Gallery of projective tuning space images: Difference between revisions

Revision as of 11:55, 1 September 2014

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[[toc]]
=5-limit=

|| [[image:tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg link="file:pts-2-3-5-e2-amp-hand-annotated-gray.jpg]"]] || {2,3,5} PTS with handwritten annotations
from [[file:Erlich-MiddlePath.pdf|A Middle Path]] ||

|| [[image:tn_pts-2-3-5-e2-twtop.jpg link="file:pts-2-3-5-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150090296739367&set=gm.10150170038074482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-04)]]
projective tuning space based on {2,3,5}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||

|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo4.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247244367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
zoomed waaaay out ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo3.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247269367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
zoomed-out still more ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo2.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247314367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
zoomed-out more ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo1.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247409367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
zoomed-out ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin.jpg link="file:pts-2-3-5-e2-twtop-tlin.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152310913269367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)]]
5-limit projective tuning space showing equal temperaments and good rank-two temperaments (or "temperament families"). Concentric hexagons indicate TOP damage (of TOP tuning) up to 10 in increments of 1. ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg link="file:pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152640895264367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
zoomed-in ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg link="file:pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152640895159367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
zoomed-in more ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi3.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152640895094367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
zoomed-in still more ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi4.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152641282619367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
zoomed way in ||
|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi5.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152641282779367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
zoomed way way in ||

|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z0.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397394367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 0 ||
|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z1.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397459367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 1 ||
|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z2.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397549367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 2 ||
|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z3.jpg"]] ||[[@https://www.facebook.com/photo.php?fbid=10152643397614367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance  - Mathematical Theory (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 3 ||
|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z4.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397709367&set=oa.389536071186621&type=1&permPage=1
|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 4 ||
|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z5.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397784367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 5 ||
|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z6.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397899367&set=oa.389536071186621&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 6 ||
|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z7.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397989367&set=oa.389536071186621&type=1&permPage=1
|Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 7 ||

=11-limit subgroups=

|| [[image:tn_pts-2-3-7-e2-twtop-z0.jpg link="file:pts-2-3-7-e2-twtop-z0.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091440809367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
projective tuning space based on {2,3,7}
zoomed way out!  ||
|| [[image:tn_pts-2-3-9-e2-twtop.jpg link="file:pts-2-3-9-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091485324367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
"projective tuning space" "based on" "{2,3,9}"
numerals at "ETs" represent steps per '2'
hexagons are "contours" of "maximum Tenney-weighted damage" of "TOP tunings"

this one is not like the others! 9/3 is considered a different interval than 3, with complexity log(27) rather than log(3), and susceptibility to damage 1/log(27) instead of 1/log(3) -- whatever that means!  ||
|| [[image:tn_pts-2-3-11-e2-twtop.jpg link="file:pts-2-3-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091496424367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
projective tuning space based on {2,3,11}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
|| [[image:tn_pts-2-5-7-e2-twtop.jpg link="file:pts-2-5-7-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150090453309367&set=gm.10150170038074482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-04)]]
projective tuning space based on {2,5,7}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings  ||
|| [[image:tn_pts-2-5-9-e2-twtop.jpg link="file:pts-2-5-9-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091482029367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
projective tuning space based on {2,5,9}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
|| [[image:tn_pts-2-5-11-e2-twtop.jpg link="file:pts-2-5-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091500519367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
projective tuning space based on {2,5,11}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
|| [[image:tn_pts-2-7-9-e2-twtop.jpg link="file:pts-2-7-9-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091540804367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
projective tuning space based on {2,7,9}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
|| [[image:tn_pts-2-7-11-e2-twtop.jpg link="file:pts-2-7-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091506389367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
projective tuning space based on {2,7,11}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
|| [[image:tn_pts-2-9-11-e2-twtop.jpg link="file:pts-2-9-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091508499367&set=gm.10150170037959482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)]]
projective tuning space based on {2,9,11}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||

