Rank-3 temperament: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-02 17:33:01 UTC</tt>.<br>

: The original revision id was <tt>160092625</tt>.<br>

7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted monzo m = |-5 3.17 4.64 -2.81&gt; and call the weighted monzo |1 0 0 0&gt; for 2 t, then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2x4 matrix  U = [t, m]. If U` denotes the [[RMS tuning|Moore-Pensrose pseudoinverse]] of U, then letting Q = U`U take P = I - Q, where I is the identity matrix. P is the projection map from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector |0 0 0 0&gt; representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96 to the fifth, and the major third angled 129.84 to the fifth.

If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log2(q) for each of primes up to p, which allows us to directly project unweighted monzos.

7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted monzo m = |-5 3.17 4.64 -2.81&amp;gt; and call the weighted monzo |1 0 0 0&amp;gt; for 2 t, then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2x4 matrix  U = [t, m]. If U` denotes the &lt;a class="wiki_link" href="/RMS%20tuning"&gt;Moore-Pensrose pseudoinverse&lt;/a&gt; of U, then letting Q = U`U take P = I - Q, where I is the identity matrix. P is the projection map from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector |0 0 0 0&amp;gt; representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96 to the fifth, and the major third angled 129.84 to the fifth.&lt;br /&gt;

If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log2(q) for each of primes up to p, which allows us to directly project unweighted monzos.&lt;br /&gt;