Projective tuning space: Difference between revisions

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Projective tuning space is the ~~[https://en.wikipedia.org/wiki/Projectivization~~ projectivization] of ordinary [[Vals_and_Tuning_Space|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.

'''Projective tuning space''' is the projectivization of ordinary [[Vals_and_Tuning_Space|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.

=Quotes=

== Quotes ==

"Plot the number of steps of each ET's mapping of prime 2 on the x-axis, prime 3 on the y-axis, and prime 5 on the z-axis. Actually you divide the number of ET steps by the log of the prime to get the distance along the corresponding axis. Now put your eye at the origin and look along a line equidistant from all three axes. That's Projective Tuning Space." -- [[Paul_Erlich|Paul Erlich]]

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{| class="wikitable"

|-

| [[File:pts-2-3-5-e2-twtop-tlin.jpg|alt=pts-2-3-5-e2-twtop-tlin.jpg|x250px|pts-2-3-5-e2-twtop-tlin.jpg]]

|}

= See also =

== See also ==

* [[Gallery of projective tuning space images]]

* [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Projective tuning space|Dave Keenan & Douglas Blumeyer's run-down on projective spaces]]: a very detailed and beginner-friendly explanation (with many supplementary diagrams) of how to understand, read, and use these diagrams

* [~~[Gallery_of_projective_tuning_space_images|Gallery of projective tuning space images]~~]

== External links ==

* [~~https~~://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])

* [http://www.projectivetuningspace.com/ Projective Tuning Space Interactive Application] by Chris Kline

* [http://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])

[[Category:Regular temperament theory]]

@@ Line 1: / Line 1: @@
-Projective tuning space is the [https://en.wikipedia.org/wiki/Projectivization projectivization] of ordinary [[Vals_and_Tuning_Space|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.
+{{Wikipedia|Projectivization}}
+'''Projective tuning space''' is the projectivization of ordinary [[Vals_and_Tuning_Space|tuning space]]. If a point in tuning space does not map octaves to zero, we can divide by the value to which "2" is mapped and obtain a pure-octaves tuning which serves to represent the point in projective tuning space. If the dimension of tuning space is n, the dimension of the corresponding projective space is n-1. In particular, 5-limit projective tuning space is two-dimensional, making it easy to depict it graphically.
-=Quotes=
+== Quotes ==
 "Plot the number of steps of each ET's mapping of prime 2 on the x-axis, prime 3 on the y-axis, and prime 5 on the z-axis. Actually you divide the number of ET steps by the log of the prime to get the distance along the corresponding axis. Now put your eye at the origin and look along a line equidistant from all three axes. That's Projective Tuning Space." -- [[Paul_Erlich|Paul Erlich]]
@@ Line 10: / Line 11: @@
 {| class="wikitable"
 |-
-| | [[File:pts-2-3-5-e2-twtop-tlin.jpg|alt=pts-2-3-5-e2-twtop-tlin.jpg|x250px|pts-2-3-5-e2-twtop-tlin.jpg]]
+| [[File:pts-2-3-5-e2-twtop-tlin.jpg|alt=pts-2-3-5-e2-twtop-tlin.jpg|x250px|pts-2-3-5-e2-twtop-tlin.jpg]]
 |}
-= See also =
+== See also ==
+* [[Gallery of projective tuning space images]]
+* [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Projective tuning space|Dave Keenan & Douglas Blumeyer's run-down on projective spaces]]: a very detailed and beginner-friendly explanation (with many supplementary diagrams) of how to understand, read, and use these diagrams
-* [[Gallery_of_projective_tuning_space_images|Gallery of projective tuning space images]]
+== External links ==
-* [https://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])
+* [http://www.projectivetuningspace.com/ Projective Tuning Space Interactive Application] by Chris Kline
+* [http://www.youtube.com/watch?v=Irnq7sZzilY Recreating an iconic map of tunings and temperaments], a YouTube video by Tony Durham about PTS (also [https://www.youtube.com/watch?v=TTcXPEvDitw part 2])
 [[Category:Regular temperament theory]]