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

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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

<h4>Original Wikitext content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=[[media type="custom" key="15539114"]]=  

$$

\newcommand{\bra}[1]{\left \langle #1 \right |}

$$

&lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span class="math"&gt;&lt;span style="clip: rect(1.72em 1000em 2.742em -0.558em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 1.731em;"&gt;&lt;span class="mrow"&gt;&lt;span class="mi" style="font-family: MathJax_Math;"&gt;//test//&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

If you haven't seen monzos or vals before and are totally confused, please read the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] 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:

==1.1: A monzo can be viewed as a **VECTOR** in a **VECTOR SPACE**.==  

OK, so how do we use these things?

==1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)==  

So, if you mechanically work out the way that you'd compute how many steps 81/80 is in 12-EDO, you get 0 steps, meaning you're back at 1/1 and hence tempered out. No surprise there.

==1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)==  

Assuming you've understood my exposition thus far, you now hopefully see where all things like monzos and vals come from.

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!

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.

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.

Both of these have different strengths. Gene's approach I find to be remarkably elegant in a certain kind of way, in that multivectors are very intuitive objects to think about. Graham's approach, on the other hand, isn't quite as "elegant," but may be a bit simpler to work out and immediately glean information from - mapping matrices are a bit easier to understand than wedgies.

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$$&lt;br /&gt;

&lt;br /&gt;

  &lt;span style="display: block; text-align: center;"&gt;&lt;span class="MathJax"&gt;&lt;span class="math"&gt;&lt;span style="clip: rect(1.72em 1000em 2.742em -0.558em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 1.731em;"&gt;&lt;span class="mrow"&gt;&lt;span style="font-family: MathJax_Math;" class="mi"&gt;&lt;em&gt;test&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;

If you haven't seen monzos or vals before and are totally confused, please read the pages on &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos"&gt;Monzos&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals"&gt;Vals&lt;/a&gt; first!&lt;br /&gt;

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:&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;1.1: A monzo can be viewed as a &lt;strong&gt;VECTOR&lt;/strong&gt; in a &lt;strong&gt;VECTOR SPACE&lt;/strong&gt;.&lt;/h2&gt;

  &lt;br /&gt;

OK, so how do we use these things?&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)&lt;/h2&gt;

  &lt;br /&gt;

So, if you mechanically work out the way that you'd compute how many steps 81/80 is in 12-EDO, you get 0 steps, meaning you're back at 1/1 and hence tempered out. No surprise there.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="LECTURE 1: Vector Spaces and Dual Spaces-1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)&lt;/h2&gt;

  &lt;br /&gt;

Assuming you've understood my exposition thus far, you now hopefully see where all things like monzos and vals come from.&lt;br /&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;

&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;

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;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

Both of these have different strengths. Gene's approach I find to be remarkably elegant in a certain kind of way, in that multivectors are very intuitive objects to think about. Graham's approach, on the other hand, isn't quite as &amp;quot;elegant,&amp;quot; but may be a bit simpler to work out and immediately glean information from - mapping matrices are a bit easier to understand than wedgies.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

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