Tenney–Euclidean metrics: Difference between revisions

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== TE temperamental norm ==

Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{~~mpp~~}}''V''}}, where ''V''{{~~mpp~~}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{~~mpp~~}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{~~mpp~~}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''2''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''1 and '''m'''2, {{subsup|'''m'''|1|T}}''P'''''m'''2 defines the semidefinite form on weighted monzos, and hence {{subsup|'''b'''|1|T}}''W''{{inv}}''PW''{{inv}}'''b'''2 defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''2''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').

Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{+}}''V''}}, where ''V''{{+}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{+}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{+}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''2''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''1 and '''m'''2, {{subsup|'''m'''|1|T}}''P'''''m'''2 defines the semidefinite form on weighted monzos, and hence {{subsup|'''b'''|1|T}}''W''{{inv}}''PW''{{inv}}'''b'''2 defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''2''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').

It may be noted that {{nowrap|(''VV''{{t}}){{inv}} {{=}} (''AW''2''A''{{t}}){{inv}}}} is the inverse of the {{w|Gramian matrix}} used to compute [[TE complexity]], and hence is the corresponding Gram matrix for the dual space. Hence '''P''' represents a change of basis defined by the mapping given by the vals combined with an {{w|inner product space|inner product}} on the result. Given a monzo '''b''', ''A'''''b''' represents the tempered interval corresponding to '''b''' in a basis defined by the mapping ''A'', and {{nowrap|''P''''T'' {{=}} (''AW''2''A''{{t}}){{inv}}}} defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by ''A''.

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== Octave-equivalent TE seminorm ==

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If ''B'' is a matrix with columns of monzos spanning the commas of a regular temperament, then {{nowrap|''M'' {{=}} ''W''{{inv}}''B''}} is the corresponding weighted matrix. {{nowrap|''Q'' {{=}} ''MM''{{~~mpp~~}}}} is a projection matrix dual to {{nowrap|''P'' {{=}} ''I'' − ''Q''}}, where ''I'' is the identity matrix, and ''P'' is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then {{nowrap|''P'' {{=}} ''I'' − ''M''(''M''{{t}}''M''){{inv}}''M''{{t}}}} {{nowrap|{{=}} ''I'' − ''W''{{inv}}''B''(''B''{{t}}''W''-2''B''){{inv}}''B''{{t}}W{{inv}}}}, and {{nowrap|'''m'''{{t}}''P'''''m''' {{=}} '''b'''{{t}}''W''{{inv}}''PW''{{inv}}'''b'''}}, or {{nowrap|'''b'''{{t}}(''W''{{inv|2}} − ''W''{{inv|2}}''B''(''B''{{t}}''W''{{inv|2}}''B''){{inv}}''B''{{t}}''W''{{inv|2}})'''b'''}}, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos ''B''.

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If ''B'' is a matrix with columns of monzos spanning the commas of a regular temperament, then {{nowrap|''M'' {{=}} ''W''{{inv}}''B''}} is the corresponding weighted matrix. {{nowrap|''Q'' {{=}} ''MM''{{+}}}} is a projection matrix dual to {{nowrap|''P'' {{=}} ''I'' − ''Q''}}, where ''I'' is the identity matrix, and ''P'' is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then {{nowrap|''P'' {{=}} ''I'' − ''M''(''M''{{t}}''M''){{inv}}''M''{{t}}}} {{nowrap|{{=}} ''I'' − ''W''{{inv}}''B''(''B''{{t}}''W''-2''B''){{inv}}''B''{{t}}W{{inv}}}}, and {{nowrap|'''m'''{{t}}''P'''''m''' {{=}} '''b'''{{t}}''W''{{inv}}''PW''{{inv}}'''b'''}}, or {{nowrap|'''b'''{{t}}(''W''{{inv|2}} − ''W''{{inv|2}}''B''(''B''{{t}}''W''{{inv|2}}''B''){{inv}}''B''{{t}}''W''{{inv|2}})'''b'''}}, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos ''B''.

To define the '''octave-equivalent Tenney–Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo| 1 0 0 … 0 }} representing 2 to the matrix ''B''. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by ''A''.

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If instead we want the OETES, we may remove the first row of {{mapping| 1 2 3 | 0 -3 -5 }}, leaving just {{mapping| 0 -3 -5 }}. If we now call this 1×3 matrix ''A'', then {{nowrap|''P''''T'' {{=}} (''AW''2''A''{{t}}){{inv}}}} is a 1×1 matrix; in effect a scalar, with value {{mapping| 0.1215588 }}. Multiplying a monzo '''b''' by ''A'' on the left gives a 1×1 matrix ''A'''''b''' whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which '''b''' belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of generator steps.

For a more substantial example we need to consider at least a rank-3 temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2×4 matrix of monzos whose first row represents 2 and whose second row 225/224 is {{monzo list| 1 0 0 0 | -5 2 2 -1 }}. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is {{nowrap|''M'' {{=}} {{monzo list| 1 0 0 0 | -5 2p3 2p5 -p7 }}}}, and {{nowrap|''P'' {{=}} ''I'' − ''MM''{{~~mpp~~}}}} = [{{monzo| 1 0 0 0 }}, {{monzo| 0 4(p5)2+(p7)2 -4(p3)(p5) 2(p3)(p7) }}/''H'', {{monzo| 0 -4(p3)(p5) 4(p3)2+(p7)2 2(p5)(p7) }}/''H'', {{monzo| 0 2(p3)(p7) 2(p5)(p7) 4((p3)2+(p5)2) }}/''H''], where {{nowrap|''H'' {{=}} 4(p3)2+4(p5)2+(p7)2}}. On the other hand, we may start from the normal val list for the temperament, which is {{mapping| 1 0 0 -5 | 0 1 0 2 | 0 0 1 2 }}. Removing the first row gives {{mapping| 0 1 0 2 | 0 0 1 2 }}, and val weighting this gives {{nowrap|''C'' {{=}} {{mapping| 0 1/p3 0 2/p7 | 0 0 1/p5 2/p7 }}}}. Then {{nowrap|''P'' {{=}} ''C''{{~~mpp~~}}''C''}} is precisely the same matrix we obtained before.

For a more substantial example we need to consider at least a rank-3 temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2×4 matrix of monzos whose first row represents 2 and whose second row 225/224 is {{monzo list| 1 0 0 0 | -5 2 2 -1 }}. If we denote log2 of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is {{nowrap|''M'' {{=}} {{monzo list| 1 0 0 0 | -5 2p3 2p5 -p7 }}}}, and {{nowrap|''P'' {{=}} ''I'' − ''MM''{{+}}}} = [{{monzo| 1 0 0 0 }}, {{monzo| 0 4(p5)2 + (p7)2 -4(p3)(p5) 2(p3)(p7) }}/''H'', {{monzo| 0 -4(p3)(p5) 4(p3)2 + (p7)2 2(p5)(p7) }}/''H'', {{monzo| 0 2(p3)(p7) 2(p5)(p7) 4((p3)2 + (p5)2) }}/''H''], where {{nowrap|''H'' {{=}} 4(p3)2 + 4(p5)2 + (p7)2}}. On the other hand, we may start from the normal val list for the temperament, which is {{mapping| 1 0 0 -5 | 0 1 0 2 | 0 0 1 2 }}. Removing the first row gives {{mapping| 0 1 0 2 | 0 0 1 2 }}, and val weighting this gives {{nowrap|''C'' {{=}} {{mapping| 0 1/p3 0 2/p7 | 0 0 1/p5 2/p7 }}}}. Then {{nowrap|''P'' {{=}} ''C''{{+}}''C''}} is precisely the same matrix we obtained before.

Octaves are now projected to the origin as well as commas. We can as before form the quotient space with respect to the seminorm, and obtain a normed space in which octave-equivalent interval classes of the intervals of the temperament are the lattice points. This seminorm applied to monzos gives the OE complexity.

If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals ''R'', then the inner product on note classes in this basis is defined by the symmetric matrix {{nowrap|''S'' {{=}} (''RW''2''R''{{t}}){{inv}}}}. In the case of marvel, we obtain {{nowrap|''S'' {{=}} {{!((}}(p3)2(4(p5)2+(p7)2) -4(p3)2(p5)2{{)!}},}} {{nowrap|{{!(}}-4(p3)2(p5)2 (p5)2(4(p3)2+(p7)2){{))!}}/''H''}}. If {{nowrap|'''k''' {{=}} {{monzo| ''k''1 ''k''2 }}}} is a note class of marvel in the coordinates defined by the truncated val list ''R'', which in this case has a basis corresponding to tempered 3 and 5, then √('''k'''{{t}}''S'''''k''') gives the OE complexity of the note class.

