Subgroup basis matrix
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=<span style="background-color: #ffffff;">Basics</span>= <span style="background-color: #ffffff;">A [[Temperament Mapping Matrices (M-maps)|temperament mapping matrix]], or M-map, is a module homomorphism </span>**<span style="background-color: #ffffff;">T</span>**<span style="background-color: #ffffff;">: J -> K from the module J of JI ratios to a new module K, where K then comes to represent tempered intervals. We can also consider module homomorphisms **S:** J* -> L*, where J* is module of linear functionals on J, and where we map directly from J* to another module of linear functionals L*; this module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to [[xenharmonic/Smonzos and Svals|svals]] on a certain subgroup, and that the module L which the elements of L* act on are [[xenharmonic/Smonzos and Svals|smonzos]]. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices **subgroup mapping matrices**, or "val-maps" or **V-maps** when context demands they be distinguished from their temperamental counterparts, the [[Temperament Mapping Matrices (M-maps)|M-maps]].</span> <span style="background-color: #ffffff;">If we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix V is said to be a mapping matrix for a subgroup S of a JI module J if and only if the column module of V spans S and if V is of full column rank. Note that, unlike with M-maps, we drop the restriction that S must be saturated, so that we specifically allow for subgroups with prime powers and the like.</span> <span style="background-color: #ffffff;">The column module of any subgroup mapping matrix is the submodule of J corresponding to the subgroup S. </span><span style="background-color: #ffffff;">The row module of any subgroup mapping matrix V is the module of </span>[[xenharmonic/Smonzos and Svals|svals]] which take coefficients representing, in order, the mappings for the intervals specified by the columns of V. <span style="background-color: #ffffff;">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 S will also send vals to svals on that subgroup, but the coefficients of the svals will change to reflect the basis of V.</span> 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 S defined by the columns of V, the kernel of V consists of those vals tempering out S. 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 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 <span style="background-color: #ffffff;">Much like with temperament homomorphism, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos.</span> <span style="background-color: #ffffff;">While the thought of "tempering out a val" may seem unintuitive at first,</span> Of note is the kernel of the V-map, which consists of those vals tempering out S. 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 th <span style="background-color: #ffffff;">The column module of any mapping matrix is the module of T-tempered intervals, also known as the module of </span><span style="background-color: #ffffff;">[[xenharmonic/tmonzos and tvals|tmonzos]]</span><span style="background-color: #ffffff;"> for T. The row module of any mapping matrix for a temperament T is the submodule of all vals supporting T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows are r linearly independent vals supporting T and which form a saturated submodule of the module of vals will be a valid mapping for T.</span> <span style="background-color: #ffffff;">Note also that since all mapping matrices for T will have the same row module, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same</span><span style="background-color: #ffffff;">[[xenharmonic/Normal lists#x-Normal%20val%20lists|normal val list]]</span><span style="background-color: #ffffff;">, or more generally if they have the same Hermite normal form.</span>
Original HTML content:
<html><head><title>Subgroup Mapping Matrices (V-maps)</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="background-color: #ffffff;">Basics</span></h1> <span style="background-color: #ffffff;">A <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">temperament mapping matrix</a>, or M-map, is a module homomorphism </span><strong><span style="background-color: #ffffff;">T</span></strong><span style="background-color: #ffffff;">: J -> K from the module J of JI ratios to a new module K, where K then comes to represent tempered intervals. We can also consider module homomorphisms <strong>S:</strong> J* -> L*, where J* is module of linear functionals on J, and where we map directly from J* to another module of linear functionals L*; this module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals">svals</a> on a certain subgroup, and that the module L which the elements of L* act on are <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Smonzos%20and%20Svals">smonzos</a>. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices <strong>subgroup mapping matrices</strong>, or "val-maps" or <strong>V-maps</strong> when context demands they be distinguished from their temperamental counterparts, the <a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29">M-maps</a>.</span><br /> <br /> <span style="background-color: #ffffff;">If we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix V is said to be a mapping matrix for a subgroup S of a JI module J if and only if the column module of V spans S and if V is of full column rank. Note that, unlike with M-maps, we drop the restriction that S must be saturated, so that we specifically allow for subgroups with prime powers and the like.</span><br /> <br /> <br /> <br /> <br /> <span style="background-color: #ffffff;">The column module of any subgroup mapping matrix is the submodule of J corresponding to the subgroup S. </span><span style="background-color: #ffffff;">The row module of any subgroup mapping matrix V is the module of </span><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. <span style="background-color: #ffffff;">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 S will also send vals to svals on that subgroup, but the coefficients of the svals will change to reflect the basis of V.</span><br /> <br /> <br /> 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 S defined by the columns of V, the kernel of V consists of those vals tempering out S. 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 vals are "tempered out," we instead say that they are <strong>restricted away</strong>, as their subgroup restriction under V is the zero sval.<br /> <br /> 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 <br /> <br /> <span style="background-color: #ffffff;">Much like with temperament homomorphism, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos.</span><br /> <span style="background-color: #ffffff;">While the thought of "tempering out a val" may seem unintuitive at first,</span><br /> <br /> Of note is the kernel of the V-map, which consists of those vals tempering out S. 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 th<br /> <br /> <br /> <span style="background-color: #ffffff;">The column module of any mapping matrix is the module of T-tempered intervals, also known as the module of </span><span style="background-color: #ffffff;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/tmonzos%20and%20tvals">tmonzos</a></span><span style="background-color: #ffffff;"> for T. The row module of any mapping matrix for a temperament T is the submodule of all vals supporting T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows are r linearly independent vals supporting T and which form a saturated submodule of the module of vals will be a valid mapping for T.</span><br /> <br /> <span style="background-color: #ffffff;">Note also that since all mapping matrices for T will have the same row module, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same</span><span style="background-color: #ffffff;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Normal%20lists#x-Normal%20val%20lists">normal val list</a></span><span style="background-color: #ffffff;">, or more generally if they have the same Hermite normal form.</span></body></html>