Generator preimage: Difference between revisions

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~~For each~~ [[generator]] of a [[regular temperament]], we ~~can~~ choose one JI interval that maps to it~~. Then~~ we ~~can call~~ a ~~set of JI generators like this '''~~transversal ~~generators''' of~~ that temperament.

Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

=Definition=

Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generated by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. [[Miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].

=Technical Definition=

Given a reduced list of [[Harmonic_Limit|p-limit]] vals V, we may define a set of transversal generators for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By ''reduced'' is meant that the GCD of the elements of each of the vals is 1--or in other words, none of the vals are [[contorted]]--and that they are [[linearly independent]], so that if there are r vals, the rank of V as a matrix is r.

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q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q)

In this way the transversal generators provide a ~~[[Transversal|~~transversal]] of the p-limit, and hence the name.

In this way the transversal generators provide a transversal of the p-limit, and hence the name.

=Examples=

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For another example, consider [<1 1 1 2|, <0 2 1 1|, <0 0 2 1|] which is the [[Normal_lists|normal val list]] for breed temperament, the temperament tempering out 2401/2400. A corresponding list of transversal generators is [2, 49/40, 10/7].

=Finding the transversal ~~generators~~=

=Finding the generator preimage transversal=

Two methods for finding transversal ~~generators~~ have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel|Sintel]] in December 2021, which uses the [[Smith normal form]].

Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel|Sintel]] in December 2021, which uses the [[Smith normal form]].

== Method using the Smith Normal Form ==

So we want to find a ~~generators matrix~~ <math>G</math> for a mapping <math>M</math> where:

So we want to find a generator preimage transversal <math>T</math> for a mapping <math>M</math> where:

and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>G</math> as a prime-count vector represents an interval that <math>M</math> maps to that generator. And when this <math>G</math> has all integer entries, then these generators are all JI.

and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a [[prime-count vector]] represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.

Essentially we need a way to do:

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<math>

M^{-1}MG = M^{-1}I \\

M^{-1}MT = M^{-1}I \\

\cancel{M^{-1}M}G = M^{-1}I \\

\cancel{M^{-1}M}T = M^{-1}I \\

G = M^{-1}

T = M^{-1}

</math>

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where <math>D</math> is the Smith Normal Form of <math>M</math>, which is a diagonal matrix (~~hence~~ the name "D"). Furthermore, if <math>M</math> was defactored, then not only is <math>D</math> diagonal (only has nonzero values along its main diagonal) but the numbers along its main diagonal are all equal to 1. This type of matrix is called an orthonormal matrix, and has the property that <math>D^{T}D = I</math>, which we're going to take advantage of.

where <math>D</math> is the Smith Normal Form of <math>M</math>, which is a rectangular diagonal matrix (the name 'D' is for "diagonal"). Furthermore, if <math>M</math> was defactored, then not only is <math>D</math> diagonal (only has nonzero values along its main diagonal) but the numbers along its main diagonal are all equal to 1. This type of matrix is called an orthonormal matrix, and has the property that <math>D^{T}D = I</math>, which we're going to take advantage of.

So first let's solve the decomposition equation for <math>M</math>. First, left-multiply by <math>L^{-1}</math>:

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<math>

(L^{-1}DR^{-1})G = I

(L^{-1}DR^{-1})T = I

</math>

And we can proceed to solve this for our target <math>G</math>. The rest is busywork. First, left-multiply by <math>L</math>:

And we can proceed to solve this for our target <math>T</math>. The rest is busywork. First, left-multiply by <math>L</math>:

<math>

LL^{-1}DR^{-1}G = LI \\

LL^{-1}DR^{-1}T = LI \\

\cancel{LL^{-1}}DR^{-1}G = LI \\

\cancel{LL^{-1}}DR^{-1}T = LI \\

DR^{-1}G = L

DR^{-1}T = L

</math>

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<math>

D^{T}DR^{-1}G = D^{T}L \\

D^{T}DR^{-1}T = D^{T}L \\

\cancel{D^{T}D}R^{-1}G = D^{T}L \\

\cancel{D^{T}D}R^{-1}T = D^{T}L \\

R^{-1}G = D^{T}L

R^{-1}T = D^{T}L

</math>

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<math>

RR^{-1}G = RD^{T}L \\

RR^{-1}T = RD^{T}L \\

\cancel{RR^{-1}}G = RD^{T}L \\

\cancel{RR^{-1}}T = RD^{T}L \\

G = RD^{T}L

T = RD^{T}L

</math>

And there's our answer! It's still not necessarily giving the simplest or best JI generators, but arriving at those is an independent problem (perhaps by choosing a complexity metric and minimizing it though linear combinations with the commas and the other generators).

And there's our answer! It's still not necessarily giving the simplest or best JI generators, but arriving at those is an independent problem (perhaps by choosing a complexity metric and minimizing it though linear combinations with the commas and the other generators, or perhaps using LLL).

=== Example ===

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And so ~~some JI~~ preimages of its generators are <math>RD^{T}L</math>:

And so a transversal of the preimages of its generators are <math>RD^{T}L</math>:

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=== Wolfram Language implementation ===

{{Databox|~~jiG~~[]|

{{Databox|getGpt[]|

~~jiG~~[m_] := Module[{decomp, left, snf, right},

getGpt[m_] := Module[{decomp, left, snf, right},

decomp = SmithDecomposition[m];

left = Part[decomp, 1];

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== Method using the Hermite Normal Form ==

We can find transversal ~~generators~~ for V by the following procedure:

We can find a generator preimage transversal for V by the following procedure:

<ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma basis for V</li><li>[[Saturation|Saturate]] the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo and convert it to a rational number</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul>

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=== Wolfram Language implementation ===

{{Databox|~~transversalGenerators~~[]|

{{Databox|getGpt[]|

~~transversalGenerator~~[u_, v_, c_] := Module[{base},

getGptEntry[u_, v_, c_] := Module[{base},

base = Transpose[columnHermiteDefactor[Join[{u}, Transpose[c]]]];

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];

~~transversalGenerators~~[m_] := Module[{c},

getGptSmithMethod[m_] := Module[{c},

c = nullSpaceBasis[m];

Transpose[MapThread[~~transversalGenerator~~[#1, #1, c]&, {Map[multByLcd,Transpose[PseudoInverse[m]]],m}]]

Transpose[MapThread[getGptEntry[#1, #1, c]&, {Map[multByLcd,Transpose[PseudoInverse[m]]],m}]]

];

~~transversalGenerators~~[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)

getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)

</syntaxhighlight>}}

[[Category:generator]]

~~[[Category:theory]]~~

[[Category:Regular temperament theory]]

[[Category:todo:reduce_mathslang]]

