Normal forms: Difference between revisions

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== Normal interval list ==

=== ~~Hermite normal~~ form ===

=== Canonical form ===

Given a list of ''p''-limit intervals, we can convert it to a normal list by the following procedure:

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# Reverse the order of the elements of the monzos

# Form a matrix whose rows are these reversed monzos

# Hermite reduce this matrix to Hermite normal form

# Defactor and Hermite reduce this matrix to Hermite normal form

# Reverse (once again) the entries in each row of the Hermite normal form matrix, and use these re-reversed rows to form a new list of monzos

# Discard all of the all-zero monzos, corresponding to the unison 1/1

# Reverse the whole list (not the entries) of the remaining monzos

# Convert this list back to a list of ''p''-limit rational numbers

~~# For any number ''q'' < 1 on this list, replace ''q'' with 1/''q''~~

The result is ~~a normal~~ interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of ~~generators, each greater than zero~~, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/~~80,~~ 59049~~/57344~~]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[abstract regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma sequence of septimal meantone.

The result is the canonical form of the interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [80/81, 57344/59049]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[abstract regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma sequence of septimal meantone.

There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences due to [[enfactoring]], ~~leading to~~ the ~~following section~~ of canonical form.

There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences due to [[enfactoring]]. The canonical form of the comma list, however, requires that the normal comma sequence be defactored. There is only one such form of any temperament. For example, both [25/27, 35/36] and [25/27, 49/48] are normal, and they both characterize Beep. But the latter is enfactored, so the former is Beep's canonical form.

Normal interval lists can also be used to characterize the [[just intonation subgroups]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[chromatic pairs]], [[subgroup temperaments]] and [[just intonation subgroup]]s can be found many examples; the subgroup lists are given in a form where generators of the subgroup are separated by periods so as to flag the fact that the list defines a subgroup. An example would be the Barbados subgroup, 2.3.13/5.

=== ~~Canonical~~ form ===

=== Positive ratio form ===

~~{{See also| Canonical~~ form }}

The '''positive ratio form''' is based on the canonical form but each ratio is greater than unity. To obtain this, simply replace any ratio smaller than 1 by its reciprocal. For example, the positive ratio form of septimal meantone is [81/80, 59049/57344]

~~The~~ canonical form ~~of the comma list, however~~, ~~requires that the normal comma sequence be defactored. There is only one such form of~~ any ~~temperament~~. For example, ~~both~~ [27/25, 36/35] ~~and [27/25, 49/48] are normal, and they both characterize Beep. But the latter is enfactored, so the former is Beep's canonical form.~~

=== Minimal ratio form ===

~~In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it~~ shows ratio-wise the simplest comma sequence sufficient to define the temperament, ~~which may be called~~ the ~~Tenney-minimal~~ form.

The '''minimal ratio form''' shows ratio-wise the simplest comma sequence sufficient to define the temperament, where the ratios are rated by the product of the numerator and the denominator, i.e. the [[Benedetti height]].

This is the form shown in the "comma lists" of each temperament on this wiki.

== Maple code ==

@@ Line 53: / Line 53: @@
 == Normal interval list ==
-=== Hermite normal form ===
+=== Canonical form ===
 Given a list of ''p''-limit intervals, we can convert it to a normal list by the following procedure:
@@ Line 59: / Line 59: @@
 # Reverse the order of the elements of the monzos
 # Form a matrix whose rows are these reversed monzos
-# Hermite reduce this matrix to Hermite normal form
+# Defactor and Hermite reduce this matrix to Hermite normal form
 # Reverse (once again) the entries in each row of the Hermite normal form matrix, and use these re-reversed rows to form a new list of monzos
 # Discard all of the all-zero monzos, corresponding to the unison 1/1
 # Reverse the whole list (not the entries) of the remaining monzos
 # Convert this list back to a list of ''p''-limit rational numbers
-# For any number ''q'' &lt; 1 on this list, replace ''q'' with 1/''q''
-The result is a normal interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of generators, each greater than zero, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [81/80, 59049/57344]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[abstract regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma sequence of septimal meantone.
+The result is the canonical form of the interval list. The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any ''p''-limit group it lives inside. The normalized list contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we normalize [81/80, 126/125] we obtain [80/81, 57344/59049]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit. Because [[abstract regular temperament]]s, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma sequence of septimal meantone.
-There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences due to [[enfactoring]], leading to the following section of canonical form.
+There is only one normal comma sequence that characterizes septimal meantone. But sometimes a temperament can be characterized by multiple normal comma sequences due to [[enfactoring]]. The canonical form of the comma list, however, requires that the normal comma sequence be defactored. There is only one such form of any temperament. For example, both [25/27, 35/36] and [25/27, 49/48] are normal, and they both characterize Beep. But the latter is enfactored, so the former is Beep's canonical form.
 Normal interval lists can also be used to characterize the [[just intonation subgroups]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[chromatic pairs]], [[subgroup temperaments]] and [[just intonation subgroup]]s can be found many examples; the subgroup lists are given in a form where generators of the subgroup are separated by periods so as to flag the fact that the list defines a subgroup. An example would be the Barbados subgroup, 2.3.13/5.
-=== Canonical form ===
+=== Positive ratio form ===
-{{See also| Canonical form }}
+The '''positive ratio form''' is based on the canonical form but each ratio is greater than unity. To obtain this, simply replace any ratio smaller than 1 by its reciprocal. For example, the positive ratio form of septimal meantone is [81/80, 59049/57344]
-The canonical form of the comma list, however, requires that the normal comma sequence be defactored. There is only one such form of any temperament. For example, both [27/25, 36/35] and [27/25, 49/48] are normal, and they both characterize Beep. But the latter is enfactored, so the former is Beep's canonical form.
 === Minimal ratio form ===
-In each temperament page on this wiki, however, the "comma lists" are not normal interval lists as defined above. Instead, it shows ratio-wise the simplest comma sequence sufficient to define the temperament, which may be called the Tenney-minimal form.
+The '''minimal ratio form''' shows ratio-wise the simplest comma sequence sufficient to define the temperament, where the ratios are rated by the product of the numerator and the denominator, i.e. the [[Benedetti height]].
+This is the form shown in the "comma lists" of each temperament on this wiki.
 == Maple code ==