Tenney–Euclidean tuning: Difference between revisions

Line 1:

__FORCETOC__

'''Tenney-Euclidean tuning''' (or '''TE tuning''') is a variant of [[TOP tuning]] which employs the [[Tenney-Euclidean metrics|TE norm]]. Just as the TOP tuning minimizes the maximum Tenney-weighted (L1) error of any interval, the TE tuning minimizes the maximum TE-weighted (L2) error of any interval.

'''Tenney-Euclidean tuning''' (or '''TE tuning''') is a variant of [[TOP tuning]] which employs the [[Tenney-~~Euclidean_metrics~~|TE norm]]. Just as the TOP tuning minimizes the maximum Tenney-weighted (L1) error of any interval, the TE tuning minimizes the maximum TE-weighted (L2) error of any interval.

== Introduction ==

If we have ''k'' linearly independent [[~~Vals_and_Tuning_Space~~|vals]] of dimension ''n'', they will span a subspace of [[~~Vals_and_Tuning_Space~~|tuning space]]. This subspace defines a regular temperament of rank ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' dimensional subspace of [[~~Monzos_and_Interval_Space~~|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely.

If we have ''k'' linearly independent [[Vals and tuning space|vals]] of dimension ''n'', they will span a subspace of [[Vals and tuning space|tuning space]]. This subspace defines a regular temperament of rank ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' dimensional subspace of [[Monzos and interval space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely.

A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is the weighted RMS (root-mean-squared) tuning.

Line 17:

Line 16:

where J = pW, i.e. the JI point, in the case of TE weighting it is {{val|1 1 … 1}}. The system simply says that the sum of V''ij'' steps of generator g''i'' for all ''i'' should equal the ''j''-th targeted JI interval J''j''.

There are a number of methods to solve least square problems. One common way is to use the [[~~wikipedia~~:~~Moore%E2%80%93Penrose_pseudoinverse~~|Moore–Penrose pseudoinverse]].

There are a number of methods to solve least square problems. One common way is to use the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]].

== Computing TE tuning using pseudoinverse ==

Line 44:

Line 43:

We may also obtain the TE tuning from a projection map. P = V+V is the orthogonal projection map 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 JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.

We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose rows are weighted monzos, and I is the ''n''x''n'' identity matrix, then P = I - M+M is the same projection map as V+V 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.

We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose rows are weighted monzos, and I is the ''n''×''n'' identity matrix, then P = I - M+M is the same projection map as V+V 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.

== Pure octaves TE tuning ==

~~''See~~ also~~: [[~~POTE tuning~~]]''~~

We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = gV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia:~~Scalar_multiplication~~|scalar multiplication]], T/T1, where T1, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.

We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = gV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia: Scalar multiplication|scalar multiplication]], T/T1, where T1, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.

== ~~The~~ Frobenius projection map ==

== Frobenius projection map ==

We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[~~Fractional_monzos~~|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[~~Fractional_monzos|~~fractional monzos]]. The Frobenius projection map therefore, like the [[~~Wedgies_and_Multivals|~~wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I - Q is the same projection map as would be found if starting from a set of vals defining the same temperament.

We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection map therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I - Q is the same projection map 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 P = V+V is a [~~http~~:~~//en.wikipedia.org/wiki/~~Positive-~~definite_matrix~~ positive-semidefinite] [~~http~~:~~//en.wikipedia.org/wiki/Symmetric_matrix~~ symmetric matrix] with rational matrix entries, which exactly specifies the regular temperament defined by the vals of V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M+M.

Spelling this out, if V is a matrix whose rows are vals, then P = V+V is a [[wikipedia: Positive-definite matrix|positive-semidefinite]] [[wikipedia: Symmetric matrix|symmetric matrix]] with rational matrix entries, which exactly specifies the regular temperament defined by the vals of V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M+M.

If the vals defining V are linearly independent, then P = VT(VVT)-1V. If the rows of M are independent, then we likewise have P = I - MT(MMT)-1M.

== Examples ==

''For practical helps, see [[POTE tuning]].''

