Gallery of projective tuning space images
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author spt3125 and made on 2016-02-28 15:09:06 UTC.
- The original revision id was 576079145.
- The revision comment was: added explanation of numbers, rotation, lines etc
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Original Wikitext content:
[[toc]]
In the following images, unless otherwise specified:
* numerals represent ETs (steps per '2')
* multiple appearances of the same number are different mappings
* rotation of the ET numbers indicates octave stretch of the TOP tuning of the ET (15 cents stretch = 90° clockwise rotation)
* lines are temperaments (where a single comma is tempered out)
* hexagons are contours of maximum Tenney-weighted damage of TOP tunings
=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 [[http://lumma.org/tuning/erlich/MiddlePath2015.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.jpg link="file:pts-2-3-7-e2-twtop.jpg"]] || [[@https://www.facebook.com/photo.php?fbid=10150090453304367&set=gm.10150170038074482&type=1&permPage=1|Paul Erlich > The Xenharmonic Alliance (2011-02-04)]]
projective tuning space based on {2,3,7}
numerals at ETs represent steps per '2'
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
|| [[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 --><br />
In the following images, unless otherwise specified:<br />
<ul><li>numerals represent ETs (steps per '2')</li><li>multiple appearances of the same number are different mappings</li><li>rotation of the ET numbers indicates octave stretch of the TOP tuning of the ET (15 cents stretch = 90° clockwise rotation)</li><li>lines are temperaments (where a single comma is tempered out)</li><li>hexagons are contours of maximum Tenney-weighted damage of TOP tunings</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:1:<h1> --><h1 id="toc0"><a name="x5-limit"></a><!-- ws:end:WikiTextHeadingRule:1 -->5-limit</h1>
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<table class="wiki_table">
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<td><!-- ws:start:WikiTextLocalImageRule:239:<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:239 --><br />
</td>
<td>{2,3,5} PTS with handwritten annotations<br />
from <a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/MiddlePath2015.pdf" rel="nofollow">A Middle Path</a><br />
</td>
</tr>
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<td><!-- ws:start:WikiTextLocalImageRule:241:<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:241 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296739367&set=gm.10150170038074482&type=1&permPage=1" rel="nofollow" target="_blank">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">
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<td><!-- ws:start:WikiTextLocalImageRule:243:<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:243 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247244367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed waaaay out<br />
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<td><!-- ws:start:WikiTextLocalImageRule:245:<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:245 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247269367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out still more<br />
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<td><!-- ws:start:WikiTextLocalImageRule:247:<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:247 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247314367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out more<br />
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<td><!-- ws:start:WikiTextLocalImageRule:249:<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:249 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247409367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">Paul Erlich > The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a><br />
zoomed-out<br />
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<td><!-- ws:start:WikiTextLocalImageRule:251:<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:251 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152310913269367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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 />
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<td><!-- ws:start:WikiTextLocalImageRule:253:<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:253 --><br />
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zoomed-in<br />
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<td><!-- ws:start:WikiTextLocalImageRule:255:<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:255 --><br />
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zoomed-in more<br />
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<td><!-- ws:start:WikiTextLocalImageRule:257:<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:257 --><br />
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zoomed-in still more<br />
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<td><!-- ws:start:WikiTextLocalImageRule:259:<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:259 --><br />
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zoomed way in<br />
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<td><!-- ws:start:WikiTextLocalImageRule:261:<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:261 --><br />
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zoomed way way in<br />
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<td><!-- ws:start:WikiTextLocalImageRule:263:<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:263 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397394367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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 />
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<td><!-- ws:start:WikiTextLocalImageRule:265:<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:265 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397459367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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 />
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<td><!-- ws:start:WikiTextLocalImageRule:267:<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:267 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397549367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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 />
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<td><!-- ws:start:WikiTextLocalImageRule:269:<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:269 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397614367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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>
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<td><!-- ws:start:WikiTextLocalImageRule:271:<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:271 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397709367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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:273:<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:273 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397784367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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:275:<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:275 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397899367&set=oa.389536071186621&type=1&permPage=1" rel="nofollow" target="_blank">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:277:<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:277 --><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" target="_blank">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>
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<td><!-- ws:start:WikiTextLocalImageRule:279:<a href="/file/view/pts-2-3-7-e2-twtop.jpg"><img src="/file/view/tn_pts-2-3-7-e2-twtop.jpg/520349152/tn_pts-2-3-7-e2-twtop.jpg" alt="" title="" /></a> --><a href="/file/view/pts-2-3-7-e2-twtop.jpg"><img src="/file/view/tn_pts-2-3-7-e2-twtop.jpg/520349152/tn_pts-2-3-7-e2-twtop.jpg" alt="tn_pts-2-3-7-e2-twtop.jpg" title="tn_pts-2-3-7-e2-twtop.jpg" /></a><!-- ws:end:WikiTextLocalImageRule:279 --><br />
</td>
<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090453304367&set=gm.10150170038074482&type=1&permPage=1" rel="nofollow" target="_blank">Paul Erlich > The Xenharmonic Alliance (2011-02-04)</a><br />
projective tuning space based on {2,3,7}<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:281:<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:281 --><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" target="_blank">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>
</table>
<br />
<table class="wiki_table">
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<td><!-- ws:start:WikiTextLocalImageRule:283:<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:283 --><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" target="_blank">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 />
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</table>
<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:285:<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:285 --><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" target="_blank">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>
</table>
<br />
<table class="wiki_table">
<tr>
<td><!-- ws:start:WikiTextLocalImageRule:287:<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:287 --><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" target="_blank">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>
</table>
<br />
<table class="wiki_table">
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<td><!-- ws:start:WikiTextLocalImageRule:289:<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:289 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091482029367&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow" target="_blank">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>
</table>
<br />
<table class="wiki_table">
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<td><!-- ws:start:WikiTextLocalImageRule:291:<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:291 --><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" target="_blank">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>
</table>
<br />
<table class="wiki_table">
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<td><!-- ws:start:WikiTextLocalImageRule:293:<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:293 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091540804367&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow" target="_blank">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>
</table>
<br />
<table class="wiki_table">
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<td><!-- ws:start:WikiTextLocalImageRule:295:<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:295 --><br />
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<td><a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091506389367&set=gm.10150170037959482&type=1&permPage=1" rel="nofollow" target="_blank">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>
</table>
<br />
<table class="wiki_table">
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<td><!-- ws:start:WikiTextLocalImageRule:297:<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:297 --><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" target="_blank">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">
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<td><!-- ws:start:WikiTextLocalImageRule:299:<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:299 --><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" target="_blank">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>
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