If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals ''R'', then the inner product on note classes in this basis is defined by the symmetric matrix {{nowrap|''S'' {{=}} (''RW''2''R''{{t}}){{inv}}}}. In the case of marvel, we obtain {{nowrap|''S'' {{=}} {{!((}}(p3)2(4(p5)2 + (p7)2) -4(p3)2(p5)2{{)!}},}} {{nowrap|{{!(}}-4(p3)2(p5)2 (p5)2(4(p3)2 + (p7)2){{))!}}/''H''}}. If {{nowrap|'''k''' {{=}} {{monzo| ''k''1 ''k''2 }}}} is a note class of marvel in the coordinates defined by the truncated val list ''R'', which in this case has a basis corresponding to tempered 3 and 5, then √('''k'''{{t}}''S'''''k''') gives the OE complexity of the note class.

[[Category:Math]]

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 == TE temperamental norm ==
-Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{mpp}}''V''}}, where ''V''{{mpp}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{mpp}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{mpp}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''<sub>1</sub> and '''m'''<sub>2</sub>, {{subsup|'''m'''|1|T}}''P'''''m'''<sub>2</sub> defines the semidefinite form on weighted monzos, and hence {{subsup|'''b'''|1|T}}''W''{{inv}}''PW''{{inv}}'''b'''<sub>2</sub> defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').
+Suppose now ''A'' is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is {{nowrap|''V'' {{=}} ''AW''}}. The [[Tenney–Euclidean tuning|TE tuning]] [[projection matrix]] is then {{nowrap|''P'' {{=}} ''V''{{+}}''V''}}, where ''V''{{+}} denotes the {{w|Moore–Penrose pseudoinverse}} of ''V''. If the rows of ''V'' (or equivalently, ''A'') are linearly independent, then we have {{nowrap|''V''{{+}} {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}}}. In terms of vals, the tuning projection matrix is {{nowrap|''V''{{+}}''V'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}} {{nowrap|{{=}} ''WA''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''AW''}}. ''P'' is a {{w|positive-definite matrix|positive semidefinite matrix}}, so it defines a {{w|definite bilinear form|positive semidefinite bilinear form}}. In terms of weighted monzos '''m'''<sub>1</sub> and '''m'''<sub>2</sub>, {{subsup|'''m'''|1|T}}''P'''''m'''<sub>2</sub> defines the semidefinite form on weighted monzos, and hence {{subsup|'''b'''|1|T}}''W''{{inv}}''PW''{{inv}}'''b'''<sub>2</sub> defines a semidefinite form on unweighted monzos, in terms of the matrix {{nowrap|'''P''' {{=}} ''W''{{inv}}''PW''{{inv}}}} {{nowrap|{{=}} ''A''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''A''}}. From the semidefinite form we obtain an associated {{w|definite quadratic form|semidefinite quadratic form}} '''b'''{{t}}'''Pb''' and from this the {{w|norm (mathematics)|seminorm}} √('''b'''{{t}}'''Pb''').
 It may be noted that {{nowrap|(''VV''{{t}}){{inv}} {{=}} (''AW''<sup>2</sup>''A''{{t}}){{inv}}}} is the inverse of the {{w|Gramian matrix}} used to compute [[TE complexity]], and hence is the corresponding Gram matrix for the dual space. Hence '''P''' represents a change of basis defined by the mapping given by the vals combined with an {{w|inner product space|inner product}} on the result. Given a monzo '''b''', ''A'''''b''' represents the tempered interval corresponding to '''b''' in a basis defined by the mapping ''A'', and {{nowrap|''P''<sub>''T''</sub> {{=}} (''AW''<sup>2</sup>''A''{{t}}){{inv}}}} defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by ''A''.
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 == Octave-equivalent TE seminorm ==
-Instead of starting from a matrix of vals, we may start from a matrix of monzos. If ''B'' is a matrix with columns of monzos spanning the commas of a regular temperament, then {{nowrap|''M'' {{=}} ''W''{{inv}}''B''}} is the corresponding weighted matrix. {{nowrap|''Q'' {{=}} ''MM''{{mpp}}}} is a projection matrix dual to {{nowrap|''P'' {{=}} ''I'' − ''Q''}}, where ''I'' is the identity matrix, and ''P'' is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then {{nowrap|''P'' {{=}} ''I'' − ''M''(''M''{{t}}''M''){{inv}}''M''{{t}}}} {{nowrap|{{=}} ''I'' − ''W''{{inv}}''B''(''B''{{t}}''W''<sup>-2</sup>''B''){{inv}}''B''{{t}}W{{inv}}}}, and {{nowrap|'''m'''{{t}}''P'''''m''' {{=}} '''b'''{{t}}''W''{{inv}}''PW''{{inv}}'''b'''}}, or {{nowrap|'''b'''{{t}}(''W''{{inv|2}} − ''W''{{inv|2}}''B''(''B''{{t}}''W''{{inv|2}}''B''){{inv}}''B''{{t}}''W''{{inv|2}})'''b'''}}, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos ''B''.
+Instead of starting from a matrix of vals, we may start from a matrix of monzos. If ''B'' is a matrix with columns of monzos spanning the commas of a regular temperament, then {{nowrap|''M'' {{=}} ''W''{{inv}}''B''}} is the corresponding weighted matrix. {{nowrap|''Q'' {{=}} ''MM''{{+}}}} is a projection matrix dual to {{nowrap|''P'' {{=}} ''I'' − ''Q''}}, where ''I'' is the identity matrix, and ''P'' is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefore linearly independent, then {{nowrap|''P'' {{=}} ''I'' − ''M''(''M''{{t}}''M''){{inv}}''M''{{t}}}} {{nowrap|{{=}} ''I'' − ''W''{{inv}}''B''(''B''{{t}}''W''<sup>-2</sup>''B''){{inv}}''B''{{t}}W{{inv}}}}, and {{nowrap|'''m'''{{t}}''P'''''m''' {{=}} '''b'''{{t}}''W''{{inv}}''PW''{{inv}}'''b'''}}, or {{nowrap|'''b'''{{t}}(''W''{{inv|2}} − ''W''{{inv|2}}''B''(''B''{{t}}''W''{{inv|2}}''B''){{inv}}''B''{{t}}''W''{{inv|2}})'''b'''}}, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos ''B''.
 To define the '''octave-equivalent Tenney–Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo| 1 0 0 … 0 }} representing 2 to the matrix ''B''. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by ''A''.
@@ Line 49: / Line 49: @@
 If instead we want the OETES, we may remove the first row of {{mapping| 1 2 3 | 0 -3 -5 }}, leaving just {{mapping| 0 -3 -5 }}. If we now call this 1×3 matrix ''A'', then {{nowrap|''P''<sub>''T''</sub> {{=}} (''AW''<sup>2</sup>''A''{{t}}){{inv}}}} is a 1×1 matrix; in effect a scalar, with value {{mapping| 0.1215588 }}. Multiplying a monzo '''b''' by ''A'' on the left gives a 1×1 matrix ''A'''''b''' whose value is the number of generator steps of porcupine (of size a tempered 10/9) it takes to get to the octave class to which '''b''' belongs. Performing the multiplication and taking the square root, we conclude the OE complexity is simply proportional to this number of generator steps.
-For a more substantial example we need to consider at least a rank-3 temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2×4 matrix of monzos whose first row represents 2 and whose second row 225/224 is {{monzo list| 1 0 0 0 | -5 2 2 -1 }}. If we denote log<sub>2</sub> of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is {{nowrap|''M'' {{=}} {{monzo list| 1 0 0 0 | -5 2p3 2p5 -p7 }}}}, and {{nowrap|''P'' {{=}} ''I'' − ''MM''{{mpp}}}} =&nbsp;[{{monzo| 1 0 0 0 }}, {{monzo| 0 4(p5)<sup>2</sup>+(p7)<sup>2</sup> -4(p3)(p5) 2(p3)(p7) }}/''H'', {{monzo| 0 -4(p3)(p5) 4(p3)<sup>2</sup>+(p7)<sup>2</sup> 2(p5)(p7) }}/''H'', {{monzo| 0 2(p3)(p7) 2(p5)(p7) 4((p3)<sup>2</sup>+(p5)<sup>2</sup>) }}/''H''], where {{nowrap|''H'' {{=}} 4(p3)<sup>2</sup>+4(p5)<sup>2</sup>+(p7)<sup>2</sup>}}. On the other hand, we may start from the normal val list for the temperament, which is {{mapping| 1 0 0 -5 | 0 1 0 2 | 0 0 1 2 }}. Removing the first row gives {{mapping| 0 1 0 2 | 0 0 1 2 }}, and val weighting this gives {{nowrap|''C'' {{=}} {{mapping| 0 1/p3 0 2/p7 | 0 0 1/p5 2/p7 }}}}. Then {{nowrap|''P'' {{=}} ''C''{{mpp}}''C''}} is precisely the same matrix we obtained before.
