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Generator preimage: Difference between revisions

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→Method using the Smith Normal Form: clarify, because diagonal matrices are typically square
Cmloegcmluin (talk | contribs)
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leaving original art by Gene alone, I've revised the material I've added in the past (a brief intro, explanations of algorithms, examples, implementations) to use nomenclature I feel ls less confusing and ambiguous
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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.
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 generators preimage transversal for that temperament.


=Definition=
=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.
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)
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=
=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].
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 generators 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 generators 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 ==
== 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 generators 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:
Essentially we need a way to do:
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<math>
<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>
</math>


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<math>
<math>
(L^{-1}DR^{-1})G = I
(L^{-1}DR^{-1})T = I
</math>
</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>
<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>
</math>


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<math>
<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>
</math>


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




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=== Wolfram Language implementation ===
=== Wolfram Language implementation ===


{{Databox|jiG[]|
{{Databox|jiT[]|
<syntaxhighlight lang="mathematica">
<syntaxhighlight lang="mathematica">
jiG[m_] := Module[{decomp, left, snf, right},
jiT[m_] := Module[{decomp, left, snf, right},
   decomp = SmithDecomposition[m];
   decomp = SmithDecomposition[m];
   left = Part[decomp, 1];
   left = Part[decomp, 1];
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== Method using the Hermite Normal Form ==
== Method using the Hermite Normal Form ==


We can find transversal generators for V by the following procedure:
We can find a generators 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>       
<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 ===
=== Wolfram Language implementation ===


{{Databox|transversalGenerators[]|
{{Databox|jiT[]|
<syntaxhighlight lang="mathematica">
<syntaxhighlight lang="mathematica">
transversalGenerator[u_, v_, c_] := Module[{base},
jiTEntry[u_, v_, c_] := Module[{base},
   base = Transpose[columnHermiteDefactor[Join[{u}, Transpose[c]]]];
   base = Transpose[columnHermiteDefactor[Join[{u}, Transpose[c]]]];


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];
];


transversalGenerators[m_] := Module[{c},
jiT[m_] := Module[{c},
   c = nullSpaceBasis[m];
   c = nullSpaceBasis[m];
   Transpose[MapThread[transversalGenerator[#1, #1, c]&, {Map[multByLcd,Transpose[PseudoInverse[m]]],m}]]
   Transpose[MapThread[jiTEntry[#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 *)
jiT[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)
</syntaxhighlight>}}
</syntaxhighlight>}}


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