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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | <h2>IMPORTED REVISION FROM WIKISPACES</h2> |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> |
| : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 10:33:47 UTC</tt>.<br> | | : This revision was by author [[User:guest|guest]] and made on <tt>2012-04-27 12:29:43 UTC</tt>.<br> |
| : The original revision id was <tt>326024062</tt>.<br> | | : The original revision id was <tt>326075366</tt>.<br> |
| : The revision comment was: <tt></tt><br> | | : The revision comment was: <tt>Reverted to Apr 27, 2012 7:31 am</tt><br> |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | | 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> | | <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">**<span style="font-size: 200%;">[[Mike's Lectures On Regular Temperament Theory|MIKE'S LECTURES ON REGULAR TEMPERAMENT THEORY]]</span>** | | <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"]]= |
| [[media type="custom" key="15539114"]] | |
| [[toc]] | | [[toc]] |
| <span style="display: block; text-align: center;">
| | $$\newcommand{\bra}[1]{\left \langle #1 \right |} |
| </span>
| | \newcommand{\ket}[1]{\left |#1 \right \rangle} |
| | | \newcommand{\braket}[2]{\left \langle #1 \middle |#2 \right \rangle} \newcommand{\bratext}[1]{\left \langle \text{#1} \right |} |
| | \newcommand{\kettext}[1]{\left |\text{#1} \right \rangle} |
| | \newcommand{\brakettext}[2]{\left \langle \text{#1} \middle |\text{#2} \right \rangle} |
| | \newcommand{\v}[1]{\boldsymbol{\vec{#1}}} |
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| | \newcommand{\s}[0]{\;\,}$$ |
| =__LECTURE 1: Vector Spaces and Dual Spaces__= | | =__LECTURE 1: Vector Spaces and Dual Spaces__= |
| <span style="display: block; text-align: center;"><span class="MathJax"><span class="math"><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;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//test//</span></span></span></span></span></span> | | <span style="display: block; text-align: center;"><span class="MathJax"><span class="math"><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;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//test//</span></span></span></span></span></span> |
| 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! | | 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! |
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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 <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.821em;"><span class="mrow"><span class="mfenced"><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//abc//</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.905em 1000em 2.741em -0.544em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//a//</span></span></span></span></span>, <span class="MathJax"><span class="math"><span style="clip: rect(1.652em 1000em 2.742em -0.537em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.449em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//b//</span></span></span></span></span>, and <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.742em -0.543em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.449em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//c//</span></span></span></span></span>are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span class="mfenced"><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//abcd//</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.652em 1000em 2.741em -0.544em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//d//</span></span></span></span></span>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 \(\ket{a \s b \s c}\), 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. |
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| 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 <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//xyz//</span></span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.742em -0.542em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.577em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//x//</span></span></span></span></span>, <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.936em -0.556em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//y//</span></span></span></span></span>, and <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.742em -0.542em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.449em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//z//</span></span></span></span></span>are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//abcd//</span></span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.652em 1000em 2.741em -0.544em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//d//</span></span></span></span></span>represents the additional mapping for 7. Like with monzos, going to 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. |
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| Again, if this is confusing, please go back to the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] and read those first! | | Again, if this is confusing, please go back to the pages on [[xenharmonic/Monzos|Monzos]] and [[xenharmonic/Vals|Vals]] and read those first! |
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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**.== |
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| For instance, the syntonic comma is <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.59em;"><span class="mrow"><span class="mfenced"><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">44</span><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">1</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>. A geometric interpretation of this interval might be as a point in a space, like the point <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">4</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">4</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mtext" style="font-family: MathJax_Main; padding-left: 0.167em;">-</span><span class="mn" style="font-family: MathJax_Main;">1</span><span class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span>. 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 \(\ket{\-4 \s 4 \s \-1}\). 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: |
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| [[media type="custom" key="15537326"]] | | [[media type="custom" key="15537326"]] |
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| [[media type="custom" key="15537360"]] | | [[media type="custom" key="15537360"]] |