=3.5.7, tritave (3-1) based=

|| [[image:tn_pts-3-5-7-e3-twtop.jpg link="file:pts-3-5-7-e3-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150090296744367&set=gm.10150170038074482&type=1&permPage=1|Paul Erlich > ‎The Xenharmonic Alliance (2011-02-04)]]
projective tuning space based on {3,5,7}
numerals at ETs represent steps per '3'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||

[[media type="custom" key="26495050"]]

Original HTML content:

<html><head><title>Gallery of projective tuning space images</title></head><body><!-- ws:start:WikiTextTocRule:7:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --><div style="margin-left: 1em;"><a href="#x5-limit">5-limit</a></div>
<!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><div style="margin-left: 1em;"><a href="#x11-limit subgroups">11-limit subgroups</a></div>
<!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --><div style="margin-left: 1em;"><a href="#x3.5.7, tritave (3-1) based">3.5.7, tritave (3-1) based</a></div>
<!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --></div>
<!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="x5-limit"></a><!-- ws:end:WikiTextHeadingRule:1 -->5-limit</h1>
<br />


<table class="wiki_table">
    <tr>
        <td><!-- ws:start:WikiTextLocalImageRule:205:&lt;a href=&quot;/file/view/pts-2-3-5-e2-amp-hand-annotated-gray.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg/520336302/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-amp-hand-annotated-gray.jpg"><img src="/file/view/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg/520336302/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" alt="tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" title="tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:205 --><br />
</td>
        <td>{2,3,5} PTS with handwritten annotations<br />
from <div class="objectEmbed"><img src="/i/file_not_found.png" width="32" height="32" alt="File Not Found" /><div class="notfound"><b>File Not Found</b></div></div><br />
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<table class="wiki_table">
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        <td><!-- ws:start:WikiTextLocalImageRule:207:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop.jpg/520338504/tn_pts-2-3-5-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop.jpg/520338504/tn_pts-2-3-5-e2-twtop.jpg" alt="tn_pts-2-3-5-e2-twtop.jpg" title="tn_pts-2-3-5-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:207 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296739367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)</a><br />
projective tuning space based on {2,3,5}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
</table>