@@ Line 1: / Line 1: @@
-For each [[generator]] of a [[regular temperament]], we can choose one JI interval that maps to it. Then we can call a set of JI generators like this '''transversal generators''' of that temperament.
+Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.
-=Definition=
+Generator preimages are commonly used to describe temperaments. For example, [[meantone]] is generated by an octave and fifth, and the corresponding generator preimage is [[2/1]] and [[3/2]]. [[Miracle]] is generated by an octave and a semitone, corresponding to [[2/1]] and [[16/15]]~[[15/14]].
+=Technical Definition=
 Given a reduced list of [[Harmonic_Limit|p-limit]] vals V, we may define a set of transversal generators for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By ''reduced'' is meant that the GCD of the elements of each of the vals is 1--or in other words, none of the vals are [[contorted]]--and that they are [[linearly independent]], so that if there are r vals, the rank of V as a matrix is r.
@@ Line 8: / Line 10: @@
 q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q)
-In this way the transversal generators provide a [[Transversal|transversal]] of the p-limit, and hence the name.
+In this way the transversal generators provide a transversal of the p-limit, and hence the name.
 =Examples=
@@ Line 21: / Line 23: @@
 For another example, consider [&lt;1 1 1 2|, &lt;0 2 1 1|, &lt;0 0 2 1|] which is the [[Normal_lists|normal val list]] for breed temperament, the temperament tempering out 2401/2400. A corresponding list of transversal generators is [2, 49/40, 10/7].
-=Finding the transversal generators=
+=Finding the generator preimage transversal=
+{{todo|cleanup|inline=1|text=Add simpler algorithm}}
-Two methods for finding transversal generators have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel|Sintel]] in December 2021, which uses the [[Smith normal form]].
+Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel|Sintel]] in December 2021, which uses the [[Smith normal form]].
 == Method using the Smith Normal Form ==
-So we want to find a generators matrix <math>G</math> for a mapping <math>M</math> where:
+So we want to find a generator preimage transversal <math>T</math> for a mapping <math>M</math> where:
-<math>MG = I,</math>
+<math>MT = I,</math>
-and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>G</math> as a prime-count vector represents an interval that <math>M</math> maps to that generator. And when this <math>G</math> has all integer entries, then these generators are all JI.
+and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a [[prime-count vector]] represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.
 Essentially we need a way to do:
@@ Line 39: / Line 42: @@
 <math>
-M^{-1}MG = M^{-1}I \\
+M^{-1}MT = M^{-1}I \\
-\cancel{M^{-1}M}G = M^{-1}I \\
+\cancel{M^{-1}M}T = M^{-1}I \\
-G = M^{-1}
+T = M^{-1}
 </math>
@@ Line 56: / Line 59: @@
-where <math>D</math> is the Smith Normal Form of <math>M</math>, which is a diagonal matrix (hence the name "D"). Furthermore, if <math>M</math> was defactored, then not only is <math>D</math> diagonal (only has nonzero values along its main diagonal) but the numbers along its main diagonal are all equal to 1. This type of matrix is called an orthonormal matrix, and has the property that <math>D^{T}D = I</math>, which we're going to take advantage of.
+where <math>D</math> is the Smith Normal Form of <math>M</math>, which is a rectangular diagonal matrix (the name 'D' is for "diagonal"). Furthermore, if <math>M</math> was defactored, then not only is <math>D</math> diagonal (only has nonzero values along its main diagonal) but the numbers along its main diagonal are all equal to 1. This type of matrix is called an orthonormal matrix, and has the property that <math>D^{T}D = I</math>, which we're going to take advantage of.
 So first let's solve the decomposition equation for <math>M</math>. First, left-multiply by <math>L^{-1}</math>:
@@ Line 82: / Line 85: @@
 <math>
-(L^{-1}DR^{-1})G = I
+(L^{-1}DR^{-1})T = I
 </math>
-And we can proceed to solve this for our target <math>G</math>. The rest is busywork. First, left-multiply by <math>L</math>:
+And we can proceed to solve this for our target <math>T</math>. The rest is busywork. First, left-multiply by <math>L</math>:
 <math>
-LL^{-1}DR^{-1}G = LI \\
+LL^{-1}DR^{-1}T = LI \\
-\cancel{LL^{-1}}DR^{-1}G = LI \\
+\cancel{LL^{-1}}DR^{-1}T = LI \\
-DR^{-1}G = L
+DR^{-1}T = L
 </math>
@@ Line 100: / Line 103: @@
 <math>
-D^{T}DR^{-1}G = D^{T}L \\
+D^{T}DR^{-1}T = D^{T}L \\
-\cancel{D^{T}D}R^{-1}G = D^{T}L \\
+\cancel{D^{T}D}R^{-1}T = D^{T}L \\
-R^{-1}G = D^{T}L
+R^{-1}T = D^{T}L
 </math>
@@ Line 110: / Line 113: @@
 <math>
-RR^{-1}G = RD^{T}L \\
+RR^{-1}T = RD^{T}L \\
-\cancel{RR^{-1}}G = RD^{T}L \\
+\cancel{RR^{-1}}T = RD^{T}L \\
-G = RD^{T}L
+T = RD^{T}L
 </math>
-And there's our answer! It's still not necessarily giving the simplest or best JI generators, but arriving at those is an independent problem (perhaps by choosing a complexity metric and minimizing it though linear combinations with the commas and the other generators).
+And there's our answer! It's still not necessarily giving the simplest or best JI generators, but arriving at those is an independent problem (perhaps by choosing a complexity metric and minimizing it though linear combinations with the commas and the other generators, or perhaps using LLL).
 === Example ===
@@ Line 161: / Line 164: @@
-And so some JI preimages of its generators are <math>RD^{T}L</math>:
+And so a transversal of the preimages of its generators are <math>RD^{T}L</math>:
@@ Line 192: / Line 195: @@
 === Wolfram Language implementation ===
-{{Databox|jiG[]|
+{{Databox|getGpt[]|
 <syntaxhighlight lang="mathematica">
-jiG[m_] := Module[{decomp, left, snf, right},
+getGpt[m_] := Module[{decomp, left, snf, right},
    decomp = SmithDecomposition[m];
    left = Part[decomp, 1];
@@ Line 206: / Line 209: @@
 == Method using the Hermite Normal Form ==
-We can find transversal generators for V by the following procedure:
+We can find a generator preimage transversal for V by the following procedure:
 <ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma basis for V</li><li>[[Saturation|Saturate]] the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo and convert it to a rational number</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul>
@@ Line 396: / Line 399: @@
 === Wolfram Language implementation ===
-{{Databox|transversalGenerators[]|
+{{Databox|getGpt[]|
 <syntaxhighlight lang="mathematica">
-transversalGenerator[u_, v_, c_] := Module[{base},
+getGptEntry[u_, v_, c_] := Module[{base},
    base = Transpose[columnHermiteDefactor[Join[{u}, Transpose[c]]]];
@@ Line 404: / Line 407: @@
 ];
-transversalGenerators[m_] := Module[{c},
+getGptSmithMethod[m_] := Module[{c},
    c = nullSpaceBasis[m];
-   Transpose[MapThread[transversalGenerator[#1, #1, c]&, {Map[multByLcd,Transpose[PseudoInverse[m]]],m}]]
+   Transpose[MapThread[getGptEntry[#1, #1, c]&, {Map[multByLcd,Transpose[PseudoInverse[m]]],m}]]
 ];
-transversalGenerators[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)
+getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)
 </syntaxhighlight>}}
 [[Category:generator]]
-[[Category:theory]]
 [[Category:Regular temperament theory]]
 [[Category:todo:reduce_mathslang]]