Mathematical theory of saturation: Difference between revisions

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The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank n. Its subgroups 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 V belongs to Z^n, then it belongs to V. For the latter definition we consider Z^n to be contained in the n-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] R^n, in which case Z^n is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.

The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank n. Its subgroups 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 V belongs to Z^n, then it belongs to V. For the latter definition we consider Z^n to be contained in the n-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] R^n, in which case Z^n is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.

If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession ''are'' the same note. This is called a ''torsion'' problem. 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; this at least is an actual system of musical intervals, but disconnected. This has been called a ''contorsion'' problem.

If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession ''are'' the same note. This is called a ''torsion'' problem. 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; this at least is an actual system of musical intervals, but disconnected. This has been called a '''contorsion''' problem.

For example, consider the "temperament" with commas generated by 126/125 and 3645/3584. The group generated by the [[monzo|monzos]] |1 2 -3 1> and |-9 6 1 -1> 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 is not, and we have torsion. If we take the two vals <12 19 28 34| and <26 41 60 72| we similarly get contorsion. However, this 5- and 7-limit contorsion 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.

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-The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank n. Its subgroups 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 V belongs to Z^n, then it belongs to V. For the latter definition we consider Z^n to be contained in the n-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] R^n, in which case Z^n is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.
+The set of n-tuples of integers Z^n such that two n-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank n. Its subgroups 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 V belongs to Z^n, then it belongs to V. For the latter definition we consider Z^n to be contained in the n-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] R^n, in which case Z^n is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.
-If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession ''are'' the same note. This is called a ''torsion'' problem. 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; this at least is an actual system of musical intervals, but disconnected. This has been called a ''contorsion'' problem.
+If C represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if C isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation. For example, if (81/80)^2 = 6561/6400 is tempered out but 81/80 is not, then it is not clear how the tempered versions of 5/4 and 81/64 are related, as they are not the same note yet two of them in succession ''are'' the same note. This is called a ''torsion'' problem. 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; this at least is an actual system of musical intervals, but disconnected. This has been called a '''contorsion''' problem.
 For example, consider the "temperament" with commas generated by 126/125 and 3645/3584. The group generated by the [[monzo|monzos]] |1 2 -3 1&gt; and |-9 6 1 -1&gt; 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 is not, and we have torsion. If we take the two vals &lt;12 19 28 34| and &lt;26 41 60 72| we similarly get contorsion. However, this 5- and 7-limit contorsion 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.