<br />


<table class="wiki_table">
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        <td><!-- ws:start:WikiTextLocalImageRule:209:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo4.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg/520338468/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo4.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg/520338468/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:209 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247244367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed waaaay out<br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247269367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out still more<br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247314367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out more<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:215:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo1.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg/520338424/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo1.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg/520338424/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:215 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247409367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:217:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop-tlin.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin.jpg/520338490/tn_pts-2-3-5-e2-twtop-tlin.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop-tlin.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin.jpg/520338490/tn_pts-2-3-5-e2-twtop-tlin.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin.jpg" title="tn_pts-2-3-5-e2-twtop-tlin.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:217 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152310913269367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)</a><br />
5-limit projective tuning space showing equal temperaments and good rank-two temperaments (or &quot;temperament families&quot;). Concentric hexagons indicate TOP damage (of TOP tuning) up to 10 in increments of 1.<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:219:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg/520338370/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg/520338370/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:219 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895264367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed-in<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:221:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg/520338380/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg/520338380/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:221 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895159367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed-in more<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:223:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop-tlin-zi3.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg/520341482/tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-zi3.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg/520341482/tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:223 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895094367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed-in still more<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:225:&lt;a href=&quot;/file/view/pts-2-3-5-e2-twtop-tlin-zi4.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg/520341488/tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-zi4.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg/520341488/tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:225 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152641282619367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed way in<br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152641282779367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed way way in<br />
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<table class="wiki_table">
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        <td><!-- ws:start:WikiTextLocalImageRule:229:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z0.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg/520342320/tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z0.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg/520342320/tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:229 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397394367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 0<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:231:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z1.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg/520338284/tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z1.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg/520338284/tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:231 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397459367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 1<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:233:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z2.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg/520338296/tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z2.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg/520338296/tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:233 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397549367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 2<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:235:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z3.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg/520338304/tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z3.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg/520338304/tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:235 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397614367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance  - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 3<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:237:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z4.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg/520338322/tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z4.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg/520338322/tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:237 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397709367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 4<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:239:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z5.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg/520338338/tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z5.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg/520338338/tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:239 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397784367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 5<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:241:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z6.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg/520338350/tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z6.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg/520338350/tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:241 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397899367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 6<br />
</td>
