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 ==

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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 ==

* [[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

== External links ==

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

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

* [~~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.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])

* [https://narenratan.com/writing/projective-tuning-space/ Projective tuning space and moments of symmetry] by Naren Ratan

== Further reading ==

* Reinhold Baer, [https://books.google.com.mt/books?id=zQAWxRU8PCcC&printsec=frontcover ''Linear Algebra and Projective Geometry'']. Academic Press, 1952.

[[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 ==
@@ 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 ==
 * [[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
 == External links ==
-* [https://www.projectivetuningspace.com/ Projective Tuning Space Interactive Application] by Chris Kline
+* [http://www.projectivetuningspace.com/ Projective Tuning Space Interactive Application] by Chris Kline
-* [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.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])
+* [https://narenratan.com/writing/projective-tuning-space/ Projective tuning space and moments of symmetry] by Naren Ratan
+== Further reading ==
+* Reinhold Baer, [https://books.google.com.mt/books?id=zQAWxRU8PCcC&printsec=frontcover ''Linear Algebra and Projective Geometry'']. Academic Press, 1952.
 [[Category:Regular temperament theory]]

Projective tuning space: Difference between revisions

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