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| You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 9.487em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">19</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">28</span><span class="mo" style="font-family: MathJax_Main;">),</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.167em;">(</span><span class="mn" style="font-family: MathJax_Main;">7</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">11</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">16</span><span class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span>. 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. | | 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. |
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| **So then, what's the point?** | | **So then, what's the point?** |
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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. |
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| For example: say your covector is <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">19</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span class="mo" style="font-family: MathJax_Main;">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span class="mo" style="font-family: MathJax_Main; font-size: 70.7%;">∗</span></span></span></span></span></span></span></span>(the star means it's in the dual space), and your vector is <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">4</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">4</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mtext" style="font-family: MathJax_Main; padding-left: 0.167em;">-</span><span class="mn" style="font-family: MathJax_Main;">1</span><span class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span>, then the dot product<span style="font-size: 80%; vertical-align: super;">[[Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]]</span> of the two is <span class="MathJax"><span class="math"><span style="clip: rect(1.926em 1000em 3.069em -0.494em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 11.282em;"><span class="mrow"><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">⋅</span><span class="mtext" style="font-family: MathJax_Main; padding-left: 0.222em;">-</span><span class="mn" style="font-family: MathJax_Main;">4</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">19</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">⋅</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">4</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">28</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">⋅</span><span class="mtext" style="font-family: MathJax_Main; padding-left: 0.222em;">-</span><span class="mn" style="font-family: MathJax_Main;">1</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.278em;">0</span></span></span></span></span>. Thus, the result of <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">19</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span class="mo" style="font-family: MathJax_Main;">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span class="mo" style="font-family: MathJax_Main; font-size: 70.7%;">∗</span></span></span></span></span></span></span></span>acting on <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">4</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">4</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mtext" style="font-family: MathJax_Main; padding-left: 0.167em;">-</span><span class="mn" style="font-family: MathJax_Main;">1</span><span class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span>is <span class="MathJax"><span class="math"><span style="clip: rect(1.937em 1000em 3.009em -0.538em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 0.513em;"><span class="mrow"><span class="mn" style="font-family: MathJax_Main;">0</span></span></span></span></span>. | | 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<span style="font-size: 80%; vertical-align: super;">[[Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]]</span> 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 \(0\). |
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| 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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| One interesting way to think of covectors, since they're these dual vectors that "act on" normal vectors, is as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number. | | One interesting way to think of covectors, since they're these dual vectors that "act on" normal vectors, is as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number. |
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| In other words, you know that the action of the covector <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">19</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span class="mo" style="font-family: MathJax_Main;">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span class="mo" style="font-family: MathJax_Main; font-size: 70.7%;">∗</span></span></span></span></span></span></span></span>on any arbitrary vector <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mi" style="font-family: MathJax_Math;">//a//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//b//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//c//</span><span class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span>is going to be <span class="MathJax"><span class="math"><span style="clip: rect(1.909em 1000em 3.069em -0.494em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 6.795em;"><span class="mrow"><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mi" style="font-family: MathJax_Math;">//a//</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">19</span><span class="mi" style="font-family: MathJax_Math;">//b//</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">28</span><span class="mi" style="font-family: MathJax_Math;">//c//</span></span></span></span></span>. So, you can think of <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">19</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.167em;">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span class="mo" style="font-family: MathJax_Main;">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span class="mo" style="font-family: MathJax_Main; font-size: 70.7%;">∗</span></span></span></span></span></span></span></span>itself as a function looking something like <span class="MathJax"><span class="math"><span style="clip: rect(1.779em 1000em 3.237em -0.522em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 9.936em;"><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//f//</span><span class="mo" style="font-family: MathJax_Main;">(</span><span class="texatom"><span class="mrow"><span class="munderover"><span style="display: inline-block; height: 0px; position: relative; width: 0.577em;"><span style="clip: rect(1.893em 1000em 2.739em -0.553em); left: 0em; position: absolute; top: -2.538em;"><span class="mi" style="font-family: MathJax_Math;">**//v//**</span></span><span