Mathematical theory of regular temperaments: Difference between revisions
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Grassmannians have the structure of a smooth, homogenous [http://en.wikipedia.org/wiki/Metric_space metric space], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian '''Gr'''(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below (known as "projective tone space"). | Grassmannians have the structure of a smooth, homogenous [http://en.wikipedia.org/wiki/Metric_space metric space], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian '''Gr'''(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below (known as "projective tone space"). | ||
See also [[equivalence continuum]] for describing the space of rank-''r'' temperaments supported by a given temperament, such as an edo (rank-1 temperament), as an algebraic variety. | |||
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