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

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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.

For instance, the syntonic comma is \(\ket{\-4 \s 4 \s \-1}\). 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 widget that lets you plot vectors:

You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like \((12,19,28),\s(7,11,16)\). However, this will break the nice color properties I set up above. Also, if you put in too many entries, Wolfram Alpha has been known to break.

For example: say 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&lt;span style="font-size: 80%; vertical-align: super;"&gt;[[Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]]&lt;/span&gt; of the two is \(12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0\). Thus, the result of \((12,19,28)^*\) acting on \((\-4,4,\-1)\) is \(0\).

In other words, you know that the action of the covector \((12,19,28)^*\) on any arbitrary vector \((a,b,c)\) is going to be \(12a + 19b + 28c\). So, you can think of \((12,19,28)^*\) itself as a function looking something like \(f(\v{v}) = 12a + 19b + 28c\) for some vector of the form \((a,b,c)\). Note that the formatting on \(\v{v}\) specifies that \(\v{v}\) is a vector that's being taken in as input.

Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors \(\bratext{like this}\) and to notate vectors \(\kettext{like this}\). Physicists call this "bra-ket" notation, or sometimes "Dirac" notation. So...

* Instead of writing covectors as \((x,y,z)^*\), I'll just write \(\bra{x \s y \s z}\) from now on.

* Instead of writing vectors as \((a,b,c)\), I'll just write \(\ket{a \s b \s c}\) from now on.

But, if you really think about it, what you've also just done is evaluate the expression \(12*\-4 + 19*4 + 28*\-1\): down four (possibly tempered) octaves, up four tempered tritaves, down a tempered 5/1, see what the result is. This is the same exact thing as multiplying out \(\braket{12 \s 19 \s 28}{\-4 \s 4 \s -1}\). Looks like you've just seemingly applied \(\bra{12 \s 19 \s 18}\) to \(\ket{\-4 \s 4 \s \-1}\).

Also, hopefully, now you see that this whole \(\brakettext{covector}{vector}\) thing is just a mathematical representation of the mechanical process you'd be doing if you sat down in scala for a while and stacked and subtracted intervals and figured out what you end up arriving at in some EDO. In other words, \(\bra{12 \s 19 \s 28}\) is just a little machine that takes in JI intervals and spits out steps, and the mathematical name for this sort of little machine is "covector." Nice!

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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;

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;

For instance, the syntonic comma is \(\ket{\-4 \s 4 \s \-1}\). 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 widget that lets you plot vectors:&lt;br /&gt;

You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like \((12,19,28),\s(7,11,16)\). However, this will break the nice color properties I set up above. Also, if you put in too many entries, Wolfram Alpha has been known to break.&lt;br /&gt;

For example: say 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&lt;span style="font-size: 80%; vertical-align: super;"&gt;&lt;a class="wiki_link" href="/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1"&gt;{1}&lt;/a&gt;&lt;/span&gt; of the two is \(12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0\). Thus, the result of \((12,19,28)^*\) acting on \((\-4,4,\-1)\) is \(0\).&lt;br /&gt;

In other words, you know that the action of the covector \((12,19,28)^*\) on any arbitrary vector \((a,b,c)\) is going to be \(12a + 19b + 28c\). So, you can think of \((12,19,28)^*\) itself as a function looking something like \(f(\v{v}) = 12a + 19b + 28c\) for some vector of the form \((a,b,c)\). Note that the formatting on \(\v{v}\) specifies that \(\v{v}\) is a vector that's being taken in as input.&lt;br /&gt;

Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors \(\bratext{like this}\) and to notate vectors \(\kettext{like this}\). Physicists call this &amp;quot;bra-ket&amp;quot; notation, or sometimes &amp;quot;Dirac&amp;quot; notation. So...&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;Instead of writing covectors as \((x,y,z)^*\), I'll just write \(\bra{x \s y \s z}\) from now on.&lt;/li&gt;&lt;li&gt;Instead of writing vectors as \((a,b,c)\), I'll just write \(\ket{a \s b \s c}\) from now on.&lt;/li&gt;&lt;li&gt;When I want to denote the dot product of a covector and a vector, I'll write it as \braket{x \s y \s z}{a \s b \s c}.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

But, if you really think about it, what you've also just done is evaluate the expression \(12*\-4 + 19*4 + 28*\-1\): down four (possibly tempered) octaves, up four tempered tritaves, down a tempered 5/1, see what the result is. This is the same exact thing as multiplying out \(\braket{12 \s 19 \s 28}{\-4 \s 4 \s -1}\). Looks like you've just seemingly applied \(\bra{12 \s 19 \s 18}\) to \(\ket{\-4 \s 4 \s \-1}\).&lt;br /&gt;

Also, hopefully, now you see that this whole \(\brakettext{covector}{vector}\) thing is just a mathematical representation of the mechanical process you'd be doing if you sat down in scala for a while and stacked and subtracted intervals and figured out what you end up arriving at in some EDO. In other words, \(\bra{12 \s 19 \s 28}\) is just a little machine that takes in JI intervals and spits out steps, and the mathematical name for this sort of little machine is &amp;quot;covector.&amp;quot; Nice!&lt;br /&gt;