Temperament merging: Difference between revisions

Line 138:

\end{array} \right]

\text{which ~~normalizes to~~}

\text{which in normal form is}

\left[ \begin{array} {rrr}

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Note that we've only ~~''normalized'' here so far, that is,~~ put it into [[Hermite normal form]]; we've done this to illustrate that one of the vectors is now entirely zeros (highlighted in red). This means that the matrix was nullity-deficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments tempered out some of the same commas, and so when we merged them, even though the input temperaments required 2 vectors each to represent, their merged result doesn't require all 4 vectors; it can be completely represented using only 3.

Note that we've only put it into [[Hermite normal form]]; we've done this to illustrate that one of the vectors is now entirely zeros (highlighted in red). This means that the matrix was nullity-deficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments tempered out some of the same commas, and so when we merged them, even though the input temperaments required 2 vectors each to represent, their merged result doesn't require all 4 vectors; it can be completely represented using only 3.

Once we fully [[canonical form|canonicalize]], though, the all-zero row(s) are removed, and we end up with:

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The greatest factor of this matrix is 2, because we can produce the row {{map|24 38 56}} as a coprime linear combination of its rows (that's {{map|5 8 12}} + {{map|19 30 44}}), and the entries of this row have a GCD of 2, so in other words this matrix is 2-enfactored. If we merely ~~normalize~~ it ~~(again, using the~~ Hermite normal form), we receive:

The greatest factor of this matrix is 2, because we can produce the row {{map|24 38 56}} as a coprime linear combination of its rows (that's {{map|5 8 12}} + {{map|19 30 44}}), and the entries of this row have a GCD of 2, so in other words this matrix is 2-enfactored. If we merely put it into Hermite normal form, we receive:

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which is a 2-enfactored meantone mapping, and it reveals the greatest factor as the GCD of the second row. But if we fully canonicalize it (defactor, and ~~normalize~~), then we get:

which is a 2-enfactored meantone mapping, and it reveals the greatest factor as the GCD of the second row. But if we fully canonicalize it (defactor, and put into normal form), then we get:

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=== Non-canonicalizing definition ===

By some definitions of the & operator, the [[defactoring]] part of canonicalization is not included — for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like 5&19 to represent 2-enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely ~~normalizes~~ the result and removes any all-zero rows or columns resulting from grade-deficiencies.

By some definitions of the & operator, the [[defactoring]] part of canonicalization is not included — for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like 5&19 to represent 2-enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely puts the result into normal form and removes any all-zero rows or columns resulting from grade-deficiencies.

== Parallel intersections ==

@@ Line 138: / Line 138: @@
 \end{array} \right]
-\text{which normalizes to}
+\text{which in normal form is}
 \left[ \begin{array} {rrr}
@@ Line 152: / Line 152: @@
-Note that we've only ''normalized'' here so far, that is, put it into [[Hermite normal form]]; we've done this to illustrate that one of the vectors is now entirely zeros (highlighted in red). This means that the matrix was nullity-deficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments tempered out some of the same commas, and so when we merged them, even though the input temperaments required 2 vectors each to represent, their merged result doesn't require all 4 vectors; it can be completely represented using only 3.
+Note that we've only put it into [[Hermite normal form]]; we've done this to illustrate that one of the vectors is now entirely zeros (highlighted in red). This means that the matrix was nullity-deficient, or in layperson's terms, contained redundant commas. In other words, these two temperaments tempered out some of the same commas, and so when we merged them, even though the input temperaments required 2 vectors each to represent, their merged result doesn't require all 4 vectors; it can be completely represented using only 3.
 Once we fully [[canonical form|canonicalize]], though, the all-zero row(s) are removed, and we end up with:
@@ Line 200: / Line 200: @@
-The greatest factor of this matrix is 2, because we can produce the row {{map|24 38 56}} as a coprime linear combination of its rows (that's {{map|5 8 12}} + {{map|19 30 44}}), and the entries of this row have a GCD of 2, so in other words this matrix is 2-enfactored. If we merely normalize it (again, using the Hermite normal form), we receive:
+The greatest factor of this matrix is 2, because we can produce the row {{map|24 38 56}} as a coprime linear combination of its rows (that's {{map|5 8 12}} + {{map|19 30 44}}), and the entries of this row have a GCD of 2, so in other words this matrix is 2-enfactored. If we merely put it into Hermite normal form, we receive:
@@ Line 213: / Line 213: @@
-which is a 2-enfactored meantone mapping, and it reveals the greatest factor as the GCD of the second row. But if we fully canonicalize it (defactor, and normalize), then we get:
+which is a 2-enfactored meantone mapping, and it reveals the greatest factor as the GCD of the second row. But if we fully canonicalize it (defactor, and put into normal form), then we get:
@@ Line 230: / Line 230: @@
 === Non-canonicalizing definition ===
-By some definitions of the & operator, the [[defactoring]] part of canonicalization is not included — for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like 5&19 to represent 2-enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely normalizes the result and removes any all-zero rows or columns resulting from grade-deficiencies.
+By some definitions of the & operator, the [[defactoring]] part of canonicalization is not included — for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like 5&19 to represent 2-enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely puts the result into normal form and removes any all-zero rows or columns resulting from grade-deficiencies.
 == Parallel intersections ==