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        <td><!-- ws:start:WikiTextLocalImageRule:243:&lt;a href=&quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z7.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg/520338364/tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z7.jpg"><img src="/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg/520338364/tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg" alt="tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg" title="tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:243 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397989367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 7<br />
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<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="x11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:3 -->11-limit subgroups</h1>
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<table class="wiki_table">
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        <td><!-- ws:start:WikiTextLocalImageRule:245:&lt;a href=&quot;/file/view/pts-2-3-7-e2-twtop-z0.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-7-e2-twtop-z0.jpg/520338520/tn_pts-2-3-7-e2-twtop-z0.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-7-e2-twtop-z0.jpg"><img src="/file/view/tn_pts-2-3-7-e2-twtop-z0.jpg/520338520/tn_pts-2-3-7-e2-twtop-z0.jpg" alt="tn_pts-2-3-7-e2-twtop-z0.jpg" title="tn_pts-2-3-7-e2-twtop-z0.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:245 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091440809367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,3,7}<br />
zoomed way out!<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:247:&lt;a href=&quot;/file/view/pts-2-3-9-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-9-e2-twtop.jpg/520338534/tn_pts-2-3-9-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-9-e2-twtop.jpg"><img src="/file/view/tn_pts-2-3-9-e2-twtop.jpg/520338534/tn_pts-2-3-9-e2-twtop.jpg" alt="tn_pts-2-3-9-e2-twtop.jpg" title="tn_pts-2-3-9-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:247 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091485324367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
&quot;projective tuning space&quot; &quot;based on&quot; &quot;{2,3,9}&quot;<br />
numerals at &quot;ETs&quot; represent steps per '2'<br />
hexagons are &quot;contours&quot; of &quot;maximum Tenney-weighted damage&quot; of &quot;TOP tunings&quot;<br />
<br />
this one is not like the others! 9/3 is considered a different interval than 3, with complexity log(27) rather than log(3), and susceptibility to damage 1/log(27) instead of 1/log(3) -- whatever that means!<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:249:&lt;a href=&quot;/file/view/pts-2-3-11-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-3-11-e2-twtop.jpg/520338558/tn_pts-2-3-11-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-3-11-e2-twtop.jpg"><img src="/file/view/tn_pts-2-3-11-e2-twtop.jpg/520338558/tn_pts-2-3-11-e2-twtop.jpg" alt="tn_pts-2-3-11-e2-twtop.jpg" title="tn_pts-2-3-11-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:249 --><br />
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        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091496424367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,3,11}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
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        <td><!-- ws:start:WikiTextLocalImageRule:251:&lt;a href=&quot;/file/view/pts-2-5-7-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-5-7-e2-twtop.jpg/520338576/tn_pts-2-5-7-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-5-7-e2-twtop.jpg"><img src="/file/view/tn_pts-2-5-7-e2-twtop.jpg/520338576/tn_pts-2-5-7-e2-twtop.jpg" alt="tn_pts-2-5-7-e2-twtop.jpg" title="tn_pts-2-5-7-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:251 --><br />
</td>
        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090453309367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)</a><br />
projective tuning space based on {2,5,7}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
    <tr>
        <td><!-- ws:start:WikiTextLocalImageRule:253:&lt;a href=&quot;/file/view/pts-2-5-9-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-5-9-e2-twtop.jpg/520338594/tn_pts-2-5-9-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-5-9-e2-twtop.jpg"><img src="/file/view/tn_pts-2-5-9-e2-twtop.jpg/520338594/tn_pts-2-5-9-e2-twtop.jpg" alt="tn_pts-2-5-9-e2-twtop.jpg" title="tn_pts-2-5-9-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:253 --><br />
</td>
        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091482029367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,5,9}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
    <tr>
        <td><!-- ws:start:WikiTextLocalImageRule:255:&lt;a href=&quot;/file/view/pts-2-5-11-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-5-11-e2-twtop.jpg/520338602/tn_pts-2-5-11-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-5-11-e2-twtop.jpg"><img src="/file/view/tn_pts-2-5-11-e2-twtop.jpg/520338602/tn_pts-2-5-11-e2-twtop.jpg" alt="tn_pts-2-5-11-e2-twtop.jpg" title="tn_pts-2-5-11-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:255 --><br />
</td>
        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091500519367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,5,11}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
    <tr>
        <td><!-- ws:start:WikiTextLocalImageRule:257:&lt;a href=&quot;/file/view/pts-2-7-9-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-7-9-e2-twtop.jpg/520338622/tn_pts-2-7-9-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-7-9-e2-twtop.jpg"><img src="/file/view/tn_pts-2-7-9-e2-twtop.jpg/520338622/tn_pts-2-7-9-e2-twtop.jpg" alt="tn_pts-2-7-9-e2-twtop.jpg" title="tn_pts-2-7-9-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:257 --><br />
</td>
        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091540804367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,7,9}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
    <tr>
        <td><!-- ws:start:WikiTextLocalImageRule:259:&lt;a href=&quot;/file/view/pts-2-7-11-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-7-11-e2-twtop.jpg/520338634/tn_pts-2-7-11-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-7-11-e2-twtop.jpg"><img src="/file/view/tn_pts-2-7-11-e2-twtop.jpg/520338634/tn_pts-2-7-11-e2-twtop.jpg" alt="tn_pts-2-7-11-e2-twtop.jpg" title="tn_pts-2-7-11-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:259 --><br />
</td>
        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091506389367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,7,11}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
    <tr>
        <td><!-- ws:start:WikiTextLocalImageRule:261:&lt;a href=&quot;/file/view/pts-2-9-11-e2-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-2-9-11-e2-twtop.jpg/520338644/tn_pts-2-9-11-e2-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-2-9-11-e2-twtop.jpg"><img src="/file/view/tn_pts-2-9-11-e2-twtop.jpg/520338644/tn_pts-2-9-11-e2-twtop.jpg" alt="tn_pts-2-9-11-e2-twtop.jpg" title="tn_pts-2-9-11-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:261 --><br />
</td>
        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091508499367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,9,11}<br />
numerals at ETs represent steps per '2'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="x3.5.7, tritave (3-1) based"></a><!-- ws:end:WikiTextHeadingRule:5 -->3.5.7, tritave (3-1) based</h1>
<br />