style="left: 0.066em; position: absolute; top: -4.1em;"><span class="mo" style="font-family: MathJax_Main;">**⃗** </span></span></span></span></span></span><span class="mo" style="font-family: MathJax_Main;">)</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.278em;">12</span><span class="mi" style="font-family: MathJax_Math;">//a//</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">19</span><span class="mi" style="font-family: MathJax_Math;">//b//</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">28</span><span class="mi" style="font-family: MathJax_Math;">//c//</span></span></span></span></span>for some vector of the form <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mi" style="font-family: MathJax_Math;">//a//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//b//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//c//</span><span class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span>. Note that the formatting on <span class="MathJax"><span class="math"><span style="clip: rect(2.984em 1000em 4.2em -1.053em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -4em; width: 0.577em;"><span class="mrow"><span class="texatom"><span class="mrow"><span class="munderover"><span style="display: inline-block; height: 0px; position: relative; width: 0.577em;"><span style="clip: rect(1.893em 1000em 2.739em -0.553em); left: 0em; position: absolute; top: -2.538em;"><span class="mi" style="font-family: MathJax_Math;">**//v//**</span></span><span style="left: 0.066em; position: absolute; top: -4.1em;"><span class="mo" style="font-family: MathJax_Main;">**⃗** </span></span></span></span></span></span></span></span></span></span>specifies that <span class="MathJax"><span class="math"><span style="clip: rect(2.984em 1000em 4.2em -1.053em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -4em; width: 0.577em;"><span class="mrow"><span class="texatom"><span class="mrow"><span class="munderover"><span style="display: inline-block; height: 0px; position: relative; width: 0.577em;"><span style="clip: rect(1.893em 1000em 2.739em -0.553em); left: 0em; position: absolute; top: -2.538em;"><span class="mi" style="font-family: MathJax_Math;">**//v//**</span></span><span style="left: 0.066em; position: absolute; top: -4.1em;"><span class="mo" style="font-family: MathJax_Main;">**⃗** </span></span></span></span></span></span></span></span></span></span>is a vector that's being taken in as input. | | 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. |
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| Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too. | | Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too. |
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| Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.974em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mtext" style="font-family: MathJax_Main;">like this</span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>and to notate vectors <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.974em;"><span class="mrow"><span class="mfenced"><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mtext" style="font-family: MathJax_Main;">like this</span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>. Physicists call this "bra-ket" notation, or sometimes "Dirac" notation. So... | | 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 <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.526em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mi" style="font-family: MathJax_Math;">//x//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//y//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//z//</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span class="mo" style="font-family: MathJax_Main;">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span class="mo" style="font-family: MathJax_Main; font-size: 70.7%;">∗</span></span></span></span></span></span></span></span>, I'll just write <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//xyz//</span></span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>from now on. | | * 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 <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span class="mo" style="font-family: MathJax_Main;">(</span><span class="mi" style="font-family: MathJax_Math;">//a//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//b//</span><span class="mo" style="font-family: MathJax_Main;">,</span><span class="mi" style="font-family: MathJax_Math; padding-left: 0.167em;">//c//</span><span class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span>, I'll just write <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.821em;"><span class="mrow"><span class="mfenced"><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span class="mi" style="font-family: MathJax_Math;">//abc//</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>from now on. | | * Instead of writing vectors as \((a,b,c)\), I'll just write \(\ket{a \s b \s c}\) from now on. |
| * 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}. | | * 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}. |
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| 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. | | 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. |
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| But, if you really think about it, what you've also just done is evaluate the expression <span class="MathJax"><span class="math"><span style="clip: rect(1.926em 1000em 3.069em -0.494em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 10.256em;"><span class="mrow"><span class="mn" style="font-family: MathJax_Main;">12</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">∗</span><span class="mtext" style="font-family: MathJax_Main; padding-left: 0.222em;">-</span><span class="mn" style="font-family: MathJax_Main;">4</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">19</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">∗</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">4</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">+</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">28</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">∗</span><span class="mtext" style="font-family: MathJax_Main; padding-left: 0.222em;">-</span><span class="mn" style="font-family: MathJax_Main;">1</span></span></span></span></span>: 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 <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 8.654em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span class="mn" style="font-family: MathJax_Main;">121928</span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">44</span><span class="mo" style="font-family: MathJax_Main; padding-left: 0.222em;">−</span><span class="mn" style="font-family: MathJax_Main; padding-left: 0.222em;">1</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>. Looks like you've just seemingly applied <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 4.487em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span class="mn" style="font-family: MathJax_Main;">121918</span></span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>to <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.59em;"><span class="mrow"><span class="mfenced"><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">44</span><span class="mtext" style="font-family: MathJax_Main;">-</span><span class="mn" style="font-family: MathJax_Main;">1</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>. | | 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 \(\ket{\-4 \s 4 \s \-1}\). |
