Gallery of projective tuning space images: Difference between revisions

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**Imported revision 570874871 - Original comment: **
m Fix a couple of typos
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
In the following images, unless otherwise specified:
: This revision was by author [[User:clumma|clumma]] and made on <tt>2015-12-29 13:49:10 UTC</tt>.<br>
* Numerals represent [[edo|equal temperaments]] (steps per [[octave]])
: The original revision id was <tt>570874871</tt>.<br>
* Multiple appearances of the same number are different mappings
: The revision comment was: <tt></tt><br>
* Rotation of the ET numbers indicates [[octave stretch]] of the [[TOP]] tuning of the equal temperament (15 cents stretch = 90° clockwise rotation)
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
* Lines are [[temperaments]] (where a single [[comma]] is [[tempered out]])
<h4>Original Wikitext content:</h4>
* Hexagons are contours of maximum Tenney-weighted damage of TOP tunings
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]
=5-limit=


|| [[image:tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg link="file:pts-2-3-5-e2-amp-hand-annotated-gray.jpg]"]] || {2,3,5} PTS with handwritten annotations
== 5-limit ==
from [[http://lumma.org/tuning/erlich/MiddlePath2015.pdf | A Middle Path]] ||
 
{| class="wikitable"
|-
| | [[File:pts-2-3-5-e2-amp-hand-annotated-gray.jpg|200px]]
| | {2,3,5} PTS with handwritten annotations
 
from [http://lumma.org/tuning/erlich/MiddlePath2015.pdf A Middle Path]
|}
 
{| class="wikitable"
|-
| | [[File:pts-2-3-5-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150090296739367&set=gm.10150170038074482&type=1&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}
projective tuning space based on {2,3,5}
numerals at ETs represent steps per '2'
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)]]
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
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)]]
 
zoomed-out still more ||
{| class="wikitable"
|| [[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 ||
| | [[File:pts-2-3-5-e2-twtop-tlin-zo4.jpg|157px]]
|| [[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)]]
| | [https://www.facebook.com/photo.php?fbid=10152632247244367&set=oa.389536071186621&type=1&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)]]
zoomed waaaay out
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&amp;set=oa.389536071186621&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]]
| | [[File:pts-2-3-5-e2-twtop-tlin-zo3.jpg|200px]]
zoomed-in ||
| | [https://www.facebook.com/photo.php?fbid=10152632247269367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]
|| [[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 ||
zoomed-out still 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)]]
|-
zoomed-in still more ||
| | [[File:pts-2-3-5-e2-twtop-tlin-zo2.jpg|200px]]
|| [[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)]]
| | [https://www.facebook.com/photo.php?fbid=10152632247314367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]
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)]]
zoomed-out more
zoomed way way in ||
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-zo1.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152632247409367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)]
 
zoomed-out
|-
| | [[File:pts-2-3-5-e2-twtop-tlin.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152310913269367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎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.
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152640895264367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]
 
zoomed-in
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152640895159367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]
 
zoomed-in more
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-zi3.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152640895094367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]
 
zoomed-in still more
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-zi4.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152641282619367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]
 
zoomed way in
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-zi5.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152641282779367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)]
 
zoomed way way in
|}
 
{| class="wikitable"
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z0.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152643397394367&set=oa.389536071186621&type=1&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
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)]]
zoom level 0
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z1.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152643397459367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]
 
showing all three elements of each 5-limit breed
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)]]
zoom level 1
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z2.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152643397549367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory  (2014-08-26)]
 
showing all three elements of each 5-limit breed
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)]]
zoom level 2
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z3.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152643397614367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance  - Mathematical Theory (2014-08-26)]
 
showing all three elements of each 5-limit breed
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
zoom level 3
|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
|-
showing all three elements of each 5-limit breed
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z4.jpg|200px]]
zoom level 4 ||
| | [https://www.facebook.com/photo.php?fbid=10152643397709367&set=oa.389536071186621&type=1&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
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)]]
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
|Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]]
showing all three elements of each 5-limit breed
zoom level 7 ||


=11-limit subgroups=
zoom level 4
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z5.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10152643397784367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]


|| [[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&amp;set=gm.10150170038074482&amp;type=1&amp;permPage=1|Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]]
showing all three elements of each 5-limit breed
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&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)]]
zoom level 5
"projective tuning space" "based on" "{2,3,9}"
|-
numerals at "ETs" represent steps per '2'
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z6.jpg|200px]]
hexagons are "contours" of "maximum Tenney-weighted damage" of "TOP tunings"
| | [https://www.facebook.com/photo.php?fbid=10152643397899367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]


