Mathematical theory of saturation: Difference between revisions

Line 1:

The set of n-tuples of integers <math>\mathbb{Z}^n</math> such that two <math>n</math>-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank <math>n</math>. Its subgroups have the property of '''saturation''' if for any element <math>a</math> of <math>\mathbb{Z}^n</math>, if an integer multiple <math>m·a</math> of <math>a</math> belongs to a sublattice <math>V</math>, then <math>a</math> already belongs to <math>V</math>. Another way to put it is that if some linear combination with rational coefficients <math>q_1v_1 + \dots + q_kv_k</math> of elements of <math>V</math> belongs to <math>\mathbb{Z}^n</math>, then it belongs to <math>V</math>. For the latter definition we consider <math>\mathbb{Z}^n</math> to be contained in the <math>n</math>-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] <math>\mathbb{R}^n</math>, in which case <math>\mathbb{Z}^n</math> is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.

The set of n-tuples of integers <math>\mathbb{Z}^n</math> such that two <math>n</math>-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank <math>n</math>. Its subgroups have the property of '''[[saturation]]''' if for any element <math>a</math> of <math>\mathbb{Z}^n</math>, if an integer multiple <math>m·a</math> of <math>a</math> belongs to a sublattice <math>V</math>, then <math>a</math> already belongs to <math>V</math>. Another way to put it is that if some linear combination with rational coefficients <math>q_1v_1 + \dots + q_kv_k</math> of elements of <math>V</math> belongs to <math>\mathbb{Z}^n</math>, then it belongs to <math>V</math>. For the latter definition we consider <math>\mathbb{Z}^n</math> to be contained in the <math>n</math>-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] <math>\mathbb{R}^n</math>, in which case <math>\mathbb{Z}^n</math> is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.

If <math>C</math> represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if <math>C</math> isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation (some JI intervals cannot be reached by a generator in the tempered lattice). 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 <math>V</math> 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 (cannot be reached by tempering a JI interval); this at least is an actual system of musical intervals, but disconnected. This has been called a '''contorsion''' problem.

If <math>C</math> represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if <math>C</math> isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation (some JI intervals cannot be reached by a generator in the tempered lattice). 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 <math>V</math> 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 (cannot be reached by tempering a JI interval); 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]] {{vector|1 2 -3 1}} and {{vector|-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 {{map|12 19 28 34}} and {{map|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.

@@ Line 1: / Line 1: @@
-The set of n-tuples of integers <span><math>\mathbb{Z}^n</math></span> such that two <span><math>n</math></span>-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank <span><math>n</math></span>. Its subgroups have the property of '''saturation''' if for any element <span><math>a</math></span> of <span><math>\mathbb{Z}^n</math></span>, if an integer multiple <span><math>m·a</math></span> of <span><math>a</math></span> belongs to a sublattice <span><math>V</math></span>, then <span><math>a</math></span> already belongs to <span><math>V</math></span>. Another way to put it is that if some linear combination with rational coefficients <span><math>q_1v_1 + \dots + q_kv_k</math></span> of elements of <span><math>V</math></span> belongs to <span><math>\mathbb{Z}^n</math></span>, then it belongs to <span><math>V</math></span>. For the latter definition we consider <span><math>\mathbb{Z}^n</math></span> to be contained in the <span><math>n</math></span>-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] <span><math>\mathbb{R}^n</math></span>, in which case <span><math>\mathbb{Z}^n</math></span> is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.
+The set of n-tuples of integers <span><math>\mathbb{Z}^n</math></span> such that two <span><math>n</math></span>-tuples can be added coordinatewise is the [http://en.wikipedia.org/wiki/Free_abelian_group free abelian group] of rank <span><math>n</math></span>. Its subgroups have the property of '''[[saturation]]''' if for any element <span><math>a</math></span> of <span><math>\mathbb{Z}^n</math></span>, if an integer multiple <span><math>m·a</math></span> of <span><math>a</math></span> belongs to a sublattice <span><math>V</math></span>, then <span><math>a</math></span> already belongs to <span><math>V</math></span>. Another way to put it is that if some linear combination with rational coefficients <span><math>q_1v_1 + \dots + q_kv_k</math></span> of elements of <span><math>V</math></span> belongs to <span><math>\mathbb{Z}^n</math></span>, then it belongs to <span><math>V</math></span>. For the latter definition we consider <span><math>\mathbb{Z}^n</math></span> to be contained in the <span><math>n</math></span>-dimensional [http://en.wikipedia.org/wiki/Vector_space real vector space] <span><math>\mathbb{R}^n</math></span>, in which case <span><math>\mathbb{Z}^n</math></span> is often called the [http://en.wikipedia.org/wiki/Integer_lattice integer lattice], or grid lattice.
-If <span><math>C</math></span> represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if <span><math>C</math></span> isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation (some JI intervals cannot be reached by a generator in the tempered lattice). 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 <span><math>V</math></span> 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 (cannot be reached by tempering a JI interval); this at least is an actual system of musical intervals, but disconnected. This has been called a '''contorsion''' problem.
+If <span><math>C</math></span> represents the commas (nullspace or kernel) of a supposed regular temperament, i.e. the intervals it tempers out, then if <span><math>C</math></span> isn't saturated the supposed temperament it defines may be regarded as pathological, as it has notes with no clear interpretation (some JI intervals cannot be reached by a generator in the tempered lattice). 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 <span><math>V</math></span> 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 (cannot be reached by tempering a JI interval); 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]] {{vector|1 2 -3 1}} and {{vector|-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 {{map|12 19 28 34}} and {{map|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.