+For a more substantial example we need to consider at least a rank-3 temperament, so let us turn to 7-limit marvel, the 7-limit temperament tempering out 225/224. The 2×4 matrix of monzos whose first row represents 2 and whose second row 225/224 is {{monzo list| 1 0 0 0 | -5 2 2 -1 }}. If we denote log<sub>2</sub> of the odd primes by p3, p5, p7 etc, then the monzo weighting of this matrix is {{nowrap|''M'' {{=}} {{monzo list| 1 0 0 0 | -5 2p3 2p5 -p7 }}}}, and {{nowrap|''P'' {{=}} ''I'' − ''MM''{{+}}}} =&nbsp;[{{monzo| 1 0 0 0 }}, {{monzo| 0 4(p5)<sup>2</sup> + (p7)<sup>2</sup> -4(p3)(p5) 2(p3)(p7) }}/''H'', {{monzo| 0 -4(p3)(p5) 4(p3)<sup>2</sup> + (p7)<sup>2</sup> 2(p5)(p7) }}/''H'', {{monzo| 0 2(p3)(p7) 2(p5)(p7) 4((p3)<sup>2</sup> + (p5)<sup>2</sup>) }}/''H''], where {{nowrap|''H'' {{=}} 4(p3)<sup>2</sup> + 4(p5)<sup>2</sup> + (p7)<sup>2</sup>}}. On the other hand, we may start from the normal val list for the temperament, which is {{mapping| 1 0 0 -5 | 0 1 0 2 | 0 0 1 2 }}. Removing the first row gives {{mapping| 0 1 0 2 | 0 0 1 2 }}, and val weighting this gives {{nowrap|''C'' {{=}} {{mapping| 0 1/p3 0 2/p7 | 0 0 1/p5 2/p7 }}}}. Then {{nowrap|''P'' {{=}} ''C''{{+}}''C''}} is precisely the same matrix we obtained before.
 Octaves are now projected to the origin as well as commas. We can as before form the quotient space with respect to the seminorm, and obtain a normed space in which octave-equivalent interval classes of the intervals of the temperament are the lattice points. This seminorm applied to monzos gives the OE complexity.
-If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals ''R'', then the inner product on note classes in this basis is defined by the symmetric matrix {{nowrap|''S'' {{=}} (''RW''<sup>2</sup>''R''{{t}}){{inv}}}}. In the case of marvel, we obtain {{nowrap|''S'' {{=}} {{!((}}(p3)<sup>2</sup>(4(p5)<sup>2</sup>+(p7)<sup>2</sup>) -4(p3)<sup>2</sup>(p5)<sup>2</sup>{{)!}},}} {{nowrap|{{!(}}-4(p3)<sup>2</sup>(p5)<sup>2</sup> (p5)<sup>2</sup>(4(p3)<sup>2</sup>+(p7)<sup>2</sup>){{))!}}/''H''}}. If {{nowrap|'''k''' {{=}} {{monzo| ''k''<sub>1</sub> ''k''<sub>2</sub> }}}} is a note class of marvel in the coordinates defined by the truncated val list ''R'', which in this case has a basis corresponding to tempered 3 and 5, then √('''k'''{{t}}''S'''''k''') gives the OE complexity of the note class.
+If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals ''R'', then the inner product on note classes in this basis is defined by the symmetric matrix {{nowrap|''S'' {{=}} (''RW''<sup>2</sup>''R''{{t}}){{inv}}}}. In the case of marvel, we obtain {{nowrap|''S'' {{=}} {{!((}}(p3)<sup>2</sup>(4(p5)<sup>2</sup> + (p7)<sup>2</sup>) -4(p3)<sup>2</sup>(p5)<sup>2</sup>{{)!}},}} {{nowrap|{{!(}}-4(p3)<sup>2</sup>(p5)<sup>2</sup> (p5)<sup>2</sup>(4(p3)<sup>2</sup> + (p7)<sup>2</sup>){{))!}}/''H''}}. If {{nowrap|'''k''' {{=}} {{monzo| ''k''<sub>1</sub> ''k''<sub>2</sub> }}}} is a note class of marvel in the coordinates defined by the truncated val list ''R'', which in this case has a basis corresponding to tempered 3 and 5, then √('''k'''{{t}}''S'''''k''') gives the OE complexity of the note class.
 [[Category:Math]]