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!  ||
showing all three elements of each 5-limit breed


|| [[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)]]
zoom level 6
projective tuning space based on {2,3,11}
|-
numerals at ETs represent steps per '2'
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z7.jpg|200px]]
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
| | [https://www.facebook.com/photo.php?fbid=10152643397989367&set=oa.389536071186621&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)]


|| [[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)]]
showing all three elements of each 5-limit breed
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)]]
zoom level 7
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)]]
== 11-limit subgroups ==
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)]]
{| class="wikitable"
projective tuning space based on {2,7,9}
|-
numerals at ETs represent steps per '2'
| | [[File:pts-2-3-7-e2-twtop.jpg|200px]]
hexagons are contours of maximum Tenney-weighted damage of TOP tunings ||
| | [https://www.facebook.com/photo.php?fbid=10150090453304367&set=gm.10150170038074482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]


|| [[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 {2,3,7}
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)]]
projective tuning space based on {2,9,11}
numerals at ETs represent steps per '2'
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=
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
|-
| | [[File:pts-2-3-7-e2-twtop-z0.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091440809367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]


|| [[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 {2,3,7}
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"]]
zoomed way out!
</pre></div>
|}
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Gallery of projective tuning space images&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:7:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x5-limit"&gt;5-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x11-limit subgroups"&gt;11-limit subgroups&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x3.5.7, tritave (3-1) based"&gt;3.5.7, tritave (3-1) based&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x5-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;5-limit&lt;/h1&gt;
&lt;br /&gt;


{| class="wikitable"
|-
| | [[File:pts-2-3-9-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091485324367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]


&lt;table class="wiki_table"&gt;
"projective tuning space" "based on" "{2,3,9}"
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:227:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-amp-hand-annotated-gray.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg/520336302/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-amp-hand-annotated-gray.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg/520336302/tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" alt="tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" title="tn_pts-2-3-5-e2-amp-hand-annotated-gray.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:227 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;{2,3,5} PTS with handwritten annotations&lt;br /&gt;
from [[&lt;!-- ws:start:WikiTextUrlRule:461:http://lumma.org/tuning/erlich/MiddlePath2015.pdf --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/erlich/MiddlePath2015.pdf" rel="nofollow"&gt;http://lumma.org/tuning/erlich/MiddlePath2015.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:461 --&gt; | A Middle Path]]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
numerals at "ETs" represent steps per '2'


hexagons are "contours" of "maximum Tenney-weighted damage" of "TOP tunings"


&lt;table class="wiki_table"&gt;
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!
    &lt;tr&gt;
|}
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:229:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop.jpg/520338504/tn_pts-2-3-5-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop.jpg/520338504/tn_pts-2-3-5-e2-twtop.jpg" alt="tn_pts-2-3-5-e2-twtop.jpg" title="tn_pts-2-3-5-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:229 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296739367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
{| class="wikitable"
|-
| | [[File:pts-2-3-11-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091496424367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]


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


&lt;table class="wiki_table"&gt;
numerals at ETs represent steps per '2'
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:231:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo4.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg/520338468/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo4.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg/520338468/tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo4.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:231 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247244367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:233:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo3.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg/520338452/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo3.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg/520338452/tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo3.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:233 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247269367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:235:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo2.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg/520338442/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo2.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg/520338442/tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo2.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:235 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247314367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:237:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zo1.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg/520338424/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zo1.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg/520338424/tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" alt="tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" title="tn_pts-2-3-5-e2-twtop-tlin-zo1.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:237 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152632247409367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:239:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin.jpg/520338490/tn_pts-2-3-5-e2-twtop-tlin.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:239 --&gt;&lt;br /&gt;
&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" target="_blank"&gt;Paul Erlich &amp;gt; ‎The Xenharmonic Alliance - Mathematical Theory (2014-04-10)&lt;/a&gt;&lt;br /&gt;
5-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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:241:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg/520338370/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:241 --&gt;&lt;br /&gt;
&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" 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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:243:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg/520338380/tn_pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:243 --&gt;&lt;br /&gt;
&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" 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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:245:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zi3.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg/520341482/tn_pts-2-3-5-e2-twtop-tlin-zi3.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zi3.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:245 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:247:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zi4.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg/520341488/tn_pts-2-3-5-e2-twtop-tlin-zi4.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zi4.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:247 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=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;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:249:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-twtop-tlin-zi5.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg/520341498/tn_pts-2-3-5-e2-twtop-tlin-zi5.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-twtop-tlin-zi5.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:249 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=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;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
|}


{| class="wikitable"
|-
| | [[File:pts-2-5-7-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150090453309367&set=gm.10150170038074482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]


