Vals and tuning space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 307431086 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 307445566 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 16:30:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>307445566</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 26: | Line 26: | ||
Norms may be placed on the monzos in various ways, turning them into [[http://mathworld.wolfram.com/PointLattice.html|lattices]] in a vector space. Given a vector space norm on a space of ket vectors, the [[http://mathworld.wolfram.com/DualNormedSpace.html|dual vector space norm]] on the space of bra vectors is defined as the least quantity ||V|| making | Norms may be placed on the monzos in various ways, turning them into [[http://mathworld.wolfram.com/PointLattice.html|lattices]] in a vector space. Given a vector space norm on a space of ket vectors, the [[http://mathworld.wolfram.com/DualNormedSpace.html|dual vector space norm]] on the space of bra vectors is defined as the least quantity ||V|| making | ||
**|<V|M>| | **|<V|M>| ≤ ||V|| ||M||** | ||
to be always true. The dual of the [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]] is the [[http://mathworld.wolfram.com/L-Infinity-Norm.html|Linfty norm]], and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is //Tenney-Euclidean tuning space//. The Euclidean norm on a val v is given by | to be always true. The dual of the [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]] is the [[http://mathworld.wolfram.com/L-Infinity-Norm.html|Linfty norm]], and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is //Tenney-Euclidean tuning space//. The Euclidean norm on a val v is given by | ||
| Line 71: | Line 71: | ||
Norms may be placed on the monzos in various ways, turning them into <a class="wiki_link_ext" href="http://mathworld.wolfram.com/PointLattice.html" rel="nofollow">lattices</a> in a vector space. Given a vector space norm on a space of ket vectors, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/DualNormedSpace.html" rel="nofollow">dual vector space norm</a> on the space of bra vectors is defined as the least quantity ||V|| making<br /> | Norms may be placed on the monzos in various ways, turning them into <a class="wiki_link_ext" href="http://mathworld.wolfram.com/PointLattice.html" rel="nofollow">lattices</a> in a vector space. Given a vector space norm on a space of ket vectors, the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/DualNormedSpace.html" rel="nofollow">dual vector space norm</a> on the space of bra vectors is defined as the least quantity ||V|| making<br /> | ||
<br /> | <br /> | ||
<strong>|&lt;V|M&gt;| | <strong>|&lt;V|M&gt;| ≤ ||V|| ||M||</strong><br /> | ||
<br /> | <br /> | ||
to be always true. The dual of the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a> is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L-Infinity-Norm.html" rel="nofollow">Linfty norm</a>, and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is <em>Tenney-Euclidean tuning space</em>. The Euclidean norm on a val v is given by<br /> | to be always true. The dual of the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a> is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L-Infinity-Norm.html" rel="nofollow">Linfty norm</a>, and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is <em>Tenney-Euclidean tuning space</em>. The Euclidean norm on a val v is given by<br /> | ||