:''For practical helps, see [[POTE tuning]].''

The val for 5-limit 12et is {{val|12 19 28}}. In weighted coordinates, that becomes v12 = {{val|12 19/log23 28/log25}} ~ {{val|12.0 11.988 12.059}}. If we take this to be a 1×3 matrix and take the pseudoinverse, we get the 3×1 matrix v12+ ~ [{{monzo|0.027706 0.027677 0.027842}}]. Then P = v12+v12 is a projection map onto the one-dimensional subspace whose single basis vector is v12. We find that v12P equals v12; on the other hand, if we take the monzo for 81/80, which is {{monzo|-4 4 -1}}; and monzo-weight it to {{monzo|-4 4log23 -log25}} and multiply (either side, the matrix is symmetric) by P, we get the zero vector, corresponding to the unison.

Line 93:

Line 92:

This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et and 22et to themselves.

[[Category:~~math~~]]

[[Category:Math]]

[[Category:~~theory~~]]

[[Category:Theory]]

[[Category:~~todo:improve_definition~~]]

[[Category:Tuning]]

[[Category:~~tuning~~]]

[[Category:Tuning technique]]

~~[[Category:tuning_technique]]~~

@@ Line 1: / Line 1: @@
 __FORCETOC__
+'''Tenney-Euclidean tuning''' (or '''TE tuning''') is a variant of [[TOP tuning]] which employs the [[Tenney-Euclidean metrics|TE norm]]. Just as the TOP tuning minimizes the maximum Tenney-weighted (L1) error of any interval, the TE tuning minimizes the maximum TE-weighted (L2) error of any interval.
-'''Tenney-Euclidean tuning''' (or '''TE tuning''') is a variant of [[TOP tuning]] which employs the [[Tenney-Euclidean_metrics|TE norm]]. Just as the TOP tuning minimizes the maximum Tenney-weighted (L1) error of any interval, the TE tuning minimizes the maximum TE-weighted (L2) error of any interval.
 == Introduction ==
-If we have ''k'' linearly independent [[Vals_and_Tuning_Space|vals]] of dimension ''n'', they will span a subspace of [[Vals_and_Tuning_Space|tuning space]]. This subspace defines a regular temperament of rank ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' dimensional subspace of [[Monzos_and_Interval_Space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely.
+If we have ''k'' linearly independent [[Vals and tuning space|vals]] of dimension ''n'', they will span a subspace of [[Vals and tuning space|tuning space]]. This subspace defines a regular temperament of rank ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' dimensional subspace of [[Monzos and interval space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely.
 A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is the weighted RMS (root-mean-squared) tuning.
@@ Line 17: / Line 16: @@
 where J = pW, i.e. the JI point, in the case of TE weighting it is {{val|1 1 … 1}}. The system simply says that the sum of V<sub>''ij''</sub> steps of generator g<sub>''i''</sub> for all ''i'' should equal the ''j''-th targeted JI interval J<sub>''j''</sub>.
-There are a number of methods to solve least square problems. One common way is to use the [[wikipedia:Moore%E2%80%93Penrose_pseudoinverse|Moore–Penrose pseudoinverse]].
+There are a number of methods to solve least square problems. One common way is to use the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]].
 == Computing TE tuning using pseudoinverse ==
@@ Line 44: / Line 43: @@
 We may also obtain the TE tuning from a projection map. P = V<sup>+</sup>V is the orthogonal projection map 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 JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.
-We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose rows are weighted monzos, and I is the ''n''x''n'' identity matrix, then P = I - M<sup>+</sup>M is the same projection map as V<sup>+</sup>V 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.
+We may find the same projection map starting from a list of weighted monzos rather than vals. If M is a rank ''n'' matrix whose rows are weighted monzos, and I is the ''n''×''n'' identity matrix, then P = I - M<sup>+</sup>M is the same projection map as V<sup>+</sup>V 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.
 == Pure octaves TE tuning ==
-''See also: [[POTE tuning]]''
+{{see also| POTE tuning }}
-We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = gV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia:Scalar_multiplication|scalar multiplication]], T/T<sub>1</sub>, where T<sub>1</sub>, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.
+We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = gV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia: Scalar multiplication|scalar multiplication]], T/T<sub>1</sub>, where T<sub>1</sub>, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.
-== The Frobenius projection map ==
+== Frobenius projection map ==
-We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional_monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[Fractional_monzos|fractional monzos]]. The Frobenius projection map therefore, like the [[Wedgies_and_Multivals|wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I - Q is the same projection map as would be found if starting from a set of vals defining the same temperament.
+We may also do the same things starting from unweighted vals. This leads to a different tuning, the [[Fractional monzos|Frobenius tuning]], which is perfectly functional but has less theoretical justification than TE tuning. However, if greater weight needs to be attached to the larger primes than TE tuning attaches, Frobenius tuning may be preferred; people who feel that larger primes require more tuning care than smaller ones may well prefer it. However the main value of unweighted vals is that the pseudoinverse and projection map have rational entries, so that the rows of the matrix are [[fractional monzos]]. The Frobenius projection map therefore, like the [[wedgie]], defines a completely canonical object not depending on any arbitrary definition (eg how Hermite normal form or LLL reduction is specifically defined) which corresponds 1-1 with temperaments, and which does not depend on whether the monzos or vals from which it is computed are [[Saturation|saturated]]. It may be found starting either from a set of vals or a set of commas, since if Q is the projection map found by treating monzos in the same way as vals, P = I - Q is the same projection map 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 P = V<sup>+</sup>V is a [http://en.wikipedia.org/wiki/Positive-definite_matrix positive-semidefinite] [http://en.wikipedia.org/wiki/Symmetric_matrix symmetric matrix] with rational matrix entries, which exactly specifies the regular temperament defined by the vals of V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M<sup>+</sup>M.
+Spelling this out, if V is a matrix whose rows are vals, then P = V<sup>+</sup>V is a [[wikipedia: Positive-definite matrix|positive-semidefinite]] [[wikipedia: Symmetric matrix|symmetric matrix]] with rational matrix entries, which exactly specifies the regular temperament defined by the vals of V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M<sup>+</sup>M.
 If the vals defining V are linearly independent, then P = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>V. If the rows of M are independent, then we likewise have P = I - M<sup>T</sup>(MM<sup>T</sup>)<sup>-1</sup>M.
 == Examples ==
-''For practical helps, see [[POTE tuning]].''
+:''For practical helps, see [[POTE tuning]].''
 The val for 5-limit 12et is {{val|12 19 28}}. In weighted coordinates, that becomes v<sub>12</sub> = {{val|12 19/log<sub>2</sub>3 28/log<sub>2</sub>5}} ~ {{val|12.0 11.988 12.059}}. If we take this to be a 1×3 matrix and take the pseudoinverse, we get the 3×1 matrix v<sub>12</sub><sup>+</sup> ~ [{{monzo|0.027706 0.027677 0.027842}}]. Then P = v<sub>12</sub><sup>+</sup>v<sub>12</sub> is a projection map onto the one-dimensional subspace whose single basis vector is v<sub>12</sub>. We find that v<sub>12</sub>P equals v<sub>12</sub>; on the other hand, if we take the monzo for 81/80, which is {{monzo|-4 4 -1}}; and monzo-weight it to {{monzo|-4 4log<sub>2</sub>3 -log<sub>2</sub>5}} and multiply (either side, the matrix is symmetric) by P, we get the zero vector, corresponding to the unison.
@@ Line 93: / Line 92: @@
 This sends monzos for 50/49, 64/63 etc. to the unison monzo, and vals for 10et, 12et and 22et to themselves.
-[[Category:math]]
+[[Category:Math]]
-[[Category:theory]]
+[[Category:Theory]]
-[[Category:todo:improve_definition]]
+[[Category:Tuning]]
-[[Category:tuning]]
+[[Category:Tuning technique]]
-[[Category:tuning_technique]]
+{{todo|improve definition}}