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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. |
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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. |
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| Also, hopefully, now you see that this whole <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 7.372em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span class="mtext" style="font-family: MathJax_Main;">covector</span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span><span class="mtext" style="font-family: MathJax_Main;">vector</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>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, <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 4.487em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span class="mn" style="font-family: MathJax_Main;">121928</span></span><span class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>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! | | 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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| 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. |
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| [[media type="custom" key="15540544"]]</pre></div> | | [[media type="custom" key="15540544"]]</pre></div> |
| <h4>Original HTML content:</h4> | | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Mike's Lecture on Vector Spaces and Dual Spaces</title></head><body><strong><span style="font-size: 200%;"><a class="wiki_link" href="/Mike%27s%20Lectures%20On%20Regular%20Temperament%20Theory">MIKE'S LECTURES ON REGULAR TEMPERAMENT THEORY</a></span></strong><br /> | | <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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Mike's Lecture on Vector Spaces and Dual Spaces</title></head><body><!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/15539114?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;15539114&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> |
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| <!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 1em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces">LECTURE 1: Vector Spaces and Dual Spaces</a></div> | | <!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 1em;"><a href="#toc0"></a></div> |
| <!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 2em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces-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.</a></div> | | <!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 1em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces">LECTURE 1: Vector Spaces and Dual Spaces</a></div> |
| <!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 2em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)">1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)</a></div> | | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 2em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces-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.</a></div> |
| <!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 2em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces-1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)">1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)</a></div> | | <!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 2em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)">1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)</a></div> |
| <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --></div> | | <!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><div style="margin-left: 2em;"><a href="#LECTURE 1: Vector Spaces and Dual Spaces-1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)">1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)</a></div> |
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| | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc1"><a name="LECTURE 1: Vector Spaces and Dual Spaces"></a><!-- ws:end:WikiTextHeadingRule:6 --><u>LECTURE 1: Vector Spaces and Dual Spaces</u></h1> |
| <span style="display: block; text-align: center;"><span class="MathJax"><span class="math"><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;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>test</em></span></span></span></span></span></span><br /> | | <span style="display: block; text-align: center;"><span class="MathJax"><span class="math"><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;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>test</em></span></span></span></span></span></span><br /> |
| If you haven't seen monzos or vals before and are totally confused, please read the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> first!<br /> | | If you haven't seen monzos or vals before and are totally confused, please read the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> first!<br /> |
| <br /> | | <br /> |
| If you have seen it, then to review, a <strong>monzo</strong> is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &quot;prime&quot; intervals. It does so by directly representing an interval's prime factorization. A 5-limit monzo looks like <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.821em;"><span class="mrow"><span class="mfenced"><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>abc</em></span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.905em 1000em 2.741em -0.544em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>a</em></span></span></span></span></span>, <span class="MathJax"><span class="math"><span style="clip: rect(1.652em 1000em 2.742em -0.537em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.449em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>b</em></span></span></span></span></span>, and <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.742em -0.543em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.449em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>c</em></span></span></span></span></span>are the exponents for primes 2, 3, and 5, respectively. A 7-limit JI monzo looks like <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span class="mfenced"><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>abcd</em></span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.652em 1000em 2.741em -0.544em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>d</em></span></span></span></span></span>represents the additional exponent for 7. The 11-limit gets you another coefficient and so on.<br /> | | If you have seen it, then to review, a <strong>monzo</strong> is a way to represent a JI interval that shows how it decomposes into a combination of simpler, &quot;prime&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.<br /> |
