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

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==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''.==
==1.1: A monzo can be viewed as a '''VECTOR''' in a '''VECTOR SPACE'''<ref>Technically, monzos don't form a vector space but a "'''Z'''-module", because monzos only take integer coefficients and '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. One difference is that modules are not necessarily free, unlike vector spaces, and we want temperaments to be free '''Z'''-modules. (For practical purposes, this difference just means that we have to be careful not to temper out a power of a comma, or take a wedgie out of a [[contorsion|contorted]] val like 24p in the 5-limit.)</ref>.==


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:
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This is all well and good by itself, but it doesn't mean anything unless you understand how covectors interact with vectors. Covectors are mathematical objects that are thought to ''act on'' vectors. When a covector "acts on" a vector, the interaction occurs by you taking the '''dot product''' of the two vectors.
This is all well and good by itself, but it doesn't mean anything unless you understand how covectors interact with vectors. Covectors are mathematical objects that are thought to ''act on'' vectors. When a covector "acts on" a vector, the interaction occurs by you taking the '''dot product''' of the two vectors.


For example: say your covector is <math>(12,19,28)^*</math> (the star means it's in the dual space), and your vector is <math>(\-4,4,\-1)</math>, then the dot product[[Mike's_Lecture_on_Vector_Spaces_and_Dual_Spaces#ref1|{1}]] of the two is <math>12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0</math>. Thus, the result of <math>(12,19,28)^*</math> acting on <math>(\-4,4,\-1)</math> is <math>0</math>.
For example: say your covector is <math>(12,19,28)^*</math> (the star means it's in the dual space), and your vector is <math>(\-4,4,\-1)</math>, then the dot product<ref>Note that some have raised technical concerns about this operation being called the "dot product," insisting that the dot product is something that's only done between two vectors, or two covectors, but never between one covector and one vector. Another term that's sometimes been used for this product in the "'''bracket product'''", for reasons we don't need to get into here. However, confusingly, the term bracket product has also been used for the ordinary dot product, and it's also very common to hear people call the thing I'm calling the dot product above. It's best at this point to just know that the two terms are out there. I'm going to continue calling it the dot product since its' something more people are familiar with.</ref> of the two is <math>12 \cdot \-4 + 19 \cdot 4 + 28 \cdot \-1 = 0</math>. Thus, the result of <math>(12,19,28)^*</math> acting on <math>(\-4,4,\-1)</math> is <math>0</math>.


The action of a covector on a vector must, of course, be pictured as the different colored arrows lining up and exploding and spitting out a single number, or something. Wolfram unfortunately doesn't let me do nice explosion effects, so you'll have to imagine it.
The action of a covector on a vector must, of course, be pictured as the different colored arrows lining up and exploding and spitting out a single number, or something. Wolfram unfortunately doesn't let me do nice explosion effects, so you'll have to imagine it.
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So you already knew that intervals like 81/80, if placed in monzo form, can be viewed as vectors. In light of the above example, it's now starting to look pretty clear that equal temperament mappings, if placed in val form, can be viewed as covectors.
So you already knew that intervals like 81/80, if placed in monzo form, can be viewed as vectors. In light of the above example, it's now starting to look pretty clear that equal temperament mappings, if placed in val form, can be viewed as covectors.


In fact, this is exactly what's going on. Vals are simply covectors acting on the space of monzos. Once you invent monzos and decide that they're vectors, the laws of mathematics set themselves up so that you get vals as covectors for free.[[Mike's_Lecture_on_Vector_Spaces_and_Dual_Spaces#ref2|{2}]] Very nice!
In fact, this is exactly what's going on. Vals are simply covectors acting on the space of monzos. Once you invent monzos and decide that they're vectors, the laws of mathematics set themselves up so that you get vals as covectors for free.<ref>As we'll soon see, vals aren't the only sorts of covectors there are. There's another way to interpret the space of covectors, which is as "'''tuning maps'''" which assign an actual tuning value in cents to the intervals in a temperament. But we're not there yet, so all you need to concern yourself with at this point is vals as covectors and monzos as vectors.
</ref> Very nice!


But where do we go with this realization?
But where do we go with this realization?
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Well, if there's one thing mathematicians know a lot of random crap about, it's vectors and covectors. They've been exploring manipulations of these sorts of objects for literally thousands of years under the moniker of "linear algebra." These objects may be new to music theory, but they're definitely not new to math. And since mathematics has concerned itself for such a long time with manipulating covectors and vectors, we can simply take some of the techniques that mathematicians have developed to do so, and apply them here, in a musical context.
Well, if there's one thing mathematicians know a lot of random crap about, it's vectors and covectors. They've been exploring manipulations of these sorts of objects for literally thousands of years under the moniker of "linear algebra." These objects may be new to music theory, but they're definitely not new to math. And since mathematics has concerned itself for such a long time with manipulating covectors and vectors, we can simply take some of the techniques that mathematicians have developed to do so, and apply them here, in a musical context.


One of the manipulations that would appear to be totally useless is a deceptively simple extension of all of this called '''exterior algebra'''. It introduces a single product, called the '''wedge product''', and you can multiply vectors together using this in a similar sort of way that you'd multiply polynomials on paper in high school algebra or something. It also introduces a new type of object, called a '''multivector''' - defined as the product of vectors. Turns out it's not useless at all - these will be seen to represent temperaments.[[Mike's_Lecture_on_Vector_Spaces_and_Dual_Spaces#ref2|{2}]]
One of the manipulations that would appear to be totally useless is a deceptively simple extension of all of this called '''exterior algebra'''. It introduces a single product, called the '''wedge product''', and you can multiply vectors together using this in a similar sort of way that you'd multiply polynomials on paper in high school algebra or something. It also introduces a new type of object, called a '''multivector''' - defined as the product of vectors. Turns out it's not useless at all - these will be seen to represent temperaments.<ref>This is one of those things that seems totally useless unless you're [[Gene_Ward_Smith|Gene Smith]], at which point you realize that multivals represent higher-rank temperaments, and that this random field of mathematics has a rather musical interpretation.</ref>


Another manipulation that might appear to be useless is, rather than to generalize vectors to multivectors, to generalize them to '''matrices''' - we can view vectors/monzos as column matrices, and covectors/vals as row matrices, and arrive at many of the same concepts above, but from this different approach. Instead of multiplying vectors together with a special multiplication operation, we can concatenate them into matrices which also represent temperaments.
Another manipulation that might appear to be useless is, rather than to generalize vectors to multivectors, to generalize them to '''matrices''' - we can view vectors/monzos as column matrices, and covectors/vals as row matrices, and arrive at many of the same concepts above, but from this different approach. Instead of multiplying vectors together with a special multiplication operation, we can concatenate them into matrices which also represent temperaments.
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[0] - Technically, monzos don't form a vector space but a "'''Z'''-module", because monzos only take integer coefficients and '''Z''', the set of integers, is a ring but not a field. Similarly vals are a dual '''Z'''-module to the monzos. One difference is that modules are not necessarily free, unlike vector spaces, and we want temperaments to be free '''Z'''-modules. (For practical purposes, this difference just means that we have to be careful not to temper out a power of a comma, or take a wedgie out of a [[contorsion|contorted]] val like 24p in the 5-limit.)
<references/>
 
[1] - Note that some have raised technical concerns about this operation being called the "dot product," insisting that the dot product is something that's only done between two vectors, or two covectors, but never between one covector and one vector. Another term that's sometimes been used for this product in the "'''bracket product'''", for reasons we don't need to get into here. However, confusingly, the term bracket product has also been used for the ordinary dot product, and it's also very common to hear people call the thing I'm calling the dot product above. It's best at this point to just know that the two terms are out there. I'm going to continue calling it the dot product since its' something more people are familiar with.
 
[2] - As we'll soon see, vals aren't the only sorts of covectors there are. There's another way to interpret the space of covectors, which is as "'''tuning maps'''" which assign an actual tuning value in cents to the intervals in a temperament. But we're not there yet, so all you need to concern yourself with at this point is vals as covectors and monzos as vectors.
 
[3] - This is one of those things that seems totally useless unless you're [[Gene_Ward_Smith|Gene Smith]], at which point you realize that multivals represent higher-rank temperaments, and that this random field of mathematics has a rather musical interpretation.
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