Tenney–Euclidean tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 158799561 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 158939035 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-08- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-08-29 15:59:33 UTC</tt>.<br> | ||
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We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J. | We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J. | ||
We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank n matrix whose rows are weighted monzos, and I is the nxn identity matrix, then P = I - M`M is the same projection map as V`V so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are. | |||
===Pure octaves RMS tuning=== | ===Pure octaves RMS tuning=== | ||
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We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.<br /> | We may also obtain the TOP-RMS tuning from a projection map. P = V`V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether torsion problems have been removed. Those are removed automatically. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.<br /> | ||
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We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank n matrix whose rows are weighted monzos, and I is the nxn identity matrix, then P = I - M`M is the same projection map as V`V so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-RMS tuning-Pure octaves RMS tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->Pure octaves RMS tuning</h3> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-RMS tuning-Pure octaves RMS tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->Pure octaves RMS tuning</h3> | ||