| <br /> | | <br /> |
| On the other hand, a <strong>val</strong> 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 <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>xyz</em></span></span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.742em -0.542em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.577em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>x</em></span></span></span></span></span>, <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.936em -0.556em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>y</em></span></span></span></span></span>, and <span class="MathJax"><span class="math"><span style="clip: rect(1.904em 1000em 2.742em -0.542em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.449em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>z</em></span></span></span></span></span>are the number of steps along the chain that primes 2, 3, and 5 map to, respectively. A 7-limit val looks like <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>abcd</em></span></span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>, where <span class="MathJax"><span class="math"><span style="clip: rect(1.652em 1000em 2.741em -0.544em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.538em; width: 0.513em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>d</em></span></span></span></span></span>represents the additional mapping for 7. Like with monzos, going to the 11-limit gets you another coefficient and so on.<br /> | | On the other hand, a <strong>val</strong> 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.<br /> |
| <br /> | | <br /> |
| Again, if this is confusing, please go back to the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> and read those first!<br /> | | Again, if this is confusing, please go back to the pages on <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Monzos">Monzos</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vals">Vals</a> and read those first!<br /> |
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc1"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."></a><!-- ws:end:WikiTextHeadingRule:6 -->1.1: A monzo can be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.</h2> | | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc2"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.1: A monzo can be viewed as a VECTOR** in a **VECTOR SPACE."></a><!-- ws:end:WikiTextHeadingRule:8 -->1.1: A monzo can be viewed as a <strong>VECTOR</strong> in a <strong>VECTOR SPACE</strong>.</h2> |
| <br /> | | <br /> |
| For instance, the syntonic comma is <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.59em;"><span class="mrow"><span class="mfenced"><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">44</span><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">1</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>. A geometric interpretation of this interval might be as a point in a space, like the point <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">4</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">4</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">1</span><span style="font-family: MathJax_Main;" class="mo">)</span></span></span></span></span>. 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:<br /> | | 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:<br /> |
| <br /> | | <br /> |
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| You can also plot more than one vector or covector by putting in a list of vectors separated by commas, something like <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 9.487em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">19</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">28</span><span style="font-family: MathJax_Main;" class="mo">),</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mn">7</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">11</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">16</span><span style="font-family: MathJax_Main;" class="mo">)</span></span></span></span></span>. 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.<br /> | | 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.<br /> |
| <br /> | | <br /> |
| <strong>So then, what's the point?</strong><br /> | | <strong>So then, what's the point?</strong><br /> |
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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 <em>act on</em> vectors. When a covector &quot;acts on&quot; a vector, the interaction occurs by you taking the <strong>dot product</strong> of the two vectors.<br /> | | 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 <em>act on</em> vectors. When a covector &quot;acts on&quot; a vector, the interaction occurs by you taking the <strong>dot product</strong> of the two vectors.<br /> |
| <br /> | | <br /> |
| For example: say your covector is <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">19</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span style="font-family: MathJax_Main;" class="mo">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span style="font-family: MathJax_Main; font-size: 70.7%;" class="mo">∗</span></span></span></span></span></span></span></span>(the star means it's in the dual space), and your vector is <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">4</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">4</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">1</span><span style="font-family: MathJax_Main;" class="mo">)</span></span></span></span></span>, then the dot product<span style="font-size: 80%; vertical-align: super;"><a class="wiki_link" href="/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1">{1}</a></span> of the two is <span class="MathJax"><span class="math"><span style="clip: rect(1.926em 1000em 3.069em -0.494em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 