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 [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. | ||
=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]] | "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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= See also = | == See also == | ||
* [[Gallery of projective tuning space images]] | |||
* [ | == External links == | ||
* [https://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]) | * [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]) | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||