Tenney–Euclidean tuning: Difference between revisions
Wikispaces>clumma **Imported revision 535153046 - Original comment: ** |
Wikispaces>clumma **Imported revision 582561047 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:clumma|clumma]] and made on <tt> | : This revision was by author [[User:clumma|clumma]] and made on <tt>2016-05-09 14:56:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>582561047</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 40: | Line 40: | ||
=Computing TE tuning using pseudoinverses= | =Computing TE tuning using pseudoinverses= | ||
Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that the vals are free of [Saturation|contorsion]. If J is the JI point, <1 1 ... 1|, then JV` gives the TE tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number. | Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that the vals are free of [[Saturation|contorsion]]. If J is the JI point, <1 1 ... 1|, then JV` gives the TE tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number. | ||
We may also obtain the TE 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 contorsion has been removed. 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 TE 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 contorsion has been removed. 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. | ||
| Line 117: | Line 117: | ||
<ul><li>If A is invertible, its inverse is A`</li><li>If A has rational entries, so does A`</li><li>A`` = A</li><li>The pseudoinverse of the transpose is the transpose of the pseudoinverse</li><li>AA` is the orthogonal projection map onto the space spanned by the columns of A</li><li>A`A is the orthogonal projection map onto the space spanned by the rows of A</li><li>I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A</li><li>If the rows of A are linearly independent, then A` = A*(AA*)^(-1), where A* is the transpose of A. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.</li><li>uA` is the nearest point to u in the subspace spanned by the rows of A; A`v is the nearest point to v in the space spanned by the columns of A.</li></ul><br /> | <ul><li>If A is invertible, its inverse is A`</li><li>If A has rational entries, so does A`</li><li>A`` = A</li><li>The pseudoinverse of the transpose is the transpose of the pseudoinverse</li><li>AA` is the orthogonal projection map onto the space spanned by the columns of A</li><li>A`A is the orthogonal projection map onto the space spanned by the rows of A</li><li>I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A</li><li>If the rows of A are linearly independent, then A` = A*(AA*)^(-1), where A* is the transpose of A. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.</li><li>uA` is the nearest point to u in the subspace spanned by the rows of A; A`v is the nearest point to v in the space spanned by the columns of A.</li></ul><br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Computing TE tuning using pseudoinverses"></a><!-- ws:end:WikiTextHeadingRule:6 -->Computing TE tuning using pseudoinverses</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Computing TE tuning using pseudoinverses"></a><!-- ws:end:WikiTextHeadingRule:6 -->Computing TE tuning using pseudoinverses</h1> | ||
Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that the vals are free of | Suppose V is a matrix whose rows consist of vals in the weighted basis. No assumption need be made that the rows are linearly independent or that the vals are free of <a class="wiki_link" href="/Saturation">contorsion</a>. If J is the JI point, &lt;1 1 ... 1|, then JV` gives the TE tuning in the sense that it gives (not necessarily independent) generators which correspond to the rows of V. How many of each generator to take to map a rational number contained in the prime limit in question is determined by applying the val corresponding to the generator to the rational number.<br /> | ||
<br /> | <br /> | ||
We may also obtain the TE 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 contorsion has been removed. 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 TE 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 contorsion has been removed. 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 /> | ||