<table class="wiki_table">
    <tr>
        <td><!-- ws:start:WikiTextLocalImageRule:263:&lt;a href=&quot;/file/view/pts-3-5-7-e3-twtop.jpg&quot;&gt;&lt;img src=&quot;/file/view/tn_pts-3-5-7-e3-twtop.jpg/520338662/tn_pts-3-5-7-e3-twtop.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/a&gt; --><a href="/file/view/pts-3-5-7-e3-twtop.jpg"><img src="/file/view/tn_pts-3-5-7-e3-twtop.jpg/520338662/tn_pts-3-5-7-e3-twtop.jpg" alt="tn_pts-3-5-7-e3-twtop.jpg" title="tn_pts-3-5-7-e3-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:263 --><br />
</td>
        <td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296744367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1" rel="nofollow" target="_blank">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)</a><br />
projective tuning space based on {3,5,7}<br />
numerals at ETs represent steps per '3'<br />
hexagons are contours of maximum Tenney-weighted damage of TOP tunings<br />
</td>
    </tr>
</table>

<br />
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@@ Line 1: / Line 1: @@
 <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:spt3125|spt3125]] and made on <tt>2014-09-01 11:53:21 UTC</tt>.<br>
+: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-09-01 11:55:52 UTC</tt>.<br>
-: The original revision id was <tt>520347754</tt>.<br>
+: The original revision id was <tt>520347958</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>
@@ Line 9: / Line 9: @@
 =5-limit=
-|| [[image:tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg link="file:pts-2-3-5-e2-amp-hand-annotated-gray.jpg]"]] || {2,3,5} PTS with handwritten annotations, from [[file:Erlich-MiddlePath.pdf|A Middle Path]] ||
+|| [[image:tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg link="file:pts-2-3-5-e2-amp-hand-annotated-gray.jpg]"]] || {2,3,5} PTS with handwritten annotations
+from [[file:Erlich-MiddlePath.pdf|A Middle Path]] ||
-|| [[image:tn_pts-2-3-5-e2-twtop.jpg link="file:pts-2-3-5-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150090296739367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]]
+|| [[image:tn_pts-2-3-5-e2-twtop.jpg link="file:pts-2-3-5-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150090296739367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]]
 projective tuning space based on {2,3,5}
 numerals at ETs represent steps per '2'
 hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo4.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152632247244367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo4.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247244367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
 zoomed waaaay out ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo3.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152632247269367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo3.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247269367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
 zoomed-out still more ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo2.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152632247314367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo2.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247314367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
 zoomed-out more ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo1.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152632247409367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg link="file:pts-2-3-5-e2-twtop-tlin-zo1.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152632247409367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]]
 zoomed-out ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin.jpg link="file:pts-2-3-5-e2-twtop-tlin.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152310913269367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin.jpg link="file:pts-2-3-5-e2-twtop-tlin.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152310913269367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)]]
 -limit projective tuning space showing equal temperaments and good rank-two temperaments (or "temperament families"). Concentric hexagons indicate TOP damage (of TOP tuning) up to 10 in increments of 1. ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg link="file:pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152640895264367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg link="file:pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152640895264367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
 zoomed-in ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg link="file:pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152640895159367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg link="file:pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152640895159367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
 zoomed-in more ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi3.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152640895094367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi3.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152640895094367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
 zoomed-in still more ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi4.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152641282619367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi4.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152641282619367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
 zoomed way in ||
-|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi5.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152641282779367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
+|| [[image:tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg link="file:pts-2-3-5-e2-twtop-tlin-zi5.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152641282779367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
 zoomed way way in ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z0.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152643397394367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z0.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397394367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
 showing all three elements of each 5-limit breed
 zoom level 0 ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z1.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152643397459367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]]
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z1.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397459367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]]
 showing all three elements of each 5-limit breed
 zoom level 1 ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z2.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152643397549367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]]
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z2.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397549367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]]
 showing all three elements of each 5-limit breed
