901 --><br />
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        <td>11.675<br />
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