Mike's lecture on vector spaces and dual spaces: Difference between revisions

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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-: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 09:33:51 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 09:34:24 UTC</tt>.<br>
-: The original revision id was <tt>325998346</tt>.<br>
+: The original revision id was <tt>325998586</tt>.<br>
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 If you have seen it, 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 \(\ket{a \s b \s c}\), where \(a\), \(b\), and \(c\) are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like \(\ket{a \s b \s c \s d}\), where \(d\) represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.
-On the other hand, a **val** is a way to represent how JI intervals map to tempered steps along a chain of generator. A val does this by specifying the mapping for the primes, and in so doing ends up specifying the mapping for every JI interval as well: since every interval is a combination of primes, then we can find the mapping for any interval in some val by simply adding and subtracting the mapping for the primes in such a way that the original interval is recreated. A 5-limit val looks like \(\bra{x \s y \s z}\), where \(x\), \(y\), and \(z\) are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like \(\bra{a \s b \s c \s d}\), where \(\d\) represents the additional mapping for 7. Like with monzos, going to the 11-limit gets you another coefficient and so on.
+On the other hand, a **val** is a way to represent how JI intervals map to tempered steps along a chain of generator. A val does this by specifying the mapping for the primes, and in so doing ends up specifying the mapping for every JI interval as well: since every interval is a combination of primes, then we can find the mapping for any interval in some val by simply adding and subtracting the mapping for the primes in such a way that the original interval is recreated. A 5-limit val looks like \(\bra{x \s y \s z}\), where \(x\), \(y\), and \(z\) are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like \(\bra{a \s b \s c \s d}\), where \(d\) represents the additional mapping for 7. Like with monzos, going to the 11-limit gets you another coefficient and so on.
 Again, if this is confusing, please go back to the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] and read those first!
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 If you have seen it, then to review, a &lt;strong&gt;monzo&lt;/strong&gt; is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &amp;quot;prime&amp;quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like \(\ket{a \s b \s c}\), where \(a\), \(b\), and \(c\) are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like \(\ket{a \s b \s c \s d}\), where \(d\) represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.&lt;br /&gt;
 &lt;br /&gt;
-On the other hand, a &lt;strong&gt;val&lt;/strong&gt; is a way to represent how JI intervals map to tempered steps along a chain of generator. A val does this by specifying the mapping for the primes, and in so doing ends up specifying the mapping for every JI interval as well: since every interval is a combination of primes, then we can find the mapping for any interval in some val by simply adding and subtracting the mapping for the primes in such a way that the original interval is recreated. A 5-limit val looks like \(\bra{x \s y \s z}\), where \(x\), \(y\), and \(z\) are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like \(\bra{a \s b \s c \s d}\), where \(\d\) represents the additional mapping for 7. Like with monzos, going to the 11-limit gets you another coefficient and so on.&lt;br /&gt;
+On the other hand, a &lt;strong&gt;val&lt;/strong&gt; is a way to represent how JI intervals map to tempered steps along a chain of generator. A val does this by specifying the mapping for the primes, and in so doing ends up specifying the mapping for every JI interval as well: since every interval is a combination of primes, then we can find the mapping for any interval in some val by simply adding and subtracting the mapping for the primes in such a way that the original interval is recreated. A 5-limit val looks like \(\bra{x \s y \s z}\), where \(x\), \(y\), and \(z\) are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like \(\bra{a \s b \s c \s d}\), where \(d\) represents the additional mapping for 7. Like with monzos, going to the 11-limit gets you another coefficient and so on.&lt;br /&gt;
 &lt;br /&gt;
 Again, if this is confusing, please go back to the pages on &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos"&gt;Monzos&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals"&gt;Vals&lt;/a&gt; and read those first!&lt;br /&gt;