11.282em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">⋅</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">4</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">19</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">⋅</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">4</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">28</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">⋅</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">1</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" class="mo">=</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" class="mn">0</span></span></span></span></span>. Thus, the result of <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">19</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span style="font-family: MathJax_Main;" class="mo">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span style="font-family: MathJax_Main; font-size: 70.7%;" class="mo">∗</span></span></span></span></span></span></span></span>acting on <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.718em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">4</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">4</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">1</span><span style="font-family: MathJax_Main;" class="mo">)</span></span></span></span></span>is <span class="MathJax"><span class="math"><span style="clip: rect(1.937em 1000em 3.009em -0.538em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 0.513em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mn">0</span></span></span></span></span>.<br /> | | 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<span style="font-size: 80%; vertical-align: super;"><a class="wiki_link" href="/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1">{1}</a></span> 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 \(0\).<br /> |
| <br /> | | <br /> |
| 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.<br /> | | 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.<br /> |
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc2"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"></a><!-- ws:end:WikiTextHeadingRule:8 -->1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)</h2> | | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc3"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)"></a><!-- ws:end:WikiTextHeadingRule:10 -->1.2: Covectors mean stuff. (OR: YOU DON'T KNOW MONZO)</h2> |
| <br /> | | <br /> |
| One interesting way to think of covectors, since they're these dual vectors that &quot;act on&quot; normal vectors, is as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number.<br /> | | One interesting way to think of covectors, since they're these dual vectors that &quot;act on&quot; normal vectors, is as functions - they take in a vector as input, multiply each coefficient of the vector by the corresponding coefficient of the covector, sum them up, and spit out a number.<br /> |
| <br /> | | <br /> |
| In other words, you know that the action of the covector <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">19</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span style="font-family: MathJax_Main;" class="mo">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span style="font-family: MathJax_Main; font-size: 70.7%;" class="mo">∗</span></span></span></span></span></span></span></span>on any arbitrary vector <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Math;" class="mi"><em>a</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>b</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>c</em></span><span style="font-family: MathJax_Main;" class="mo">)</span></span></span></span></span>is going to be <span class="MathJax"><span class="math"><span style="clip: rect(1.909em 1000em 3.069em -0.494em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 6.795em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Math;" class="mi"><em>a</em></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">19</span><span style="font-family: MathJax_Math;" class="mi"><em>b</em></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">28</span><span style="font-family: MathJax_Math;" class="mi"><em>c</em></span></span></span></span></span>. So, you can think of <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 5.064em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">19</span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Main; padding-left: 0.167em;" class="mn">28</span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span style="font-family: MathJax_Main;" class="mo">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span style="font-family: MathJax_Main; font-size: 70.7%;" class="mo">∗</span></span></span></span></span></span></span></span>itself as a function looking something like <span class="MathJax"><span class="math"><span style="clip: rect(1.779em 1000em 3.237em -0.522em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 9.936em;"><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>f</em></span><span style="font-family: MathJax_Main;" class="mo">(</span><span class="texatom"><span class="mrow"><span class="munderover"><span style="display: inline-block; height: 0px; position: relative; width: 0.577em;"><span style="clip: rect(1.893em 1000em 2.739em -0.553em); left: 0em; position: absolute; top: -2.538em;"><span style="font-family: MathJax_Math;" class="mi"><strong><em>v</em></strong></span></span><span style="left: 0.066em; position: absolute; top: -4.1em;"><span style="font-family: MathJax_Main;" class="mo"><strong>⃗</strong> </span></span></span></span></span></span><span style="font-family: MathJax_Main;" class="mo">)</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" class="mo">=</span><span style="font-family: MathJax_Main; padding-left: 0.278em;" class="mn">12</span><span style="font-family: MathJax_Math;" class="mi"><em>a</em></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">19</span><span style="font-family: MathJax_Math;" class="mi"><em>b</em></span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">28</span><span style="font-family: MathJax_Math;" class="mi"><em>c</em></span></span></span></span></span>for some vector of the form <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Math;" class="mi"><em>a</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>b</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>c</em></span><span style="font-family: MathJax_Main;" class="mo">)</span></span></span></span></span>. Note that the formatting on <span class="MathJax"><span class="math"><span style="clip: rect(2.984em 1000em 4.2em -1.053em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -4em; width: 0.577em;"><span class="mrow"><span class="texatom"><span class="mrow"><span class="munderover"><span style="display: inline-block; height: 0px; position: relative; width: 0.577em;"><span style="clip: rect(1.893em 1000em 2.739em -0.553em); left: 0em; position: absolute; top: -2.538em;"><span style="font-family: MathJax_Math;" class="mi"><strong><em>v</em></strong></span></span><span style="left: 0.066em; position: absolute; top: -4.1em;"><span style="font-family: MathJax_Main;" class="mo"><strong>⃗</strong> </span></span></span></span></span></span></span></span></span></span>specifies that <span class="MathJax"><span class="math"><span style="clip: rect(2.984em 1000em 4.2em -1.053em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -4em; width: 0.577em;"><span class="mrow"><span class="texatom"><span class="mrow"><span class="munderover"><span style="display: inline-block; height: 0px; position: relative; width: 0.577em;"><span style="clip: rect(1.893em 1000em 2.739em -0.553em); left: 0em; position: absolute; top: -2.538em;"><span style="font-family: MathJax_Math;" class="mi"><strong><em>v</em></strong></span></span><span style="left: 0.066em; position: absolute; top: -4.1em;"><span style="font-family: MathJax_Main;" class="mo"><strong>⃗</strong> </span></span></span></span></span></span></span></span></span></span>is a vector that's being taken in as input.<br /> | | 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.<br /> |
