Mike's lecture on vector spaces and dual spaces: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 325968020 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 325968126 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 08: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 08:05:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>325968126</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> | ||
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If you haven't seen monzos or vals before and are totally confused, please read the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] first! | If you haven't seen monzos or vals before and are totally confused, please read the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] first! | ||
If you have, then to review, a **monzo** is a way to represent a JI interval that shows how it decomposes into a combination of simpler, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like [[media type="custom" key="15538076"]], where a, b, and c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like [[media type="custom" key=" | If you have, then to review, a **monzo** is a way to represent a JI interval that shows how it decomposes into a combination of simpler, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like [[media type="custom" key="15538076"]], where a, b, and c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like [[media type="custom" key="15538090"]], where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on. | ||
Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this: | Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this: | ||
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==1.1: A monzo can be viewed as a **VECTOR** in a **VECTOR SPACE**.== | ==1.1: A monzo can be viewed as a **VECTOR** in a **VECTOR SPACE**.== | ||
For instance, the syntonic comma is [[media type="custom" key=" | For instance, the syntonic comma is [[media type="custom" key="15538088"]]. A geometric interpretation of this interval might be as a point in a space, like the point [[media type="custom" key="15538024"]]. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors: | ||
[[media type="custom" key="15537326"]] | [[media type="custom" key="15537326"]] | ||
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If you haven't seen monzos or vals before and are totally confused, please read the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> first!<br /> | If you haven't seen monzos or vals before and are totally confused, please read the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> first!<br /> | ||
<br /> | <br /> | ||
If you have, then to review, a <strong>monzo</strong> is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &quot;prime&quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like <!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15538076?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15538076&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\kettext{a b c}\)<!-- ws:end:WikiTextMediaRule:1 -->, where a, b, and c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <!-- ws:start:WikiTextMediaRule:2:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/ | If you have, then to review, a <strong>monzo</strong> is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &quot;prime&quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like <!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15538076?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15538076&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\kettext{a b c}\)<!-- ws:end:WikiTextMediaRule:1 -->, where a, b, and c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <!-- ws:start:WikiTextMediaRule:2:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15538090?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15538090&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\kettext{a b c d}\)<!-- ws:end:WikiTextMediaRule:2 -->, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.<br /> | ||
<br /> | <br /> | ||
Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this:<br /> | Assuming you understand that, then we've reached our first new idea, which will help us gain a geometric intuition into what some of these abstract entities mean. That idea is this:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc2"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."></a><!-- ws:end:WikiTextHeadingRule:13 -->1.1: A monzo can be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.</h2> | <!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc2"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."></a><!-- ws:end:WikiTextHeadingRule:13 -->1.1: A monzo can be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.</h2> | ||
<br /> | <br /> | ||
For instance, the syntonic comma is <!-- ws:start:WikiTextMediaRule:3:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/ | For instance, the syntonic comma is <!-- ws:start:WikiTextMediaRule:3:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15538088?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15538088&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\kettext{-4 4 -1}\)<!-- ws:end:WikiTextMediaRule:3 -->. A geometric interpretation of this interval might be as a point in a space, like the point <!-- ws:start:WikiTextMediaRule:4:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15538024?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15538024&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->(-4, 4, -1)<!-- ws:end:WikiTextMediaRule:4 -->. You'd plot this point by going -4 steps on the x axis, 4 steps on the y axis, and -1 steps on the z-axis. And if you really want to think of it like a vector in the sense that some high school or college algebra courses teach it, you can also draw an arrow with a big arrowhead from the origin that connects to this point. Here's a widget that lets you plot vectors:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextMediaRule:5:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537326?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537326&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" id="WolframAlphaScriptf5af8de6802460753a75a4692d255641" src="http://www.wolframalpha.com/widget/widget.jsp?id=f5af8de6802460753a75a4692d255641&amp;output=lightbox"> | <!-- ws:start:WikiTextMediaRule:5:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15537326?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537326&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" id="WolframAlphaScriptf5af8de6802460753a75a4692d255641" src="http://www.wolframalpha.com/widget/widget.jsp?id=f5af8de6802460753a75a4692d255641&amp;output=lightbox"> | ||