Tenney–Euclidean tuning: Difference between revisions

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== Computation using pseudoinverse ==

The Moore–Penrose pseudoinverse, denoted ''A''{{~~mpp~~}}, is a generalization of the inverse matrix with which it shares a lot of properties. To name a few:

The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. To name a few:

* If ''A'' is square and invertible, then its pseudoinverse is equal to its inverse; that is, {{nowrap|''A''{{~~mpp~~}} {{=}} ''A''{{inv}}}}

* If ''A'' is square and invertible, then its pseudoinverse is equal to its inverse; that is, {{nowrap|''A''{{+}} {{=}} ''A''{{inv}}}}

* If ''A'' has rational entries, so does ''A''{{~~mpp~~}}

* If ''A'' has rational entries, so does ''A''{{+}}

* {{nowrap|(''A''{{~~mpp~~}}){{~~mpp~~}} {{=}} ''A''}}

* {{nowrap|(''A''{{+}}){{+}} {{=}} ''A''}}

* {{nowrap|(''A''{{t}}){{~~mpp~~}} {{=}} (''A''{{~~mpp~~}}){{t}}}}, where ''A''{{t}} is the transpose of ''A''

* {{nowrap|(''A''{{t}}){{+}} {{=}} (''A''{{+}}){{t}}}}, where ''A''{{t}} is the transpose of ''A''

* ''AA''{{~~mpp~~}} is the orthogonal projection matrix that maps onto the space spanned by the columns of ''A''

* ''AA''{{+}} is the orthogonal projection matrix that maps onto the space spanned by the columns of ''A''

* ''A''{{~~mpp~~}}''A'' is the orthogonal projection matrix that maps onto the space spanned by the rows of ''A''

* ''A''{{+}}''A'' is the orthogonal projection matrix that maps onto the space spanned by the rows of ''A''

* {{nowrap|''I'' − ''A''{{~~mpp~~}}''A''}}, where ''I'' is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of ''A''

* {{nowrap|''I'' − ''A''{{+}}''A''}}, where ''I'' is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of ''A''

* If the rows of ''A'' are linearly independent, then {{nowrap|''A''{{~~mpp~~}} {{=}} ''A''{{t}}(''AA''{{~~mpp~~}}){{inv}}}}. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.

* If the rows of ''A'' are linearly independent, then {{nowrap|''A''{{+}} {{=}} ''A''{{t}}(''AA''{{+}}){{inv}}}}. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.

* ''uA''{{~~mpp~~}} is the nearest point to ''u'' in the subspace spanned by the rows of ''A''; ''A''{{~~mpp~~}}''v'' is the nearest point to ''v'' in the space spanned by the columns of ''A''.

* ''uA''{{+}} is the nearest point to ''u'' in the subspace spanned by the rows of ''A''; ''A''{{+}}''v'' is the nearest point to ''v'' in the space spanned by the columns of ''A''.

In the pseudoinverse method, the (not necessarily independent) TE generators which correspond to the rows of ''V'' are given by

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<math>\displaystyle T = GV = J_W V_W^+ V</math>

We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''''W''{{~~mpp~~}}''V''''W''}} is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does ''P''. However, ''P'' is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JIP: {{nowrap|''T'' {{=}} ''JP''}} where ''J'' is the JIP, which is the nearest point in the subspace corresponding to the temperament to ''J''.

We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''''W''{{+}}''V''''W''}} is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does ''P''. However, ''P'' is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JIP: {{nowrap|''T'' {{=}} ''JP''}} where ''J'' is the JIP, which is the nearest point in the subspace corresponding to the temperament to ''J''.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''''W'' is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''''W''{{subsup|''M''|''W''|+}}}} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''''W'' so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.

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However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection matrix therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Mathematical theory of saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if ''Q'' is the projection matrix found by treating monzos in the same way as vals, {{nowrap|''P'' {{=}} ''I'' − ''Q''}} is the same projection matrix as would be found if starting from a set of vals defining the same temperament.

Spelling this out, if ''V'' is a matrix whose rows are vals, then {{nowrap|''P'' {{=}} ''V''{{~~mpp~~}}''V''}} is a {{w|Positive-definite matrix|positive-semidefinite}} {{w|symmetric matrix}} with rational matrix entries, which exactly specifies the regular temperament defined by the vals of ''V''. If ''M'' is a matrix with columns of monzos which spans the subspace of interval space containing the commas, then this same matrix ''P'' is given by {{nowrap|''I'' − ''MM''{{~~mpp~~}}}}.

