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 <math>\ket{a \s b \s c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <math>\ket{a \s b \s c \s d}</math>, where <math>d</math> represents the additional exponent for 7. The 11-limit gets you another coefficient and so on. | 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 <math>\ket{a \s b \s c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <math>\ket{a \s b \s c \s d}</math>, where <math>d</math> 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 | On the other hand, a '''val''' is a way to represent how JI intervals map to tempered steps along a chain of generators. 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 , where , , and are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like , where represents the additional mapping for 7. Like with monzos, going to the 11-limit gets you another coefficient and so on.<math>\bra{x \s y \s z}</math><math>x</math><math>y</math><math>z</math><math>\bra{a \s b \s c \s d}</math><math>d</math>Again, if this is confusing, please go back to the pages on [[monzos|Monzos]] and [[Vals|Vals]] and read those first! | ||
Again, if this is confusing, please go back to the pages on [[monzos|Monzos]] and [[Vals|Vals]] and read those first! | |||
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, which will enable us to rediscover vals and monzos in a much stronger mathematical and geometric context, 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, which will enable us to rediscover vals and monzos in a much stronger mathematical and geometric context, is this: | ||
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For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. 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: | For instance, the syntonic comma is <math>\ket{\-4 \s 4 \s \-1}</math>. A geometric interpretation of this interval might be as a point in a space, like the point <math>(\-4,4,\-1)</math>. 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: | ||
Keep in mind that Wolfram Alpha is very fragile, so if you try to do anything fancy, it's going to break. But, 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: | Keep in mind that Wolfram Alpha is very fragile, so if you try to do anything fancy, it's going to break. But, 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, 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." I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here: | Now, 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." I can't get the exact colors I mentioned here, but I've cheated a bit to get Wolfram to change the colors, so you can plot covectors here: | ||