Gallery of projective tuning space images: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-09-01 11:53:21 UTC</tt>.

: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-09-01 11:55:52 UTC</tt>.

: The original revision id was <tt>~~520347754~~</tt>.

: The original revision id was <tt>520347958</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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=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&set=gm.10150170038074482&type=1&permPage=1|Paul Erlich > ‎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&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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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

|| [[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)]]

|| [[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)]]

|| [[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

|| [[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

Line 66:

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=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)]]

|| [[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)]]

|| [[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'

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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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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)]]

|| [[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'

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&set=gm.10150170038074482&type=1&permPage=1|Paul Erlich > ‎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&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'

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<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">File Not Found</div></div>

<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">File Not Found</div></div>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)</a>

<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>

projective tuning space based on {2,3,5}

numerals at ETs represent steps per '2'

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)</a>

<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>

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.

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a>

<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>

zoomed-in

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)</a>

<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>

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)</a>

<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>

showing all three elements of each 5-limit breed

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

projective tuning space based on {2,3,7}

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

&quot;projective tuning space&quot; &quot;based on&quot; &quot;{2,3,9}&quot;

numerals at &quot;ETs&quot; represent steps per '2'

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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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

projective tuning space based on {2,3,11}

numerals at ETs represent steps per '2'

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<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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)</a>

<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>

projective tuning space based on {2,5,7}

numerals at ETs represent steps per '2'

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<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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

projective tuning space based on {2,5,9}

numerals at ETs represent steps per '2'

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<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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

projective tuning space based on {2,5,11}

numerals at ETs represent steps per '2'

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<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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

projective tuning space based on {2,7,9}

numerals at ETs represent steps per '2'

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<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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

projective tuning space based on {2,7,11}

numerals at ETs represent steps per '2'

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<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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)</a>

<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>

projective tuning space based on {2,9,11}

numerals at ETs represent steps per '2'

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<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">Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)</a>

<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>

projective tuning space based on {3,5,7}

numerals at ETs represent steps per '3'

@@ 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: @@
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 &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: @@
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 &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: @@
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 &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: @@
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 &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: @@
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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=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: @@
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 &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;
-         &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"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)&lt;/a&gt;&lt;br /&gt;
+         &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;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"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)&lt;/a&gt;&lt;br /&gt;
+         &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;
 &lt;/td&gt;
@@ Line 195: / Line 197: @@
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 &lt;/td&gt;
-         &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"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)&lt;/a&gt;&lt;br /&gt;
+         &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;
 &lt;/td&gt;
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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=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;
 &lt;/td&gt;
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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=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;
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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=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" 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;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;
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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;
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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;
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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;
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+         &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;
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-         &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;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: @@
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 &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: @@
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 &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: @@
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 &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: @@
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 &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: @@
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 &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: @@
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 &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;