Douglas Blumeyer's RTT How-To: Difference between revisions

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To be clear, canonical form isn’t necessary to avoid ambiguity: you will never find a comma basis that could represent more than one temperament.

I've seen many specialized matrix forms used in RTT for problems like this, such as [https://en.wikipedia.org/wiki/Smith_normal_form Smith normal form~~], [[IRREF]~~], [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form], [http://home.iitk.ac.in/~rksr/html/03CANONICALFACTORIZATIONS.htm Hermite Canonical Form], and maybe others. What we will be using here, is ~~a potentially new form recently developed by Dave Keenan tentatively called~~ "integer canonical form".<ref>Historically, Hermite Normal Form was used as if it gave a unique identifier for each temperament, when in fact it did not. It failed to remove possibly-hidden common factors in the columns of comma bases and rows of map bases, leading to situations which were named "[[torsion]]" and "[[contorsion]]", respectively. Hermite Canonical Form ~~is closer, but not quite there, because~~ it reduces the matrix's diagonals to all 1's or 0's, which loses important information for temperaments with non-octave periods. ~~Our integer~~ canonical form removes such common factors without compromising non-octave periods, therefore truly giving a unique identifier, and also eliminating the need to deal with torsion or contorsion~~. Preliminary tests suggest that our method gives equivalent results to IRREF, but this has not yet been proven~~.</ref>.

I've seen many specialized matrix forms used in RTT for problems like this, such as [https://en.wikipedia.org/wiki/Smith_normal_form Smith normal form], [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form], [http://home.iitk.ac.in/~rksr/html/03CANONICALFACTORIZATIONS.htm Hermite Canonical Form], and maybe others. What we will be using here is "[[IRREF|integer canonical form]]".<ref>Historically, Hermite Normal Form was used as if it gave a unique identifier for each temperament, when in fact it did not. It failed to remove possibly-hidden common factors in the columns of comma bases and rows of map bases, leading to situations which were named "[[torsion]]" and "[[contorsion]]", respectively. Hermite Canonical Form goes too far it reduces the matrix's diagonals to all 1's or 0's, which loses important information for temperaments with non-octave periods. Integer canonical form removes such common factors without compromising non-octave periods, therefore truly giving a unique identifier, and also eliminating the need to deal with torsion or contorsion.</ref>.

For example, the canonical form of meantone is:

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!output

|-

|<code>~~nullSpaceBasis~~[m_] := Transpose[~~NullSpace~~[m]]</code>

|<code>antiTranspose[m_]:=Transpose[Reverse[m,{1,2}]]</code>

<code>~~vFlip~~[m_] := ~~Reverse~~[m]</code>

<code>multByLcd[row_]:=Apply[LCM,Denominator[row]]*row</code>

<code>~~hFlip~~[m_] := ~~Reverse~~[m,2]</code>

<code>canonicalForm[m_]:=Map[multByLcd,RowReduce[m]]</code>

<code>~~antiNullSpaceBasis~~[m_] := ~~hFlip~~[~~Transpose~~[~~nullSpaceBasis[Transpose[vFlip~~[m]]]]]</code>

<code>columnCanonicalForm[m_]:=antiTranspose[canonicalForm[antiTranspose[m]]]</code>

<code>~~mapBasisHermiteCanonicalForm~~[~~m_] := antiNullSpaceBasis[nullSpaceBasis[m]~~]</code>

<code>canonicalForm[{{5,8,12},{7,11,16}}]</code>

<code>~~commaBasisHermiteCanonicalForm[m_] := nullSpaceBasis[antiNullSpaceBasis[m]]</code>~~