 zoom level 2 ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z3.jpg"]] ||[[https://www.facebook.com/photo.php?fbid=10152643397614367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance  - Mathematical Theory (2014-08-26)]]
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z3.jpg"]] ||[[@https://www.facebook.com/photo.php?fbid=10152643397614367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance  - Mathematical Theory (2014-08-26)]]
 showing all three elements of each 5-limit breed
 zoom level 3 ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z4.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152643397709367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z4.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397709367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1
 |Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
 showing all three elements of each 5-limit breed
 zoom level 4 ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z5.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152643397784367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z5.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397784367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
 showing all three elements of each 5-limit breed
 zoom level 5 ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z6.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152643397899367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z6.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397899367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
 showing all three elements of each 5-limit breed
 zoom level 6 ||
-|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z7.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10152643397989367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1
+|| [[image:tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg link="file:pts-2-3-5-e2-breed-twtop-tlin-z7.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10152643397989367&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1
 |Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
 showing all three elements of each 5-limit breed
@@ Line 66: / Line 67: @@
 =11-limit subgroups=
-|| [[image:tn_pts-2-3-7-e2-twtop-z0.jpg link="file:pts-2-3-7-e2-twtop-z0.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091440809367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-3-7-e2-twtop-z0.jpg link="file:pts-2-3-7-e2-twtop-z0.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091440809367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 projective tuning space based on {2,3,7}
 zoomed way out!  ||
-|| [[image:tn_pts-2-3-9-e2-twtop.jpg link="file:pts-2-3-9-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091485324367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-3-9-e2-twtop.jpg link="file:pts-2-3-9-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091485324367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 "projective tuning space" "based on" "{2,3,9}"
 numerals at "ETs" represent steps per '2'
@@ Line 75: / Line 76: @@
 this one is not like the others! 9/3 is considered a different interval than 3, with complexity log(27) rather than log(3), and susceptibility to damage 1/log(27) instead of 1/log(3) -- whatever that means!  ||
-|| [[image:tn_pts-2-3-11-e2-twtop.jpg link="file:pts-2-3-11-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091496424367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-3-11-e2-twtop.jpg link="file:pts-2-3-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091496424367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 projective tuning space based on {2,3,11}
 numerals at ETs represent steps per '2'
 hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
-|| [[image:tn_pts-2-5-7-e2-twtop.jpg link="file:pts-2-5-7-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150090453309367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]]
+|| [[image:tn_pts-2-5-7-e2-twtop.jpg link="file:pts-2-5-7-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150090453309367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]]
 projective tuning space based on {2,5,7}
 numerals at ETs represent steps per '2'
 hexagons are contours of maximum Tenney-weighted damage of TOP tunings  ||
-|| [[image:tn_pts-2-5-9-e2-twtop.jpg link="file:pts-2-5-9-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091482029367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-5-9-e2-twtop.jpg link="file:pts-2-5-9-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091482029367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 projective tuning space based on {2,5,9}
 numerals at ETs represent steps per '2'
 hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
-|| [[image:tn_pts-2-5-11-e2-twtop.jpg link="file:pts-2-5-11-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091500519367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-5-11-e2-twtop.jpg link="file:pts-2-5-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091500519367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 projective tuning space based on {2,5,11}
 numerals at ETs represent steps per '2'
 hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
-|| [[image:tn_pts-2-7-9-e2-twtop.jpg link="file:pts-2-7-9-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091540804367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-7-9-e2-twtop.jpg link="file:pts-2-7-9-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091540804367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 projective tuning space based on {2,7,9}
 numerals at ETs represent steps per '2'
 hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
-|| [[image:tn_pts-2-7-11-e2-twtop.jpg link="file:pts-2-7-11-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091506389367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-7-11-e2-twtop.jpg link="file:pts-2-7-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091506389367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 projective tuning space based on {2,7,11}
 numerals at ETs represent steps per '2'
 hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
-|| [[image:tn_pts-2-9-11-e2-twtop.jpg link="file:pts-2-9-11-e2-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150091508499367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
+|| [[image:tn_pts-2-9-11-e2-twtop.jpg link="file:pts-2-9-11-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150091508499367&amp;set=gm.10150170037959482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]]
 projective tuning space based on {2,9,11}
 numerals at ETs represent steps per '2'
@@ Line 106: / Line 107: @@
 =3.5.7, tritave (3-1) based=
-|| [[image:tn_pts-3-5-7-e3-twtop.jpg link="file:pts-3-5-7-e3-twtop.jpg"]] || [[https://www.facebook.com/photo.php?fbid=10150090296744367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]]
+|| [[image:tn_pts-3-5-7-e3-twtop.jpg link="file:pts-3-5-7-e3-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150090296744367&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]]
 projective tuning space based on {3,5,7}
 numerals at ETs represent steps per '3'
@@ Line 126: / Line 127: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:205:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-amp-hand-annotated-gray.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg/520336302/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-amp-hand-annotated-gray.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg/520336302/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" alt="tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" title="tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:205 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;{2,3,5} PTS with handwritten annotations, from &lt;div class="objectEmbed"&gt;&lt;img src="/i/file_not_found.png" width="32" height="32" alt="File Not Found" /&gt;&lt;div class="notfound"&gt;&lt;b&gt;File Not Found&lt;/b&gt;&lt;/div&gt;&lt;/div&gt;&lt;br /&gt;
+         &lt;td&gt;{2,3,5} PTS with handwritten annotations&lt;br /&gt;
+from &lt;div class="objectEmbed"&gt;&lt;img src="/i/file_not_found.png" width="32" height="32" alt="File Not Found" /&gt;&lt;div class="notfound"&gt;&lt;b&gt;File Not Found&lt;/b&gt;&lt;/div&gt;&lt;/div&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 138: / Line 140: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:207:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop.jpg/520338504/tn_pts-2-3-5-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop.jpg/520338504/tn_pts-2-3-5-e2-twtop.jpg" alt="tn_pts-2-3-5-e2-twtop.jpg" title="tn_pts-2-3-5-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:207 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296739367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-04)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296739367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-04)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,3,5}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 153: / Line 155: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:209:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo4.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg/520338468/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo4.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg/520338468/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:209 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247244367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247244367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
 zoomed waaaay out&lt;br /&gt;
 &lt;/td&gt;
@@ Line 160: / Line 162: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:211:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo3.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg/520338452/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo3.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg/520338452/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:211 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247269367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247269367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
 zoomed-out still more&lt;br /&gt;
 &lt;/td&gt;
@@ Line 167: / Line 169: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:213:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo2.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg/520338442/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo2.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg/520338442/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:213 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247314367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247314367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
 zoomed-out more&lt;br /&gt;
 &lt;/td&gt;
@@ Line 174: / Line 176: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:215:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo1.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg/520338424/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo1.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg/520338424/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:215 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247409367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247409367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)&lt;/a&gt;&lt;br /&gt;
 zoomed-out&lt;br /&gt;
 &lt;/td&gt;
@@ Line 181: / Line 183: @@
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 &lt;/td&gt;
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152310913269367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)&lt;/a&gt;&lt;br /&gt;
 -limit projective tuning space showing equal temperaments and good rank-two temperaments (or &amp;quot;temperament families&amp;quot;). Concentric hexagons indicate TOP damage (of TOP tuning) up to 10 in increments of 1.&lt;br /&gt;
 &lt;/td&gt;
@@ Line 188: / Line 190: @@
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895264367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)&lt;/a&gt;&lt;br /&gt;
 zoomed-in&lt;br /&gt;
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@@ Line 195: / Line 197: @@
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 &lt;/td&gt;
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895159367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)&lt;/a&gt;&lt;br /&gt;
 zoomed-in more&lt;br /&gt;
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@@ Line 202: / Line 204: @@
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895094367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)&lt;/a&gt;&lt;br /&gt;
 zoomed-in still more&lt;br /&gt;
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@@ Line 209: / Line 211: @@
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152641282619367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)&lt;/a&gt;&lt;br /&gt;
 zoomed way in&lt;br /&gt;
 &lt;/td&gt;
@@ Line 216: / Line 218: @@
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152641282779367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)&lt;/a&gt;&lt;br /&gt;
 zoomed way way in&lt;br /&gt;
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@@ Line 229: / Line 231: @@
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-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397394367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397394367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 0&lt;br /&gt;
@@ Line 237: / Line 239: @@
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397459367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397459367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 1&lt;br /&gt;
@@ Line 245: / Line 247: @@
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 &lt;/td&gt;
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397549367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 2&lt;br /&gt;
@@ Line 253: / Line 255: @@
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 &lt;/td&gt;
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397614367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance  - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 3&lt;br /&gt;
