Mike's lecture on vector spaces and dual spaces
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=LECTURE 1: Vector Spaces and Dual Spaces=
If you haven't seen monzos 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 |a b c>, where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like |a b c d>, 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:
A monzo can also be viewed as a **VECTOR** in a **VECTOR SPACE**.
For instance, the syntonic comma is |-4 4 -1>. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). 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 picture of that:
{{#widget:WolframAlpha|id=207d0ac77d88b7c7f3b28a5f309715d|theme=orange}}
Paul's "A Middle Path" paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf
Now, here's the interesting part: in linear algebra, every vector space has a "dual space," which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called "covectors."
Covectors can "interact" with vectors, or rather "act on" them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something.
In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c).
OK, what the hell does all of that mean? END LESSON 1Original HTML content:
<html><head><title>Mike's Lecture on Vector Spaces and Dual Spaces</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:0 -->LECTURE 1: Vector Spaces and Dual Spaces</h1> <br /> If you haven't seen monzos 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 /> 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, "prime" intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like |a b c>, where a b c are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like |a b c d>, where d represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.<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 /> <br /> A monzo can also be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.<br /> <br /> For instance, the syntonic comma is |-4 4 -1>. A geometric interpretation of this interval might be as a point in a space, like the point (-4, 4, -1). 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 picture of that:<br /> <br /> <tt>#widget:WolframAlpha|id=207d0ac77d88b7c7f3b28a5f309715d|theme=orange</tt><br /> <br /> Paul's "A Middle Path" paper has so many good plots of this that I might as well just point anyone interested to take a look at it over there: <!-- ws:start:WikiTextUrlRule:21:http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf --><a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf</a><!-- ws:end:WikiTextUrlRule:21 --><br /> Now, here's the interesting part: in linear algebra, every vector space has a "dual space," which of course must be thought of as a bizarro universe for the vector space in which the background is black and the arrows and points are white. The elements in this space are called "covectors."<br /> Covectors can "interact" with vectors, or rather "act on" them, by taking the dot product of the covector and a vector. So for instance, if your covector is (12, 19, 28)* (the star means it's in the dual space), and your vector is (-4, 4, -1), then the dot product of the two is 12*-4 + 19*4 * 28*-1 = 0. This must of course be pictured as the black and white arrows lining up and exploding and spitting out a single number, or something.<br /> In a drier sense, a covector can also be thought of as a type of function that takes in a vector and spits out a number. So (12, 19, 28)* can also be thought of as f(v) = 12a + 19b + 28c for some vector of the form (a, b, c).<br /> OK, what the hell does all of that mean? END LESSON 1</body></html>