Spelling this out, if ''V'' is a matrix whose rows are vals, then {{nowrap|''P'' {{=}} ''V''{{+}}''V''}} is a {{w|Positive-definite matrix|positive-semidefinite}} {{w|symmetric matrix}} with rational matrix entries, which exactly specifies the regular temperament defined by the vals of ''V''. If ''M'' is a matrix with columns of monzos which spans the subspace of interval space containing the commas, then this same matrix ''P'' is given by {{nowrap|''I'' − ''MM''{{+}}}}.

If the vals defining ''V'' are linearly independent, then {{nowrap|''P'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}}. If the columns of ''M'' are independent, then we likewise have {{nowrap|''P'' {{=}} ''I'' − ''M''(''M''{{t}}''M''){{inv}}''M''{{t}}}}.

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</math>

Then the pseudoinverse ''V''{{~~mpp~~}} is approximately

Then the pseudoinverse ''V''{{+}} is approximately

<math>\displaystyle

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Multiplying {{val| 12 19 28 }} by ''P''''F'' gives {{val| 12 19 28 }} again. Multiplying the monzos for 81/80, 128/125, 648/625 etc. gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, {{val| 19 30 44 }}, by PF gives {{val| 24360 38570 56840 }}/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is {{monzo| -1 1 0 }}; times F gives the fractional monzo corresponding to (284 3133 5196)1/1289, which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of PF, which is to say by {{monzo| 1 0 0 }}PF, of 1196.778 cents.

We can do the same thing with a matrix ''V'' with rows consisting of the vals for 7-limit 12 and 22 equal; then ''V''{{~~mpp~~}}''V'', the Frobenius projection matrix for pajara, is

We can do the same thing with a matrix ''V'' with rows consisting of the vals for 7-limit 12 and 22 equal; then ''V''{{+}}''V'', the Frobenius projection matrix for pajara, is

