Mathematical theory of saturation
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The set of n-tuples of integers Z^n contained in the n-dimensional [[http://en.wikipedia.org/wiki/Vector_space|real vector space]] R^n is often called the [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]], or grid lattice. Two lattice points consisting of n-tuples of integers can be added coordinatewise, making Z^n a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] of rank n. Its subgroups, which are also sublattices, have the property of //saturation// if for any element a of Z^n, if an integer multiple m*a of a belongs to a sublattice V, then a already belongs to V. Another way to put it is that if some linear combination with rational coefficients q1*v1 + ... + qk*vk of elements of vk belongs to Z^n, then it belongs to V. If V represents the commas (nullspace or kernel) of a regular temperament, ie the intervals it tempers out, then if V 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 lattice of vals of the temperament, and is not saturated, then we obtain a temperament 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 lattice these generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the lattice. Hence (81/80)^2 is tempered out, but 81/80 isn't, and you get two parallel meantones 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".
Original HTML content:
<html><head><title>Saturation</title></head><body>The set of n-tuples of integers Z^n contained in the n-dimensional <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">real vector space</a> R^n is often called the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow">integer lattice</a>, or grid lattice. Two lattice points consisting of n-tuples of integers can be added coordinatewise, making Z^n a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> of rank n. Its subgroups, which are also sublattices, have the property of <em>saturation</em> if for any element a of Z^n, if an integer multiple m*a of a belongs to a sublattice V, then a already belongs to V. Another way to put it is that if some linear combination with rational coefficients q1*v1 + ... + qk*vk of elements of vk belongs to Z^n, then it belongs to V.<br /> <br /> If V represents the commas (nullspace or kernel) of a regular temperament, ie the intervals it tempers out, then if V 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 lattice of vals of the temperament, and is not saturated, then we obtain a temperament in which all of the notes cannot be reached by tempered intervals.<br /> <br /> For example, consider the temperament with commas generated by 126/125 and 3645/3584. The lattice these generate is not saturated, since (126/125)*(3645/3584) = (81/80)^2, but 81/80 does not belong to the lattice. Hence (81/80)^2 is tempered out, but 81/80 isn't, and you get two parallel meantones 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".</body></html>