Mike's lecture on vector spaces and dual spaces: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 325960332 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 325960438 - 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 07: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 07:32:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>325960438</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="15537816"]], where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like [[media type="custom" key="15537820"]], where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on. | 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="15537816"]], where \(a\) b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like [[media type="custom" key="15537820"]], 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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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 /> | ||
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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/15537816?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537816&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\ket{\text{a b c}}\)<!-- ws:end:WikiTextMediaRule:1 -->, where a b 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/15537820?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537820&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\ket{\text{a b c d}}\)<!-- ws:end:WikiTextMediaRule:2 -->, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.<br /> | 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/15537816?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537816&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\ket{\text{a b c}}\)<!-- ws:end:WikiTextMediaRule:1 -->, where \(a\) b 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/15537820?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15537820&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; -->\(\ket{\text{a b c d}}\)<!-- ws:end:WikiTextMediaRule:2 -->, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.<br /> | ||
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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 /> | ||