Regular temperament: Difference between revisions

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Abstract regular temperaments can be identified with [http://en.wikipedia.org/wiki/Rational_point rational points] on an [http://en.wikipedia.org/wiki/Algebraic_variety algebraic variety] known as a [http://en.wikipedia.org/wiki/Grassmannian Grassmannian]. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian '''Gr'''(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space '''R'''^n. This has an embedding into a real vector space known as the [http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding Plücker embedding], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on '''Gr'''(r, n), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian '''Gr'''(r, n) can be identified with real symmetric projection matrices with trace r. The rational symmetric projection matrices with trace r are precisely the Frobenius projections, so under this identification it is clear they represent rational points on '''Gr'''(r, n). A rational projection matrix of trace r which is not symmetric is still a tuning map; minimax and least squares tunings provide examples of this.

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.

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").

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 Abstract regular temperaments can be identified with [http://en.wikipedia.org/wiki/Rational_point rational points] on an [http://en.wikipedia.org/wiki/Algebraic_variety algebraic variety] known as a [http://en.wikipedia.org/wiki/Grassmannian Grassmannian]. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian '''Gr'''(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space '''R'''^n. This has an embedding into a real vector space known as the [http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding Plücker embedding], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on '''Gr'''(r, n), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian '''Gr'''(r, n) can be identified with real symmetric projection matrices with trace r. The rational symmetric projection matrices with trace r are precisely the Frobenius projections, so under this identification it is clear they represent rational points on '''Gr'''(r, n). A rational projection matrix of trace r which is not symmetric is still a tuning map; minimax and least squares tunings provide examples of this.
-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.
+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").
 [[File:dualzoom.gif|alt=dualzoom.gif|dualzoom.gif]]