| <br /> | | <br /> |
| Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too.<br /> | | Thus, covectors are little mathematical machines - they take in vectors, do some simple dot product-ish stuff, and output a scalar. Simple! We'll see that they turn up again and again and again in tuning theory, too.<br /> |
| <br /> | | <br /> |
| Before we go on, however, let's clean up the notation a bit. In physics, the notation commonly used is to notate covectors <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.974em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span style="font-family: MathJax_Main;" class="mtext">like this</span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>and to notate vectors <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.974em;"><span class="mrow"><span class="mfenced"><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span style="font-family: MathJax_Main;" class="mtext">like this</span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>. Physicists call this &quot;bra-ket&quot; notation, or sometimes &quot;Dirac&quot; notation. So...<br /> | | 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 &quot;bra-ket&quot; notation, or sometimes &quot;Dirac&quot; notation. So...<br /> |
| <ul><li>Instead of writing covectors as <span class="MathJax"><span class="math"><span style="clip: rect(1.797em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.526em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Math;" class="mi"><em>x</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>y</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>z</em></span><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 0.844em;"><span style="clip: rect(1.853em 1000em 3.237em -0.522em); left: 0em; position: absolute; top: -2.795em;"><span style="font-family: MathJax_Main;" class="mo">)</span></span><span style="left: 0.385em; position: absolute; top: -2.695em;"><span style="font-family: MathJax_Main; font-size: 70.7%;" class="mo">∗</span></span></span></span></span></span></span></span>, I'll just write <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>xyz</em></span></span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>from now on.</li><li>Instead of writing vectors as <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.483em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.949em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mo">(</span><span style="font-family: MathJax_Math;" class="mi"><em>a</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>b</em></span><span style="font-family: MathJax_Main;" class="mo">,</span><span style="font-family: MathJax_Math; padding-left: 0.167em;" class="mi"><em>c</em></span><span style="font-family: MathJax_Main;" class="mo">)</span></span></span></span></span>, I'll just write <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 2.821em;"><span class="mrow"><span class="mfenced"><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span style="font-family: MathJax_Math;" class="mi"><em>abc</em></span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>from now on.</li><li>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}.</li></ul><br /> | | <ul><li>Instead of writing covectors as \((x,y,z)^*\), I'll just write \(\bra{x \s y \s z}\) from now on.</li><li>Instead of writing vectors as \((a,b,c)\), I'll just write \(\ket{a \s b \s c}\) from now on.</li><li>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}.</li></ul><br /> |
| Now then, how do we use covectors? Or, rather, how do we recognize when our thought processes are using them, even if we never realized before these thought processes involved things called covectors? Well, let's say you're going to ask a harmlessly ordinary question like &quot;does 81/80 vanish in 12-EDO?&quot;<br /> | | Now then, how do we use covectors? Or, rather, how do we recognize when our thought processes are using them, even if we never realized before these thought processes involved things called covectors? Well, let's say you're going to ask a harmlessly ordinary question like &quot;does 81/80 vanish in 12-EDO?&quot;<br /> |
| <br /> | | <br /> |
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| 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.<br /> | | 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.<br /> |
| <br /> | | <br /> |
| But, if you really think about it, what you've also just done is evaluate the expression <span class="MathJax"><span class="math"><span style="clip: rect(1.926em 1000em 3.069em -0.494em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 10.256em;"><span class="mrow"><span style="font-family: MathJax_Main;" class="mn">12</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">∗</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">4</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">19</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">∗</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">4</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">+</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">28</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">∗</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">1</span></span></span></span></span>: 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 <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 8.654em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span style="font-family: MathJax_Main;" class="mn">121928</span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">44</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mo">−</span><span style="font-family: MathJax_Main; padding-left: 0.222em;" class="mn">1</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>. Looks like you've just seemingly applied <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 4.487em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span style="font-family: MathJax_Main;" class="mn">121918</span></span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>to <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.458em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 3.59em;"><span class="mrow"><span class="mfenced"><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span class="mrow"><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">44</span><span style="font-family: MathJax_Main;" class="mtext">-</span><span style="font-family: MathJax_Main;" class="mn">1</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>.<br /> | | 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 \(\ket{\-4 \s 4 \s \-1}\).<br /> |