<code>columnCanonicalForm[{{-4,-10},{4,-1},{-1,5}}]</code>

~~<code>mapBasisHermiteCanonicalForm[{{5,8,12},{7,11,16}}]</code>~~

~~<code>commaBasisHermiteCanonicalForm~~[{{-4,-10},{4,-1},{-1,5}}]</code>

|{{1,0,-4},{0,1,4}}

{{-44,-30},{0,19},{19,0}}

@@ Line 1,087: / Line 1,087: @@
 To be clear, canonical form isn’t necessary to avoid ambiguity: you will never find a comma basis that could represent more than one temperament.
-I've seen many specialized matrix forms used in RTT for problems like this, such as [https://en.wikipedia.org/wiki/Smith_normal_form Smith normal form], [[IRREF]], [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form], [http://home.iitk.ac.in/~rksr/html/03CANONICALFACTORIZATIONS.htm Hermite Canonical Form], and maybe others. What we will be using here, is a potentially new form recently developed by Dave Keenan tentatively called "integer canonical form".<ref>Historically, Hermite Normal Form was used as if it gave a unique identifier for each temperament, when in fact it did not. It failed to remove possibly-hidden common factors in the columns of comma bases and rows of map bases, leading to situations which were named "[[torsion]]" and "[[contorsion]]", respectively. Hermite Canonical Form is closer, but not quite there, because it reduces the matrix's diagonals to all 1's or 0's, which loses important information for temperaments with non-octave periods. Our integer canonical form removes such common factors without compromising non-octave periods, therefore truly giving a unique identifier, and also eliminating the need to deal with torsion or contorsion. Preliminary tests suggest that our method gives equivalent results to IRREF, but this has not yet been proven.</ref>.
+I've seen many specialized matrix forms used in RTT for problems like this, such as [https://en.wikipedia.org/wiki/Smith_normal_form Smith normal form], [https://en.wikipedia.org/wiki/Hermite_normal_form Hermite Normal Form], [http://home.iitk.ac.in/~rksr/html/03CANONICALFACTORIZATIONS.htm Hermite Canonical Form], and maybe others. What we will be using here is "[[IRREF|integer canonical form]]".<ref>Historically, Hermite Normal Form was used as if it gave a unique identifier for each temperament, when in fact it did not. It failed to remove possibly-hidden common factors in the columns of comma bases and rows of map bases, leading to situations which were named "[[torsion]]" and "[[contorsion]]", respectively. Hermite Canonical Form goes too far it reduces the matrix's diagonals to all 1's or 0's, which loses important information for temperaments with non-octave periods. Integer canonical form removes such common factors without compromising non-octave periods, therefore truly giving a unique identifier, and also eliminating the need to deal with torsion or contorsion.</ref>.
 For example, the canonical form of meantone is:
@@ Line 1,109: / Line 1,109: @@
 !output
 |-
-|<code>nullSpaceBasis[m_] := Transpose[NullSpace[m]]</code>
+|<code>antiTranspose[m_]:=Transpose[Reverse[m,{1,2}]]</code>
-<code>vFlip[m_] := Reverse[m]</code>
+<code>multByLcd[row_]:=Apply[LCM,Denominator[row]]*row</code>
-<code>hFlip[m_] := Reverse[m,2]</code>
+<code>canonicalForm[m_]:=Map[multByLcd,RowReduce[m]]</code>
-<code>antiNullSpaceBasis[m_] := hFlip[Transpose[nullSpaceBasis[Transpose[vFlip[m]]]]]</code>
+<code>columnCanonicalForm[m_]:=antiTranspose[canonicalForm[antiTranspose[m]]]</code>
-<code>mapBasisHermiteCanonicalForm[m_] := antiNullSpaceBasis[nullSpaceBasis[m]]</code>
+<code>canonicalForm[{{5,8,12},{7,11,16}}]</code>
-<code>commaBasisHermiteCanonicalForm[m_] := nullSpaceBasis[antiNullSpaceBasis[m]]</code>
+<code>columnCanonicalForm[{{-4,-10},{4,-1},{-1,5}}]</code>
-<code>mapBasisHermiteCanonicalForm[{{5,8,12},{7,11,16}}]</code>
-<code>commaBasisHermiteCanonicalForm[{{-4,-10},{4,-1},{-1,5}}]</code>
 |{{1,0,-4},{0,1,4}}
 {{-44,-30},{0,19},{19,0}}