Subgroup basis matrix: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:56:13 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-04 12:59:58 UTC</tt>.

: The original revision id was <tt>~~355716414~~</tt>.

: The original revision id was <tt>356334190</tt>.

: The revision comment was: <tt></tt>

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

Line 13:

The column module of any subgroup mapping matrix is the submodule of J corresponding to the subgroup G. The row module of any subgroup mapping matrix V is the module of [[xenharmonic/Smonzos and Svals|svals]] which take coefficients representing, in order, the mappings for the intervals specified by the columns of V. Note that, much like with M-maps, there is not a unique mapping matrix for any subgroup: any matrix V of full-column rank which has columns that form a basis for G will also send vals to svals on that subgroup, but the coefficients of the svals will change to reflect the basis of V.

Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val n in the kernel and any other val v, (n+v)*V = ~~n*V~~ + ~~v*V~~ = 0 + ~~v*V~~ = ~~v*V~~. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are "tempered out," we instead say that they are **restricted away**, as their subgroup restriction under V is the zero sval.

Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val k in the kernel and any other val v, (k+v)∙V = k∙V + v∙V = 0 + v∙V = v∙V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are "tempered out," we instead say that they are **restricted away**, as their subgroup restriction under V is the zero sval.

As a final note, we can easily see if two V-maps represent the same subgroup by checking to see if they form the same [[Normal lists|normal interval list]], or if they have the same Hermite normal form.

Line 74:

[[math]]

If we call this matrix **M**, then the matrix multiplication **~~M*V~~** gives us the following result:

If we call this matrix **M**, then the matrix multiplication **M∙V** gives us the following result:

[[math]]

Line 88:

**The Dual Transformation**

**V** implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0>, |0 -2 1>|]. If this matrix is X, then the dual transformation can be found by multiplying ~~V*X~~, which yields

**V** implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0>, |0 -2 1>|]. If this matrix is X, then the dual transformation can be found by multiplying V∙X, which yields

[[math]]

Line 106:

The column module of any subgroup mapping matrix is the submodule of J corresponding to the subgroup G. The row module of any subgroup mapping matrix V is the module of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals">svals</a> which take coefficients representing, in order, the mappings for the intervals specified by the columns of V. Note that, much like with M-maps, there is not a unique mapping matrix for any subgroup: any matrix V of full-column rank which has columns that form a basis for G will also send vals to svals on that subgroup, but the coefficients of the svals will change to reflect the basis of V.

Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val n in the kernel and any other val v, (n+v)*V = ~~n*V~~ + ~~v*V~~ = 0 + ~~v*V~~ = ~~v*V~~. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are &quot;tempered out,&quot; we instead say that they are restricted away, as their subgroup restriction under V is the zero sval.

Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val k in the kernel and any other val v, (k+v)∙V = k∙V + v∙V = 0 + v∙V = v∙V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are &quot;tempered out,&quot; we instead say that they are restricted away, as their subgroup restriction under V is the zero sval.

As a final note, we can easily see if two V-maps represent the same subgroup by checking to see if they form the same <a class="wiki_link" href="/Normal%20lists">normal interval list</a>, or if they have the same Hermite normal form.

Line 188:

\end{array} \right] \]</script>

If we call this matrix M, then the matrix multiplication ~~M*V~~ gives us the following result:

If we call this matrix M, then the matrix multiplication M∙V gives us the following result:

<!-- ws:start:WikiTextMathRule:5:

Line 206:

The Dual Transformation

V implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0&gt;, |0 -2 1&gt;|]. If this matrix is X, then the dual transformation can be found by multiplying ~~V*X~~, which yields

V implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0&gt;, |0 -2 1&gt;|]. If this matrix is X, then the dual transformation can be found by multiplying V∙X, which yields

