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

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* Vals, straightforwardly, are elements in the "dual space" to that space - they're covectors - which is the new thing you just learned about and which you'd end up learning in college linear algebra.
* Vals, straightforwardly, are elements in the "dual space" to that space - they're covectors - which is the new thing you just learned about and which you'd end up learning in college linear algebra.


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." Very nice!
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!


Now, historically speaking, the concept of taking JI intervals and treating them like vectors isn't new. Theorists have been doing that for years and years. Neither Paul nor Gene nor Graham nor Joe Monzo nor anyone I know first came up with the concept of plotting JI intervals as vectors on a JI lattice.
Now, historically speaking, the concept of taking JI intervals and treating them like vectors isn't new. Theorists have been doing that for years and years. Neither Paul nor Gene nor Graham nor Joe Monzo nor anyone I know first came up with the concept of plotting JI intervals as vectors on a JI lattice.


The real quantum leap in thought, here, is to consider the musical interpretation of the elements in the -dual space- - the covectors. It's is one of those things that appears to be completely meaningless from a musical standpoint unless you really think about it. Someone at some point figured out all of the above and I think it's the one of the best insights ever made in music theory. Looking at the dual space is, in a sense, the logical completion of the idea of putting JI into a "lattice," as per Tenney/Fokker/Wilson/etc.
The real quantum leap in thought, here, is to consider the musical interpretation of the elements in the //dual space// -- the covectors. It's is one of those things that appears to be completely meaningless from a musical standpoint unless you really think about it. Someone at some point figured out all of the above and I think it's the one of the best and most whole-brained insights ever made in music theory. Looking at the dual space is, in a sense, the logical completion of the idea of putting JI into a "lattice," as per Tenney/Fokker/Wilson/etc.


Now, where do we go from here?
Now, where do we go from here?


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. Some of these linear-algebraic manipulations may seem to have no musical purpose at all. Others do. Others may seem to have no purpose at all, at first, and then end up having a purpose that strikes you in a flash after a little bit of thought.
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. These will be seen to represent temperaments.&lt;span style="font-size: 80%; vertical-align: super;"&gt;[[Mike's Lecture on Vector Spaces and Dual Spaces#ref2|{2}]]&lt;/span&gt;
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.&lt;span style="font-size: 80%; vertical-align: super;"&gt;[[Mike's Lecture on Vector Spaces and Dual Spaces#ref2|{2}]]&lt;/span&gt;


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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&lt;br /&gt;
&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Monzos, straightforwardly, are elements in a vector space - they're vectors - like the spaces you learned in high school algebra.&lt;/li&gt;&lt;li&gt;Vals, straightforwardly, are elements in the &amp;quot;dual space&amp;quot; to that space - they're covectors - which is the new thing you just learned about and which you'd end up learning in college linear algebra.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Monzos, straightforwardly, are elements in a vector space - they're vectors - like the spaces you learned in high school algebra.&lt;/li&gt;&lt;li&gt;Vals, straightforwardly, are elements in the &amp;quot;dual space&amp;quot; to that space - they're covectors - which is the new thing you just learned about and which you'd end up learning in college linear algebra.&lt;/li&gt;&lt;/ul&gt;&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; Very nice!&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;
&lt;br /&gt;
&lt;br /&gt;
Now, historically speaking, the concept of taking JI intervals and treating them like vectors isn't new. Theorists have been doing that for years and years. Neither Paul nor Gene nor Graham nor Joe Monzo nor anyone I know first came up with the concept of plotting JI intervals as vectors on a JI lattice.&lt;br /&gt;
Now, historically speaking, the concept of taking JI intervals and treating them like vectors isn't new. Theorists have been doing that for years and years. Neither Paul nor Gene nor Graham nor Joe Monzo nor anyone I know first came up with the concept of plotting JI intervals as vectors on a JI lattice.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The real quantum leap in thought, here, is to consider the musical interpretation of the elements in the -dual space- - the covectors. It's is one of those things that appears to be completely meaningless from a musical standpoint unless you really think about it. Someone at some point figured out all of the above and I think it's the one of the best insights ever made in music theory. Looking at the dual space is, in a sense, the logical completion of the idea of putting JI into a &amp;quot;lattice,&amp;quot; as per Tenney/Fokker/Wilson/etc.&lt;br /&gt;
The real quantum leap in thought, here, is to consider the musical interpretation of the elements in the &lt;em&gt;dual space&lt;/em&gt; -- the covectors. It's is one of those things that appears to be completely meaningless from a musical standpoint unless you really think about it. Someone at some point figured out all of the above and I think it's the one of the best and most whole-brained insights ever made in music theory. Looking at the dual space is, in a sense, the logical completion of the idea of putting JI into a &amp;quot;lattice,&amp;quot; as per Tenney/Fokker/Wilson/etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Now, where do we go from here?&lt;br /&gt;
Now, where do we go from here?&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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 &amp;quot;linear algebra.&amp;quot; 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. Some of these linear-algebraic manipulations may seem to have no musical purpose at all. Others do. Others may seem to have no purpose at all, at first, and then end up having a purpose that strikes you in a flash after a little bit of thought.&lt;br /&gt;
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 &amp;quot;linear algebra.&amp;quot; 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
One of the manipulations that would appear to be totally useless is a deceptively simple extension of all of this called &lt;strong&gt;exterior algebra&lt;/strong&gt;. It introduces a single product, called the &lt;strong&gt;wedge product&lt;/strong&gt;, 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 &lt;strong&gt;multivector&lt;/strong&gt; - defined as the product of vectors. These will be seen to represent temperaments.&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#ref2"&gt;{2}&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
One of the manipulations that would appear to be totally useless is a deceptively simple extension of all of this called &lt;strong&gt;exterior algebra&lt;/strong&gt;. It introduces a single product, called the &lt;strong&gt;wedge product&lt;/strong&gt;, 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 &lt;strong&gt;multivector&lt;/strong&gt; - defined as the product of vectors. Turns out it's not useless at all - these will be seen to represent temperaments.&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#ref2"&gt;{2}&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Another manipulation that might appear to be useless is, rather than to generalize vectors to multivectors, to generalize them to &lt;strong&gt;matrices&lt;/strong&gt; - 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.&lt;br /&gt;
Another manipulation that might appear to be useless is, rather than to generalize vectors to multivectors, to generalize them to &lt;strong&gt;matrices&lt;/strong&gt; - 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.&lt;br /&gt;
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