<math>\displaystyle

@@ Line 24: / Line 24: @@
 == Computation using pseudoinverse ==
-The Moore–Penrose pseudoinverse, denoted ''A''{{mpp}}, is a generalization of the inverse matrix with which it shares a lot of properties. To name a few:
+The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. To name a few:
-* If ''A'' is square and invertible, then its pseudoinverse is equal to its inverse; that is, {{nowrap|''A''{{mpp}} {{=}} ''A''{{inv}}}}
+* If ''A'' is square and invertible, then its pseudoinverse is equal to its inverse; that is, {{nowrap|''A''{{+}} {{=}} ''A''{{inv}}}}
-* If ''A'' has rational entries, so does ''A''{{mpp}}
+* If ''A'' has rational entries, so does ''A''{{+}}
-* {{nowrap|(''A''{{mpp}}){{mpp}} {{=}} ''A''}}
+* {{nowrap|(''A''{{+}}){{+}} {{=}} ''A''}}
-* {{nowrap|(''A''{{t}}){{mpp}} {{=}} (''A''{{mpp}}){{t}}}}, where ''A''{{t}} is the transpose of ''A''
+* {{nowrap|(''A''{{t}}){{+}} {{=}} (''A''{{+}}){{t}}}}, where ''A''{{t}} is the transpose of ''A''
-* ''AA''{{mpp}} is the orthogonal projection matrix that maps onto the space spanned by the columns of ''A''
+* ''AA''{{+}} is the orthogonal projection matrix that maps onto the space spanned by the columns of ''A''
-* ''A''{{mpp}}''A'' is the orthogonal projection matrix that maps onto the space spanned by the rows of ''A''
+* ''A''{{+}}''A'' is the orthogonal projection matrix that maps onto the space spanned by the rows of ''A''
-* {{nowrap|''I'' − ''A''{{mpp}}''A''}}, where ''I'' is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of ''A''
+* {{nowrap|''I'' − ''A''{{+}}''A''}}, where ''I'' is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of ''A''
-* If the rows of ''A'' are linearly independent, then {{nowrap|''A''{{mpp}} {{=}} ''A''{{t}}(''AA''{{mpp}}){{inv}}}}. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.
+* If the rows of ''A'' are linearly independent, then {{nowrap|''A''{{+}} {{=}} ''A''{{t}}(''AA''{{+}}){{inv}}}}. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.
-* ''uA''{{mpp}} is the nearest point to ''u'' in the subspace spanned by the rows of ''A''; ''A''{{mpp}}''v'' is the nearest point to ''v'' in the space spanned by the columns of ''A''.
+* ''uA''{{+}} is the nearest point to ''u'' in the subspace spanned by the rows of ''A''; ''A''{{+}}''v'' is the nearest point to ''v'' in the space spanned by the columns of ''A''.
 In the pseudoinverse method, the (not necessarily independent) TE generators which correspond to the rows of ''V'' are given by
@@ Line 44: / Line 44: @@
 <math>\displaystyle T = GV = J_W V_W^+ V</math>
-We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''<sub>''W''</sub>{{mpp}}''V''<sub>''W''</sub>}} is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does ''P''. However, ''P'' is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JIP: {{nowrap|''T'' {{=}} ''JP''}} where ''J'' is the JIP, which is the nearest point in the subspace corresponding to the temperament to ''J''.
+We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''<sub>''W''</sub>{{+}}''V''<sub>''W''</sub>}} is the orthogonal projection matrix that maps onto the space spanned by the rows of V. This space corresponds to the temperament, and so does ''P''. However, ''P'' is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether [[contorsion]] has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JIP: {{nowrap|''T'' {{=}} ''JP''}} where ''J'' is the JIP, which is the nearest point in the subspace corresponding to the temperament to ''J''.
 We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''<sub>''W''</sub> is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''<sub>''W''</sub>{{subsup|''M''|''W''|+}}}} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''<sub>''W''</sub> so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
@@ Line 79: / Line 79: @@
 However, the main value of unweighted vals is that the pseudoinverse and projection matrix have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection matrix therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (e.g. how Hermite normal form or LLL reduction is specifically defined) which corresponds one-to-one with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Mathematical theory of saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if ''Q'' is the projection matrix found by treating monzos in the same way as vals, {{nowrap|''P'' {{=}} ''I'' − ''Q''}} is the same projection matrix as would be found if starting from a set of vals defining the same temperament.
-Spelling this out, if ''V'' is a matrix whose rows are vals, then {{nowrap|''P'' {{=}} ''V''{{mpp}}''V''}} is a {{w|Positive-definite matrix|positive-semidefinite}} {{w|symmetric matrix}} with rational matrix entries, which exactly specifies the regular temperament defined by the vals of ''V''. If ''M'' is a matrix with columns of monzos which spans the subspace of interval space containing the commas, then this same matrix ''P'' is given by {{nowrap|''I'' − ''MM''{{mpp}}}}.
+Spelling this out, if ''V'' is a matrix whose rows are vals, then {{nowrap|''P'' {{=}} ''V''{{+}}''V''}} is a {{w|Positive-definite matrix|positive-semidefinite}} {{w|symmetric matrix}} with rational matrix entries, which exactly specifies the regular temperament defined by the vals of ''V''. If ''M'' is a matrix with columns of monzos which spans the subspace of interval space containing the commas, then this same matrix ''P'' is given by {{nowrap|''I'' − ''MM''{{+}}}}.
 If the vals defining ''V'' are linearly independent, then {{nowrap|''P'' {{=}} ''V''{{t}}(''VV''{{t}}){{inv}}''V''}}. If the columns of ''M'' are independent, then we likewise have {{nowrap|''P'' {{=}} ''I'' − ''M''(''M''{{t}}''M''){{inv}}''M''{{t}}}}.
@@ Line 107: / Line 107: @@
 </math>
-Then the pseudoinverse ''V''{{mpp}} is approximately
+Then the pseudoinverse ''V''{{+}} is approximately
 <math>\displaystyle
@@ Line 136: / Line 136: @@
 Multiplying {{val| 12 19 28 }} by ''P''<sub>''F''</sub> gives {{val| 12 19 28 }} again. Multiplying the monzos for 81/80, 128/125, 648/625 etc. gives the zero monzo, corresponding to a unison. Multiplying the val for 5-limit 19 equal, {{val| 19 30 44 }}, by P<sub>F</sub> gives {{val| 24360 38570 56840 }}/1289, which is approximately the 19 equal val. Multiplying the 5-limit monzo for 3/2, which is {{monzo| -1 1 0 }}; times F gives the fractional monzo corresponding to (2<sup>84</sup> 3<sup>133</sup> 5<sup>196</sup>)<sup>1/1289</sup>, which equates to 698.121 cents, the tempering of 3/2 in Frobenius tuning for 5-limit 12et, the tuning with octave defined by the top row of P<sub>F</sub>, which is to say by {{monzo| 1 0 0 }}P<sub>F</sub>, of 1196.778 cents.
-We can do the same thing with a matrix ''V'' with rows consisting of the vals for 7-limit 12 and 22 equal; then ''V''{{mpp}}''V'', the Frobenius projection matrix for pajara, is
+We can do the same thing with a matrix ''V'' with rows consisting of the vals for 7-limit 12 and 22 equal; then ''V''{{+}}''V'', the Frobenius projection matrix for pajara, is
 <math>\displaystyle