&lt;table class="wiki_table"&gt;
projective tuning space based on {2,5,7}
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:251:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z0.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg/520342320/tn_pts-2-3-5-e2-breed-twtop-tlin-z0.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z0.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:251 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=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;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:253:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z1.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg/520338284/tn_pts-2-3-5-e2-breed-twtop-tlin-z1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z1.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:253 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=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;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:255:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z2.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg/520338296/tn_pts-2-3-5-e2-breed-twtop-tlin-z2.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z2.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:255 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=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;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:257:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z3.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg/520338304/tn_pts-2-3-5-e2-breed-twtop-tlin-z3.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z3.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:257 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &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;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:259:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z4.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg/520338322/tn_pts-2-3-5-e2-breed-twtop-tlin-z4.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z4.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:259 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &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;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:261:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z5.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg/520338338/tn_pts-2-3-5-e2-breed-twtop-tlin-z5.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z5.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:261 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397784367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:263:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z6.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg/520338350/tn_pts-2-3-5-e2-breed-twtop-tlin-z6.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z6.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:263 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=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;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:265:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-5-e2-breed-twtop-tlin-z7.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg/520338364/tn_pts-2-3-5-e2-breed-twtop-tlin-z7.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-5-e2-breed-twtop-tlin-z7.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:265 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10152643397989367&amp;amp;set=oa.389536071186621&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
numerals at ETs represent steps per '2'
&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x11-limit subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;11-limit subgroups&lt;/h1&gt;
&lt;br /&gt;


hexagons are contours of maximum Tenney-weighted damage of TOP tunings
|}


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:267:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-7-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-7-e2-twtop.jpg/520349152/tn_pts-2-3-7-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-7-e2-twtop.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:267 --&gt;&lt;br /&gt;
| | [[File:pts-2-5-9-e2-twtop.jpg|200px]]
&lt;/td&gt;
| | [https://www.facebook.com/photo.php?fbid=10150091482029367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090453304367&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 {2,3,7}&lt;br /&gt;
numerals at ETs represent steps per '2'&lt;br /&gt;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:269:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-7-e2-twtop-z0.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-7-e2-twtop-z0.jpg/520338520/tn_pts-2-3-7-e2-twtop-z0.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-7-e2-twtop-z0.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:269 --&gt;&lt;br /&gt;
&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" 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;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
projective tuning space based on {2,5,9}


numerals at ETs represent steps per '2'


&lt;table class="wiki_table"&gt;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
    &lt;tr&gt;
|}
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:271:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-9-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-9-e2-twtop.jpg/520338534/tn_pts-2-3-9-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-9-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-9-e2-twtop.jpg/520338534/tn_pts-2-3-9-e2-twtop.jpg" alt="tn_pts-2-3-9-e2-twtop.jpg" title="tn_pts-2-3-9-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:271 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091485324367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
hexagons are &amp;quot;contours&amp;quot; of &amp;quot;maximum Tenney-weighted damage&amp;quot; of &amp;quot;TOP tunings&amp;quot;&lt;br /&gt;
&lt;br /&gt;
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!&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
{| class="wikitable"
|-
| | [[File:pts-2-5-11-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091500519367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]


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


&lt;table class="wiki_table"&gt;
numerals at ETs represent steps per '2'
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:273:&amp;lt;a href=&amp;quot;/file/view/pts-2-3-11-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-3-11-e2-twtop.jpg/520338558/tn_pts-2-3-11-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-3-11-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-3-11-e2-twtop.jpg/520338558/tn_pts-2-3-11-e2-twtop.jpg" alt="tn_pts-2-3-11-e2-twtop.jpg" title="tn_pts-2-3-11-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:273 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091496424367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
|}


{| class="wikitable"
|-
| | [[File:pts-2-7-9-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091540804367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]


&lt;table class="wiki_table"&gt;
projective tuning space based on {2,7,9}
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:275:&amp;lt;a href=&amp;quot;/file/view/pts-2-5-7-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-5-7-e2-twtop.jpg/520338576/tn_pts-2-5-7-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-5-7-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-5-7-e2-twtop.jpg/520338576/tn_pts-2-5-7-e2-twtop.jpg" alt="tn_pts-2-5-7-e2-twtop.jpg" title="tn_pts-2-5-7-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:275 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090453309367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
numerals at ETs represent steps per '2'


hexagons are contours of maximum Tenney-weighted damage of TOP tunings
|}


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:277:&amp;lt;a href=&amp;quot;/file/view/pts-2-5-9-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-5-9-e2-twtop.jpg/520338594/tn_pts-2-5-9-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-5-9-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-5-9-e2-twtop.jpg/520338594/tn_pts-2-5-9-e2-twtop.jpg" alt="tn_pts-2-5-9-e2-twtop.jpg" title="tn_pts-2-5-9-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:277 --&gt;&lt;br /&gt;
| | [[File:pts-2-7-11-e2-twtop.jpg|200px]]
&lt;/td&gt;
| | [https://www.facebook.com/photo.php?fbid=10150091506389367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]
        &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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
projective tuning space based on {2,7,11}


numerals at ETs represent steps per '2'


