Interior product: Difference between revisions

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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 and only if the rank {{nowrap|''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|abstract regular temperament]] given by a wedgie ''W''. In this case, we use {{nowrap|''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'' primes, where ''r'' is the rank of ''W''. Form the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''R''}}], where ''R'' is the ''r''-th prime number. Let ''U'' be the transpose of the pseudoinverse of this matrix, and let {{nowrap|''V'' = ''TU''}} (the matrix product), which can be taken to be an ({{nowrap|''r'' − 1}})-multimonzo. Then for any ({{nowrap|''r'' − 1}})-multival {{nowrap|''W'' ∨ ''q''}} in the abstract regular temperament, the dot product {{nowrap|(''W'' ∨ ''q'') ∙ ''V''}} gives the tuning of {{nowrap|''W'' ∨ ''q''}}. It should be noted that ''V'' with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''q''}}] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of {{nowrap|''M''meantone ∨ ''q''}}, where "Meantone" is the 7-limit wedgie, with <math>\~~bival~~{1200 + 300 * \log_{2}(5) & -1200 & 0 & 0}</math> ~~gives~~ the value in cents of the [[Quarter-comma_meantone|quarter-comma meantone]] tuning of the interval denoted by {{nowrap|''M''meantone ∨ ''q''}}.

Another application is the use of the interior product to define the intervals of the [[Abstract_regular_temperament|abstract regular temperament]] given by a wedgie ''W''. In this case, we use {{nowrap|''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'' primes, where ''r'' is the rank of ''W''. Form the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''R''}}], where ''R'' is the ''r''-th prime number. Let ''U'' be the transpose of the pseudoinverse of this matrix, and let {{nowrap|''V'' = ''TU''}} (the matrix product), which can be taken to be an ({{nowrap|''r'' − 1}})-multimonzo. Then for any ({{nowrap|''r'' − 1}})-multival {{nowrap|''W'' ∨ ''q''}} in the abstract regular temperament, the dot product {{nowrap|(''W'' ∨ ''q'') ∙ ''V''}} gives the tuning of {{nowrap|''W'' ∨ ''q''}}. It should be noted that ''V'' with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''q''}}] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of {{nowrap|''M''meantone ∨ ''q''}}, where "Meantone" is the 7-limit wedgie, with <math>\tmonzo{1200 + 300 * \log_{2}(5) & -1200 & 0 & 0}</math> giving the value in cents of the [[Quarter-comma_meantone|quarter-comma meantone]] tuning of the interval denoted by {{nowrap|''M''meantone ∨ ''q''}}.

The interior product can also be used to add a comma to a ''p''-limit temperament of rank ''r'', producing a rank-({{nowrap|''r'' − 1}}) temperament which [[support]]s it. For instance, <math>\tritmonzo{1 & 2 & -3 & -2 & 1 & -4 & -5 & 12 & 9 & -19}</math> is the wedgie for 11-limit [[Marvel_family#Marvel|Marvel temperament]]. Then:

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 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 and only if the rank {{nowrap|''r'' &minus; 1}} multival obtained by taking the interior product of the wedgie with the interval is the zero multival&mdash;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|abstract regular temperament]] given by a wedgie ''W''. In this case, we use {{nowrap|''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'' primes, where ''r'' is the rank of ''W''. Form the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''R''}}], where ''R'' is the ''r''-th prime number. Let ''U'' be the transpose of the pseudoinverse of this matrix, and let {{nowrap|''V'' = ''TU''}} (the matrix product), which can be taken to be an ({{nowrap|''r'' &minus; 1}})-multimonzo. Then for any ({{nowrap|''r'' &minus; 1}})-multival {{nowrap|''W'' ∨ ''q''}} in the abstract regular temperament, the dot product {{nowrap|(''W'' ∨ ''q'') ∙ ''V''}} gives the tuning of {{nowrap|''W'' ∨ ''q''}}. It should be noted that ''V'' with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''q''}}] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of {{nowrap|''M''<sub>meantone</sub> ∨ ''q''}}, where "Meantone" is the 7-limit wedgie, with <math>\bival{1200 + 300 * \log_{2}(5) & -1200 & 0 & 0}</math> gives the value in cents of the [[Quarter-comma_meantone|quarter-comma meantone]] tuning of the interval denoted by {{nowrap|''M''<sub>meantone</sub> ∨ ''q''}}.
+Another application is the use of the interior product to define the intervals of the [[Abstract_regular_temperament|abstract regular temperament]] given by a wedgie ''W''. In this case, we use {{nowrap|''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'' primes, where ''r'' is the rank of ''W''. Form the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''R''}}], where ''R'' is the ''r''-th prime number. Let ''U'' be the transpose of the pseudoinverse of this matrix, and let {{nowrap|''V'' = ''TU''}} (the matrix product), which can be taken to be an ({{nowrap|''r'' &minus; 1}})-multimonzo. Then for any ({{nowrap|''r'' &minus; 1}})-multival {{nowrap|''W'' ∨ ''q''}} in the abstract regular temperament, the dot product {{nowrap|(''W'' ∨ ''q'') ∙ ''V''}} gives the tuning of {{nowrap|''W'' ∨ ''q''}}. It should be noted that ''V'' with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [{{nowrap|''W'' ∨ 2}}, {{nowrap|''W'' ∨ 3}}, ..., {{nowrap|''W'' ∨ ''q''}}] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of {{nowrap|''M''<sub>meantone</sub> ∨ ''q''}}, where "Meantone" is the 7-limit wedgie, with <math>\tmonzo{1200 + 300 * \log_{2}(5) & -1200 & 0 & 0}</math> giving the value in cents of the [[Quarter-comma_meantone|quarter-comma meantone]] tuning of the interval denoted by {{nowrap|''M''<sub>meantone</sub> ∨ ''q''}}.
 The interior product can also be used to add a comma to a ''p''-limit temperament of rank ''r'', producing a rank-({{nowrap|''r'' &minus; 1}}) temperament which [[support]]s it. For instance, <math>\tritmonzo{1 & 2 & -3 & -2 & 1 & -4 & -5 & 12 & 9 & -19}</math> is the wedgie for 11-limit [[Marvel_family#Marvel|Marvel temperament]]. Then: