Interior product: Difference between revisions
Wikispaces>genewardsmith **Imported revision 278841704 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278924072 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2011-11- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-25 01:13:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278924072</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 17: | Line 17: | ||
One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero. | One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero. | ||
Another application is the use of the interior product to define the intervals of the [[abstract regular temperament]] given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [ | Another application is the use of the interior product to define the intervals of the [[abstract regular temperament]] given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W[1200+300*p5, -1200, 0, 0]2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q. It should be noted that V with this property is underdetermined, so that many possible vectors can be used to the same effect. In most cases, solving for V subject to the condition that all its coefficients except the first r are zero will work and this introduces some simplicifation; for example the dot product of Meantone∨q, where "Meantone" is the 7-limit wedgie, with [1200+300*log2(5), -1200, 0, 0] gives the value in cents of the [[quarter comma meantone]] tuning of the interval denoted by Meantone∨q. | ||
The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit [[Marvel family#Marvel|marvel temperament]]. Then Marv∨45/44 = <<4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = <<-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = <<11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = <<-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = <<5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = <<6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = <<-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = <<6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨9801/9800 = <<-12 2 -20 6 31 2 51 -52 7 86||, wizard. | The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit [[Marvel family#Marvel|marvel temperament]]. Then Marv∨45/44 = <<4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = <<-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = <<11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = <<-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = <<5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = <<6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = <<-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = <<6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨9801/9800 = <<-12 2 -20 6 31 2 51 -52 7 86||, wizard. | ||
| Line 34: | Line 34: | ||
One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.<br /> | One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.<br /> | ||
<br /> | <br /> | ||
Another application is the use of the interior product to define the intervals of the <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [ | Another application is the use of the interior product to define the intervals of the <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W[1200+300*p5, -1200, 0, 0]2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q. It should be noted that V with this property is underdetermined, so that many possible vectors can be used to the same effect. In most cases, solving for V subject to the condition that all its coefficients except the first r are zero will work and this introduces some simplicifation; for example the dot product of Meantone∨q, where &quot;Meantone&quot; is the 7-limit wedgie, with [1200+300*log2(5), -1200, 0, 0] gives the value in cents of the <a class="wiki_link" href="/quarter%20comma%20meantone">quarter comma meantone</a> tuning of the interval denoted by Meantone∨q.<br /> | ||
<br /> | <br /> | ||
The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = &lt;&lt;&lt;1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit <a class="wiki_link" href="/Marvel%20family#Marvel">marvel temperament</a>. Then Marv∨45/44 = &lt;&lt;4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = &lt;&lt;-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = &lt;&lt;11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = &lt;&lt;-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = &lt;&lt;5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = &lt;&lt;6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = &lt;&lt;-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = &lt;&lt;6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨9801/9800 = &lt;&lt;-12 2 -20 6 31 2 51 -52 7 86||, wizard.<br /> | The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = &lt;&lt;&lt;1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit <a class="wiki_link" href="/Marvel%20family#Marvel">marvel temperament</a>. Then Marv∨45/44 = &lt;&lt;4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = &lt;&lt;-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = &lt;&lt;11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = &lt;&lt;-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = &lt;&lt;5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = &lt;&lt;6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = &lt;&lt;-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = &lt;&lt;6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨9801/9800 = &lt;&lt;-12 2 -20 6 31 2 51 -52 7 86||, wizard.<br /> | ||
<br /> | <br /> | ||
The interior product is also useful in finding the temperament map given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the map. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [&lt;0 0 -1 -2 3|, &lt;0 1 0 2 -1|, &lt;0 2 -2 0 4|, &lt;0 -3 1 -4 0|, &lt;-1 0 0 5 -12|, &lt;-2 0 -5 0 -9|, &lt;3 0 12 9 0|, &lt;2 5 0 0 19|, &lt;-1 -12 0 -19 0|, &lt;4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [&lt;1 0 0 -5 12|, &lt;0 1 0 2 -1|, &lt;0 0 1 2 -3|], the normal val list for 11-limit marvel.</body></html></pre></div> | The interior product is also useful in finding the temperament map given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the map. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [&lt;0 0 -1 -2 3|, &lt;0 1 0 2 -1|, &lt;0 2 -2 0 4|, &lt;0 -3 1 -4 0|, &lt;-1 0 0 5 -12|, &lt;-2 0 -5 0 -9|, &lt;3 0 12 9 0|, &lt;2 5 0 0 19|, &lt;-1 -12 0 -19 0|, &lt;4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [&lt;1 0 0 -5 12|, &lt;0 1 0 2 -1|, &lt;0 0 1 2 -3|], the normal val list for 11-limit marvel.</body></html></pre></div> | ||