Normal forms: Difference between revisions

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Since a [[regular temperament]] can be represented by multiple equivalent [[~~Vals~~ and tuning space|val]] lists (~~aka~~ [[mapping]]s) or [[~~Monzos~~ and interval space|monzo]] lists (~~aka~~ [[comma basis|comma bases]]), it can be helpful~~—~~e.g. when comparing or cataloguing temperaments~~—~~to choose a single one of these equivalent lists to use as its unique identifier. A set of rules that are consistently able to narrow the full set of equivalent lists down to a single list for each temperament may be called a ''normal form'', and accordingly, a list which uniquely identifies a temperament in this way may be called a '''normal list'''.

Since a [[regular temperament]] can be represented by multiple equivalent [[vals and tuning space|val]] lists (a.k.a. [[mapping]]s) or [[monzos and interval space|monzo]] lists (a.k.a. [[comma basis|comma bases]]), it can be helpful – e.g. when comparing or cataloguing temperaments – to choose a single one of these equivalent lists to use as its unique identifier. A set of rules that are consistently able to narrow the full set of equivalent lists down to a single list for each temperament may be called a '''normal form''', and accordingly, a list which uniquely identifies a temperament in this way may be called a '''normal list'''.

Because several different normal forms have been developed, each temperament has several different normal lists: one for each form. These normal lists are not all necessarily different; sometimes some or all of them may be the same.

== Hermite normal form ==

The '''Hermite normal form''', or HNF for short, is an important normal form that is defined for integer matrices in the mathematical field of linear algebra.

An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.

The '''Hermite normal form''', or HNF for short, is an important normal form that is defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.

The Hermite normal form is unique. Besides that, it is also the integer analogue of the reduced echelong form, so it can be used to solve systems of equations in the integers by Gaussian elimination.

There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved.

There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows. An ''n''×''m'' integer matrix ''H'' is in (row-wise) HNF if it satisfies the following conditions:

The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.

An ''n''~~×~~''m'' integer matrix H is in (row-wise) HNF if it satisfies the following conditions:

# H is upper triangular: the entries ''h<sub>ij</sub>'' = 0 for ''i'' > ''j'', and any rows of zeros are located below the other rows.

# ''H'' is upper triangular: the entries ''h<sub>ij</sub>'' = 0 for ''i'' > ''j'', and any rows of zeros are located below the other rows.

# The first non-zero entry from the left (called the ''leading coefficient'') of any non-zero row is always stricly to the right of the leading coefficient of the row above it.

# The leading coefficients are positive.

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-Since a [[regular temperament]] can be represented by multiple equivalent [[Vals and tuning space|val]] lists (aka [[mapping]]s) or [[Monzos and interval space|monzo]] lists (aka [[comma basis|comma bases]]), it can be helpful&mdash;e.g. when comparing or cataloguing temperaments&mdash;to choose a single one of these equivalent lists to use as its unique identifier. A set of rules that are consistently able to narrow the full set of equivalent lists down to a single list for each temperament may be called a ''normal form'', and accordingly, a list which uniquely identifies a temperament in this way may be called a '''normal list'''.
+Since a [[regular temperament]] can be represented by multiple equivalent [[vals and tuning space|val]] lists (a.k.a. [[mapping]]s) or [[monzos and interval space|monzo]] lists (a.k.a. [[comma basis|comma bases]]), it can be helpful – e.g. when comparing or cataloguing temperaments – to choose a single one of these equivalent lists to use as its unique identifier. A set of rules that are consistently able to narrow the full set of equivalent lists down to a single list for each temperament may be called a '''normal form''', and accordingly, a list which uniquely identifies a temperament in this way may be called a '''normal list'''.
 Because several different normal forms have been developed, each temperament has several different normal lists: one for each form. These normal lists are not all necessarily different; sometimes some or all of them may be the same.
 == Hermite normal form ==
-{{wikipedia|Hermite normal form}}
+{{Wikipedia| Hermite normal form }}
-The '''Hermite normal form''', or HNF for short, is an important normal form that is defined for integer matrices in the mathematical field of linear algebra.
-An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.
+The '''Hermite normal form''', or HNF for short, is an important normal form that is defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.
 The Hermite normal form is unique. Besides that, it is also the integer analogue of the reduced echelong form, so it can be used to solve systems of equations in the integers by Gaussian elimination.
-There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved.
+There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows. An ''n''×''m'' integer matrix ''H'' is in (row-wise) HNF if it satisfies the following conditions:
-The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.
-An ''n''&times;''m'' integer matrix H is in (row-wise) HNF if it satisfies the following conditions:
-# H is upper triangular: the entries ''h<sub>ij</sub>'' = 0 for ''i'' > ''j'', and any rows of zeros are located below the other rows.
+# ''H'' is upper triangular: the entries ''h<sub>ij</sub>'' = 0 for ''i'' > ''j'', and any rows of zeros are located below the other rows.
 # The first non-zero entry from the left (called the ''leading coefficient'') of any non-zero row is always stricly to the right of the leading coefficient of the row above it.
 # The leading coefficients are positive.