Gallery of projective tuning space images

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In the following images, unless otherwise specified:

  • numerals represent ETs (steps per '2')
  • multiple appearances of the same number are different mappings
  • rotation of the ET numbers indicates octave stretch of the TOP tuning of the ET (15 cents stretch = 90° clockwise rotation)
  • lines are temperaments (where a single comma is tempered out)
  • hexagons are contours of maximum Tenney-weighted damage of TOP tunings

5-limit

Pts-2-3-5-e2-amp-hand-annotated-gray.jpg {2,3,5} PTS with handwritten annotations

from A Middle Path

Pts-2-3-5-e2-twtop.jpg 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

Pts-2-3-5-e2-twtop-tlin-zo4.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed waaaay out

Pts-2-3-5-e2-twtop-tlin-zo3.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed-out still more

Pts-2-3-5-e2-twtop-tlin-zo2.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed-out more

Pts-2-3-5-e2-twtop-tlin-zo1.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-22)

zoomed-out

Pts-2-3-5-e2-twtop-tlin.jpg 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.

Pts-2-3-5-e2-twtop-tlin-graybg-zi1.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed-in

Pts-2-3-5-e2-twtop-tlin-graybg-zi2.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed-in more

Pts-2-3-5-e2-twtop-tlin-zi3.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed-in still more

Pts-2-3-5-e2-twtop-tlin-zi4.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed way in

Pts-2-3-5-e2-twtop-tlin-zi5.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-25)

zoomed way way in

Pts-2-3-5-e2-breed-twtop-tlin-z0.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 0

Pts-2-3-5-e2-breed-twtop-tlin-z1.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 1

Pts-2-3-5-e2-breed-twtop-tlin-z2.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 2

Pts-2-3-5-e2-breed-twtop-tlin-z3.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 3

Pts-2-3-5-e2-breed-twtop-tlin-z4.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 4

Pts-2-3-5-e2-breed-twtop-tlin-z5.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 5

Pts-2-3-5-e2-breed-twtop-tlin-z6.jpg Paul Erlich > ‎The Xenharmonic Alliance - Mathematical Theory (2014-08-26)

showing all three elements of each 5-limit breed

zoom level 6

Pts-2-3-5-e2-breed-twtop-tlin-z7.jpg 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

Pts-2-3-7-e2-twtop.jpg 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

Pts-2-3-7-e2-twtop-z0.jpg Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)

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

zoomed way out!

Pts-2-3-9-e2-twtop.jpg 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!

Pts-2-3-11-e2-twtop.jpg 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

Pts-2-5-7-e2-twtop.jpg 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

Pts-2-5-9-e2-twtop.jpg 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

Pts-2-5-11-e2-twtop.jpg 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

Pts-2-7-9-e2-twtop.jpg 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

Pts-2-7-11-e2-twtop.jpg 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

Pts-2-9-11-e2-twtop.jpg Paul Erlich > ‎The Xenharmonic Alliance (2011-02-06)

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

numerals at ETs represent steps per '2'

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

3.5.7, tritave (3-1) based

Pts-3-5-7-e3-twtop.jpg 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