&lt;table class="wiki_table"&gt;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
    &lt;tr&gt;
|}
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:279:&amp;lt;a href=&amp;quot;/file/view/pts-2-5-11-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-5-11-e2-twtop.jpg/520338602/tn_pts-2-5-11-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-5-11-e2-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-2-5-11-e2-twtop.jpg/520338602/tn_pts-2-5-11-e2-twtop.jpg" alt="tn_pts-2-5-11-e2-twtop.jpg" title="tn_pts-2-5-11-e2-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:279 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150091500519367&amp;amp;set=gm.10150170037959482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
{| class="wikitable"
|-
| | [[File:pts-2-9-11-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091508499367&set=gm.10150170037959482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-06)]


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


&lt;table class="wiki_table"&gt;
numerals at ETs represent steps per '2'
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:281:&amp;lt;a href=&amp;quot;/file/view/pts-2-7-9-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-7-9-e2-twtop.jpg/520338622/tn_pts-2-7-9-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-7-9-e2-twtop.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:281 --&gt;&lt;br /&gt;
&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" 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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
|}


== 3.5.7 subgroup, tritave (3/1) based ==


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:283:&amp;lt;a href=&amp;quot;/file/view/pts-2-7-11-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-7-11-e2-twtop.jpg/520338634/tn_pts-2-7-11-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-7-11-e2-twtop.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:283 --&gt;&lt;br /&gt;
| [[File:pts-3-5-7-e3-twtop.jpg|200px]]
&lt;/td&gt;
| [https://www.facebook.com/photo.php?fbid=10150090296744367&set=gm.10150170038074482&type=1&permPage=1 Paul Erlich &gt; ‎The Xenharmonic Alliance (2011-02-04)]
        &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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
projective tuning space based on {3,5,7}
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:285:&amp;lt;a href=&amp;quot;/file/view/pts-2-9-11-e2-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-2-9-11-e2-twtop.jpg/520338644/tn_pts-2-9-11-e2-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-2-9-11-e2-twtop.jpg"&gt;&lt;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" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:285 --&gt;&lt;br /&gt;
&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" 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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="x3.5.7, tritave (3-1) based"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;3.5.7, tritave (3-1) based&lt;/h1&gt;
&lt;br /&gt;


numerals at ETs represent steps per '3'


&lt;table class="wiki_table"&gt;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings
    &lt;tr&gt;
|}
        &lt;td&gt;&lt;!-- ws:start:WikiTextLocalImageRule:287:&amp;lt;a href=&amp;quot;/file/view/pts-3-5-7-e3-twtop.jpg&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/tn_pts-3-5-7-e3-twtop.jpg/520338662/tn_pts-3-5-7-e3-twtop.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;a href="/file/view/pts-3-5-7-e3-twtop.jpg"&gt;&lt;img src="/file/view/tn_pts-3-5-7-e3-twtop.jpg/520338662/tn_pts-3-5-7-e3-twtop.jpg" alt="tn_pts-3-5-7-e3-twtop.jpg" title="tn_pts-3-5-7-e3-twtop.jpg" /&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextLocalImageRule:287 --&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;a class="wiki_link_ext" href="https://www.facebook.com/photo.php?fbid=10150090296744367&amp;amp;set=gm.10150170038074482&amp;amp;type=1&amp;amp;permPage=1" rel="nofollow" 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;
hexagons are contours of maximum Tenney-weighted damage of TOP tunings&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[Category:Lists]]
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[[Category:Graphs]]
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Latest revision as of 09:14, 3 December 2024

In the following images, unless otherwise specified:

  • Numerals represent equal temperaments (steps per octave)
  • Multiple appearances of the same number are different mappings
  • Rotation of the ET numbers indicates octave stretch of the TOP tuning of the equal temperament (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

{2,3,5} PTS with handwritten annotations

from A Middle Path

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

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed waaaay out

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed-out still more

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed-out more

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed-out

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.

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed-in

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed-in more

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed-in still more

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed way in

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed way way in

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 0

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 1

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 2

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 3

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 4

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 5

Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 6

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

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

Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)

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

zoomed way out!

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!

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

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

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

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

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

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

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 subgroup, tritave (3/1) based

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

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