Gallery of projective tuning space images
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author spt3125 and made on 2014-09-01 11:53:21 UTC.
- The original revision id was 520347754.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[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:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><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:<h1> --><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:<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="" title="" /></a> --><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, 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 />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:207:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296739367&set=gm.10150170038074482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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">
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:209:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247244367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed waaaay out<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:211:<a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo3.jpg"><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="" title="" /></a> --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo3.jpg"><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" /></a><!-- ws:end:WikiTextLocalImageRule:211 --><br />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247269367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out still more<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:213:<a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo2.jpg"><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="" title="" /></a> --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo2.jpg"><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" /></a><!-- ws:end:WikiTextLocalImageRule:213 --><br />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247314367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out more<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:215:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247409367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:217:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152310913269367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-04-10)</a><br />
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.<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:219:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895264367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed-in<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:221:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895159367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed-in more<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:223:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152640895094367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed-in still more<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:225:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152641282619367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed way in<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:227:<a href="/file/view/pts-2-3-5-e2-twtop-tlin-zi5.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg/520341498/tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg" alt="" title="" /></a> --><a href="/file/view/pts-2-3-5-e2-twtop-tlin-zi5.jpg"><img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg/520341498/tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:227 --><br />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152641282779367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a><br />
zoomed way way in<br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:229:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397394367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 0<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:231:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397459367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 1<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:233:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397549367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 2<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:235:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397614367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 3<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:237:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397709367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 4<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:239:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397784367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 5<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:241:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397899367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > 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>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:243:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397989367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a><br />
showing all three elements of each 5-limit breed<br />
zoom level 7<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:3:<h1> --><h1 id="toc1"><a name="x11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:3 -->11-limit subgroups</h1>
<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:245:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091440809367&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance (2011-02-06)</a><br />
projective tuning space based on {2,3,7}<br />
zoomed way out!<br />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:247:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091485324367&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > The Xenharmonic Alliance (2011-02-06)</a><br />
"projective tuning space" "based on" "{2,3,9}"<br />
numerals at "ETs" represent steps per '2'<br />
hexagons are "contours" of "maximum Tenney-weighted damage" of "TOP tunings"<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 />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:249:<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="" title="" /></a> --><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 />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091496424367&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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 />
</td>
</tr>
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:251:<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="" title="" /></a> --><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&set=gm.10150170038074482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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:<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="" title="" /></a> --><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&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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:<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="" title="" /></a> --><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&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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:<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="" title="" /></a> --><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&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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:<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="" title="" /></a> --><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&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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:<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="" title="" /></a> --><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&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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:<h1> --><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:<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="" title="" /></a> --><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&set=gm.10150170038074482&type=1&permPage=1" rel="nofollow">Paul Erlich > 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 />
<!-- ws:start:WikiTextMediaRule:0:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/26495050?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;26495050&quot;" title="Custom Media"/> --><script type="text/javascript">
/*<![CDATA[*/
document.write('<style type="text/css"> .wiki_table td { padding: 3px; } <\/style>');
/*]]>*/
</script><!-- ws:end:WikiTextMediaRule:0 --></body></html>