| <br /> | | <br /> |
| 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.<br /> | | 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.<br /> |
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| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc3"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)"></a><!-- ws:end:WikiTextHeadingRule:10 -->1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)</h2> | | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc4"><a name="LECTURE 1: Vector Spaces and Dual Spaces-1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)"></a><!-- ws:end:WikiTextHeadingRule:12 -->1.3: Why the fact that covectors mean stuff matters. (OR: PREPARE FOR WEDGIE)</h2> |
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| Assuming you've understood my exposition thus far, you now hopefully see where all things like monzos and vals come from.<br /> | | Assuming you've understood my exposition thus far, you now hopefully see where all things like monzos and vals come from.<br /> |
| <br /> | | <br /> |
| <ul><li>Monzos, straightforwardly, are elements in a vector space - they're vectors - like the spaces you learned in high school algebra.</li><li>Vals, straightforwardly, are elements in the &quot;dual space&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.</li></ul><br /> | | <ul><li>Monzos, straightforwardly, are elements in a vector space - they're vectors - like the spaces you learned in high school algebra.</li><li>Vals, straightforwardly, are elements in the &quot;dual space&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.</li></ul><br /> |
| Also, hopefully, now you see that this whole <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 7.372em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span style="font-family: MathJax_Main;" class="mtext">covector</span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span><span style="font-family: MathJax_Main;" class="mtext">vector</span></span><span class="mo"><span style="font-family: MathJax_Main;">⟩</span></span></span></span></span></span></span>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, <span class="MathJax"><span class="math"><span style="clip: rect(1.853em 1000em 3.237em -0.467em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.795em; width: 4.487em;"><span class="mrow"><span class="mfenced"><span class="mo"><span style="font-family: MathJax_Main;">⟨</span></span><span class="mrow"><span style="font-family: MathJax_Main;" class="mn">121928</span></span><span style="vertical-align: 0em;" class="mo"><span style="font-family: MathJax_Main;">|</span></span></span></span></span></span></span>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 &quot;covector.&quot; Nice!<br /> | | 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 &quot;covector.&quot; Nice!<br /> |
| <br /> | | <br /> |
| 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.<br /> | | 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.<br /> |
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| <span style="font-size: 90%; vertical-align: sub;"><!-- ws:start:WikiTextAnchorRule:18:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@ref1&quot; title=&quot;Anchor: ref1&quot;/&gt; --><a name="ref1"></a><!-- ws:end:WikiTextAnchorRule:18 -->[1] - Note that some have raised technical concerns about this operation being called the &quot;dot product,&quot; 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 &quot;<strong>bracket product</strong>&quot;, 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.</span><br /> | | <span style="font-size: 90%; vertical-align: sub;"><!-- ws:start:WikiTextAnchorRule:21:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@ref1&quot; title=&quot;Anchor: ref1&quot;/&gt; --><a name="ref1"></a><!-- ws:end:WikiTextAnchorRule:21 -->[1] - Note that some have raised technical concerns about this operation being called the &quot;dot product,&quot; 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 &quot;<strong>bracket product</strong>&quot;, 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.</span><br /> |
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| <span style="font-size: 90%;"><!-- ws:start:WikiTextAnchorRule:19:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@ref2&quot; title=&quot;Anchor: ref2&quot;/&gt; --><a name="ref2"></a><!-- ws:end:WikiTextAnchorRule:19 -->[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 &quot;<strong>tuning maps</strong>&quot; 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.</span><br /> | | <span style="font-size: 90%;"><!-- ws:start:WikiTextAnchorRule:22:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@ref2&quot; title=&quot;Anchor: ref2&quot;/&gt; --><a name="ref2"></a><!-- ws:end:WikiTextAnchorRule:22 -->[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 &quot;<strong>tuning maps</strong>&quot; 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.</span><br /> |
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| <span style="font-size: 90%;"><!-- ws:start:WikiTextAnchorRule:20:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@ref3&quot; title=&quot;Anchor: ref3&quot;/&gt; --><a name="ref3"></a><!-- ws:end:WikiTextAnchorRule:20 -->[3] - This is one of those things that seems totally useless unless you're <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Smith</a>, at which point you realize that multivals represent higher-rank temperaments, and that this random field of mathematics has a rather musical interpretation.</span><br /> | | <span style="font-size: 90%;"><!-- ws:start:WikiTextAnchorRule:23:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@ref3&quot; title=&quot;Anchor: ref3&quot;/&gt; --><a name="ref3"></a><!-- ws:end:WikiTextAnchorRule:23 -->[3] - This is one of those things that seems totally useless unless you're <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Smith</a>, at which point you realize that multivals represent higher-rank temperaments, and that this random field of mathematics has a rather musical interpretation.</span><br /> |
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