Mathematical theory of saturation: Difference between revisions
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If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals. | If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals. | ||
For example, consider the "temperament" with commas generated by 126/125 and 3645/3584. The group the [[monzo]]s |1 2 -3 1> and |-9 6 1 -1> | For example, consider the "temperament" with commas generated by 126/125 and 3645/3584. The group the [[monzo]]s |1 2 -3 1> and |-9 6 1 -1> generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the group. Hence (81/80)^2 is tempered out, but 81/80 isn't, and one gets two parallel [[meantone]]s not connected by any [[7-limit]] interval. Something similar happens with the two vals <12 19 28 34| and <26 41 60 72|; this however can be fixed in a way by extending to the 11-limit, and interpreting the "unobtainable" notes as notes reached in the 11-limit. Adding 245/242 to the commas (81/80 and 126/125) of septimal meantone is one way of reinterpreting the situation. In musical contexts, the first kind of problem has been called a "torsion problem" and the second kind "contorsion". | ||
Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to [[http://en.wikipedia.org/wiki/Smith_normal_form|Smith normal form]]. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such. | Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to [[http://en.wikipedia.org/wiki/Smith_normal_form|Smith normal form]]. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such. | ||
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If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals.<br /> | If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals.<br /> | ||
<br /> | <br /> | ||
For example, consider the &quot;temperament&quot; with commas generated by 126/125 and 3645/3584. The group the <a class="wiki_link" href="/monzo">monzo</a>s |1 2 -3 1&gt; and |-9 6 1 -1&gt; | For example, consider the &quot;temperament&quot; with commas generated by 126/125 and 3645/3584. The group the <a class="wiki_link" href="/monzo">monzo</a>s |1 2 -3 1&gt; and |-9 6 1 -1&gt; generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the group. Hence (81/80)^2 is tempered out, but 81/80 isn't, and one gets two parallel <a class="wiki_link" href="/meantone">meantone</a>s not connected by any <a class="wiki_link" href="/7-limit">7-limit</a> interval. Something similar happens with the two vals &lt;12 19 28 34| and &lt;26 41 60 72|; this however can be fixed in a way by extending to the 11-limit, and interpreting the &quot;unobtainable&quot; notes as notes reached in the 11-limit. Adding 245/242 to the commas (81/80 and 126/125) of septimal meantone is one way of reinterpreting the situation. In musical contexts, the first kind of problem has been called a &quot;torsion problem&quot; and the second kind &quot;contorsion&quot;.<br /> | ||
<br /> | <br /> | ||
Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Smith_normal_form" rel="nofollow">Smith normal form</a>. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such.<br /> | Because unsaturated subgroups of Z^n are problematic, it is useful to have a means to saturate them; that is, to find the minimal saturated subgroup of Z^n containing the given subgroup. We may do this by inverting the right-reducing matrix which in part converts a matrix of basis elements for the subgroup V to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Smith_normal_form" rel="nofollow">Smith normal form</a>. If A is a matrix with r (the rank) rows of dimension n whose rows form a basis for V, then there are two square matricies L and R, such that S = LAR, where S is the Smith normal form. The right-reducing matrix is R, the matrix multiplying A on the right. The first r rows of R generate the saturation of V. This procedure is only useful if there is a routine for finding the Smith normal form available, so we will assume there is one and not concern ourselves with the Smith form as such.<br /> | ||