View source for Gallery of projective tuning space images

__FORCETOC__
In the following images, unless otherwise specified:
* Numerals represent [[edo|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 ==

{| 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)]

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
|}

{| class="wikitable"
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-zo4.jpg|157px]]
| | [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 waaaay out
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-zo3.jpg|200px]]
| | [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)]

zoomed-out still more
|-
| | [[File:pts-2-3-5-e2-twtop-tlin-zo2.jpg|200px]]
| | [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-out more
|-
| | [[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)]

showing all three elements of each 5-limit breed

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

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

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

zoom level 3
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z4.jpg|200px]]
| | [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)]

showing all three elements of each 5-limit breed

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)]

showing all three elements of each 5-limit breed

zoom level 5
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z6.jpg|200px]]
| | [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)]

showing all three elements of each 5-limit breed

zoom level 6
|-
| | [[File:pts-2-3-5-e2-breed-twtop-tlin-z7.jpg|200px]]
| | [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)]

showing all three elements of each 5-limit breed

zoom level 7
|}

== 11-limit subgroups ==

{| class="wikitable"
|-
| | [[File:pts-2-3-7-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150090453304367&set=gm.10150170038074482&type=1&permPage=1 Paul Erlich &gt; ‎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
|-
| | [[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)]

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

zoomed way out!
|}

{| 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)]

"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!
|}

{| 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}

numerals at ETs represent steps per '2'

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)]

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
|}

{| class="wikitable"
|-
| | [[File:pts-2-5-9-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091482029367&set=gm.10150170037959482&type=1&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
|}

{| 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}

numerals at ETs represent steps per '2'

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)]

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
|}

{| class="wikitable"
|-
| | [[File:pts-2-7-11-e2-twtop.jpg|200px]]
| | [https://www.facebook.com/photo.php?fbid=10150091506389367&set=gm.10150170037959482&type=1&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
|}

{| 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}

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 ==

{| class="wikitable"
|-
| [[File:pts-3-5-7-e3-twtop.jpg|200px]]
| [https://www.facebook.com/photo.php?fbid=10150090296744367&set=gm.10150170038074482&type=1&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'

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

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