@@ Line 261: / Line 263: @@
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+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397709367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 4&lt;br /&gt;
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397784367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397784367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 5&lt;br /&gt;
@@ Line 277: / Line 279: @@
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397899367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397899367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 6&lt;br /&gt;
@@ Line 285: / Line 287: @@
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397989367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397989367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)&lt;/a&gt;&lt;br /&gt;
 showing all three elements of each 5-limit breed&lt;br /&gt;
 zoom level 7&lt;br /&gt;
@@ Line 301: / Line 303: @@
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091440809367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091440809367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,3,7}&lt;br /&gt;
 zoomed way out!&lt;br /&gt;
@@ Line 309: / Line 311: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:247:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-9-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-9-e2-twtop.jpg/520338534/tn_pts-2-3-9-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-9-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-9-e2-twtop.jpg/520338534/tn_pts-2-3-9-e2-twtop.jpg" alt="tn_pts-2-3-9-e2-twtop.jpg" title="tn_pts-2-3-9-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:247 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091485324367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091485324367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 &amp;quot;projective tuning space&amp;quot; &amp;quot;based on&amp;quot; &amp;quot;{2,3,9}&amp;quot;&lt;br /&gt;
 numerals at &amp;quot;ETs&amp;quot; represent steps per '2'&lt;br /&gt;
@@ Line 320: / Line 322: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:249:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-11-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-11-e2-twtop.jpg/520338558/tn_pts-2-3-11-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-11-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-11-e2-twtop.jpg/520338558/tn_pts-2-3-11-e2-twtop.jpg" alt="tn_pts-2-3-11-e2-twtop.jpg" title="tn_pts-2-3-11-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:249 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091496424367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091496424367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,3,11}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 329: / Line 331: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:251:&amp;lt;a href=&amp;quot;/file/view/pts-2-5-7-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-5-7-e2-twtop.jpg/520338576/tn_pts-2-5-7-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-5-7-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-5-7-e2-twtop.jpg/520338576/tn_pts-2-5-7-e2-twtop.jpg" alt="tn_pts-2-5-7-e2-twtop.jpg" title="tn_pts-2-5-7-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:251 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090453309367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-04)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090453309367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-04)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,5,7}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 338: / Line 340: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:253:&amp;lt;a href=&amp;quot;/file/view/pts-2-5-9-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-5-9-e2-twtop.jpg/520338594/tn_pts-2-5-9-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-5-9-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-5-9-e2-twtop.jpg/520338594/tn_pts-2-5-9-e2-twtop.jpg" alt="tn_pts-2-5-9-e2-twtop.jpg" title="tn_pts-2-5-9-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:253 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091482029367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091482029367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,5,9}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 347: / Line 349: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:255:&amp;lt;a href=&amp;quot;/file/view/pts-2-5-11-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-5-11-e2-twtop.jpg/520338602/tn_pts-2-5-11-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-5-11-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-5-11-e2-twtop.jpg/520338602/tn_pts-2-5-11-e2-twtop.jpg" alt="tn_pts-2-5-11-e2-twtop.jpg" title="tn_pts-2-5-11-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:255 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091500519367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091500519367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,5,11}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 356: / Line 358: @@
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091540804367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091540804367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,7,9}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 365: / Line 367: @@
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091506389367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091506389367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,7,11}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 374: / Line 376: @@
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 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091508499367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091508499367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-06)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {2,9,11}&lt;br /&gt;
 numerals at ETs represent steps per '2'&lt;br /&gt;
@@ Line 391: / Line 393: @@
          &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:263:&amp;lt;a href=&amp;quot;/file/view/pts-3-5-7-e3-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-3-5-7-e3-twtop.jpg/520338662/tn_pts-3-5-7-e3-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-3-5-7-e3-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-3-5-7-e3-twtop.jpg/520338662/tn_pts-3-5-7-e3-twtop.jpg" alt="tn_pts-3-5-7-e3-twtop.jpg" title="tn_pts-3-5-7-e3-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:263 --&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296744367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-04)&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296744367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance (2011-02-04)&lt;/a&gt;&lt;br /&gt;
 projective tuning space based on {3,5,7}&lt;br /&gt;
 numerals at ETs represent steps per '3'&lt;br /&gt;

Gallery of projective tuning space images: Difference between revisions

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