<!-- ws:start:WikiTextMathRule:6:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-31 12:56:13 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-04 12:59:58 UTC</tt>.<br>
-: The original revision id was <tt>355716414</tt>.<br>
+: The original revision id was <tt>356334190</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 13: / Line 13: @@
 &lt;span style="background-color: #ffffff;"&gt;The column module of any subgroup mapping matrix is the submodule of J corresponding to the subgroup G. The row module of any subgroup mapping matrix V is the module of &lt;/span&gt;[[xenharmonic/Smonzos and Svals|svals]] which take coefficients representing, in order, the mappings for the intervals specified by the columns of V. &lt;span style="background-color: #ffffff;"&gt;Note that, much like with M-maps, there is not a unique mapping matrix for any subgroup: any matrix V of full-column rank which has columns that form a basis for G will also send vals to svals on that subgroup, but the coefficients of the svals will change to reflect the basis of V.&lt;/span&gt;
-Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val n in the kernel and any other val v, (n+v)*V = n*V + v*V = 0 + v*V = v*V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are "tempered out," we instead say that they are **restricted away**, as their subgroup restriction under V is the zero sval.
+Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val k in the kernel and any other val v, (k+v)∙V = k∙V + v∙V = 0 + v∙V = v∙V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are "tempered out," we instead say that they are **restricted away**, as their subgroup restriction under V is the zero sval.
 As a final note, we can easily see if two V-maps represent the same subgroup by checking to see if they form the same [[Normal lists|normal interval list]], or if they have the same Hermite normal form.
@@ Line 74: / Line 74: @@
 [[math]]
-If we call this matrix **M**, then the matrix multiplication **M*V** gives us the following result:
+If we call this matrix **M**, then the matrix multiplication **M∙V** gives us the following result:
 [[math]]
@@ Line 88: / Line 88: @@
 **The Dual Transformation**
-**V** implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0&gt;, |0 -2 1&gt;|]. If this matrix is X, then the dual transformation can be found by multiplying V*X, which yields
+**V** implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0&gt;, |0 -2 1&gt;|]. If this matrix is X, then the dual transformation can be found by multiplying V∙X, which yields
 [[math]]
@@ Line 106: / Line 106: @@
 &lt;span style="background-color: #ffffff;"&gt;The column module of any subgroup mapping matrix is the submodule of J corresponding to the subgroup G. The row module of any subgroup mapping matrix V is the module of &lt;/span&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals"&gt;svals&lt;/a&gt; which take coefficients representing, in order, the mappings for the intervals specified by the columns of V. &lt;span style="background-color: #ffffff;"&gt;Note that, much like with M-maps, there is not a unique mapping matrix for any subgroup: any matrix V of full-column rank which has columns that form a basis for G will also send vals to svals on that subgroup, but the coefficients of the svals will change to reflect the basis of V.&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val n in the kernel and any other val v, (n+v)*V = n*V + v*V = 0 + v*V = v*V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are &amp;quot;tempered out,&amp;quot; we instead say that they are &lt;strong&gt;restricted away&lt;/strong&gt;, as their subgroup restriction under V is the zero sval.&lt;br /&gt;
+Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val k in the kernel and any other val v, (k+v)∙V = k∙V + v∙V = 0 + v∙V = v∙V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are &amp;quot;tempered out,&amp;quot; we instead say that they are &lt;strong&gt;restricted away&lt;/strong&gt;, as their subgroup restriction under V is the zero sval.&lt;br /&gt;
 &lt;br /&gt;
 As a final note, we can easily see if two V-maps represent the same subgroup by checking to see if they form the same &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal interval list&lt;/a&gt;, or if they have the same Hermite normal form.&lt;br /&gt;
@@ Line 188: / Line 188: @@
 \end{array} \right] \]&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-If we call this matrix &lt;strong&gt;M&lt;/strong&gt;, then the matrix multiplication &lt;strong&gt;M*V&lt;/strong&gt; gives us the following result:&lt;br /&gt;
+If we call this matrix &lt;strong&gt;M&lt;/strong&gt;, then the matrix multiplication &lt;strong&gt;M∙V&lt;/strong&gt; gives us the following result:&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:5:
@@ Line 206: / Line 206: @@
 &lt;br /&gt;
 &lt;strong&gt;The Dual Transformation&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;V&lt;/strong&gt; implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0&amp;gt;, |0 -2 1&amp;gt;|]. If this matrix is X, then the dual transformation can be found by multiplying V*X, which yields&lt;br /&gt;
+&lt;strong&gt;V&lt;/strong&gt; implies a dual transformation mapping smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0&amp;gt;, |0 -2 1&amp;gt;|]. If this matrix is X, then the dual transformation can be found by multiplying V∙X, which yields&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:6: