Temperament merging: Difference between revisions

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== Canonicalization ==

The canonicalization step is important for eliminating any redundancies that may have been introduced by merging related temperaments, such as [[rank-deficient|~~grade~~-deficiencies]] or [[enfactoring]].

The canonicalization step is important for eliminating any redundancies that may have been introduced by merging related temperaments, such as [[rank-deficient|rank-deficiencies]] or [[enfactoring]].

=== ~~Grade~~-deficiencies ===

=== Rank-deficiencies ===

Sometimes when temperaments are merged, ~~grade~~-deficiencies may occur. For example, comma-merging septimal meantone and miracle temperaments:

Sometimes when temperaments are merged, rank-deficiencies may occur. For example, comma-merging septimal meantone and miracle temperaments:

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We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was ~~nullity~~-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, 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 vectors. So once we fully [[canonical form|canonicalize]], any all-zero column(s) are removed, and we end up with:

We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was column-rank-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, 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 vectors. So once we fully [[canonical form|canonicalize]], any all-zero column(s) are removed, and we end up with:

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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 puts the result into normal form 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 Hermite normal form and removes any all-zero rows or columns resulting from rank-deficiencies.

== Parallel intersections ==

@@ Line 104: / Line 104: @@
 == Canonicalization ==
-The canonicalization step is important for eliminating any redundancies that may have been introduced by merging related temperaments, such as [[rank-deficient|grade-deficiencies]] or [[enfactoring]].
+The canonicalization step is important for eliminating any redundancies that may have been introduced by merging related temperaments, such as [[rank-deficient|rank-deficiencies]] or [[enfactoring]].
-=== Grade-deficiencies ===
+=== Rank-deficiencies ===
-Sometimes when temperaments are merged, grade-deficiencies may occur. For example, comma-merging septimal meantone and miracle temperaments:
+Sometimes when temperaments are merged, rank-deficiencies may occur. For example, comma-merging septimal meantone and miracle temperaments:
@@ Line 152: / Line 152: @@
-We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was nullity-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, 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 vectors. So once we fully [[canonical form|canonicalize]], any all-zero column(s) are removed, and we end up with:
+We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was column-rank-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, 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 vectors. So once we fully [[canonical form|canonicalize]], any all-zero column(s) are removed, and we end up with:
@@ Line 227: / Line 227: @@
 === 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 puts the result into normal form 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 Hermite normal form and removes any all-zero rows or columns resulting from rank-deficiencies.
 == Parallel intersections ==