Temperament addition: Difference between revisions

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=A note on variance=

For simplicity, this article will use the word "vector" in its general sense, that is, [[variance]]-agnostic. This means it includes either contravariant vectors (plain "vectors", such as [[prime count vector]]s) or covariant vectors ("''co''vectors", such as [[map]]s). However, the reader should assume that only one of the two types is being used at a given time, since the two variances do not mix. For more information, see [[Linear_dependence#Variance]]. The same variance-agnosticism holds for [[multivector|''multi''vector]]s in this article as well.

For simplicity, this article will use the word "vector" in its general sense, that is, [[variance]]-agnostic. This means it includes either contravariant vectors (plain "vectors", such as [[prime-count vector]]s) or covariant vectors ("''co''vectors", such as [[map]]s). However, the reader should assume that only one of the two types is being used at a given time, since the two variances do not mix. For more information, see [[Linear_dependence#Variance]]. The same variance-agnosticism holds for [[multivector|''multi''vector]]s in this article as well.

=Visualizing temperament addition=

[[File:Sum diff and wedge.png|thumb|left|300px|A and B are vectors representing temperaments. They could be maps or prime count vectors. A∧B is their wedge product and gives a higher-[[grade]] temperament that [[temperament merging|merge]]s both A and B. A+B and A-B give the sum and difference, respectively.]]

[[File:Sum diff and wedge.png|thumb|left|300px|A and B are vectors representing temperaments. They could be maps or prime-count vectors. A∧B is their wedge product and gives a higher-[[grade]] temperament that [[temperament merging|merge]]s both A and B. A+B and A-B give the sum and difference, respectively.]]

==Versus the wedge product==

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The check is that the vectors must be in [[canonical form]]. For a contravariant vector, such as the kind that represent commas, canonical form means that the trailing entry (the final non-zero entry) must be positive. For a covariant vector, such as the kind that represent mapping-rows, canonical form means that the leading entry (the first non-zero entry) must be positive.

Sometimes the canonical form of a vector is not the most popular form. For instance, the meantone comma is usually expressed in positive form, that is, with its numerator greater than its denominator, so that its cents value is positive, or in other words, it's the meantone comma upwards in pitch, not downwards. But the prime count vector for that form, 81/80, is {{vector|-4 4 -1}}, and as we can see, its trailing entry -1 is negative. So the canonical form of meantone is actually {{vector|4 -4 1}}.

Sometimes the canonical form of a vector is not the most popular form. For instance, the meantone comma is usually expressed in positive form, that is, with its numerator greater than its denominator, so that its cents value is positive, or in other words, it's the meantone comma upwards in pitch, not downwards. But the prime-count vector for that form, 81/80, is {{vector|-4 4 -1}}, and as we can see, its trailing entry -1 is negative. So the canonical form of meantone is actually {{vector|4 -4 1}}.

=<math>g_{\text{min}}>1</math>=

@@ Line 143: / Line 143: @@
 =A note on variance=
-For simplicity, this article will use the word "vector" in its general sense, that is, [[variance]]-agnostic. This means it includes either contravariant vectors (plain "vectors", such as [[prime count vector]]s) or covariant vectors ("''co''vectors", such as [[map]]s). However, the reader should assume that only one of the two types is being used at a given time, since the two variances do not mix. For more information, see [[Linear_dependence#Variance]]. The same variance-agnosticism holds for [[multivector|''multi''vector]]s in this article as well.
+For simplicity, this article will use the word "vector" in its general sense, that is, [[variance]]-agnostic. This means it includes either contravariant vectors (plain "vectors", such as [[prime-count vector]]s) or covariant vectors ("''co''vectors", such as [[map]]s). However, the reader should assume that only one of the two types is being used at a given time, since the two variances do not mix. For more information, see [[Linear_dependence#Variance]]. The same variance-agnosticism holds for [[multivector|''multi''vector]]s in this article as well.
 =Visualizing temperament addition=
-[[File:Sum diff and wedge.png|thumb|left|300px|A and B are vectors representing temperaments. They could be maps or prime count vectors. A∧B is their wedge product and gives a higher-[[grade]] temperament that [[temperament merging|merge]]s both A and B. A+B and A-B give the sum and difference, respectively.]]
+[[File:Sum diff and wedge.png|thumb|left|300px|A and B are vectors representing temperaments. They could be maps or prime-count vectors. A∧B is their wedge product and gives a higher-[[grade]] temperament that [[temperament merging|merge]]s both A and B. A+B and A-B give the sum and difference, respectively.]]
 ==Versus the wedge product==
@@ Line 231: / Line 231: @@
 The check is that the vectors must be in [[canonical form]]. For a contravariant vector, such as the kind that represent commas, canonical form means that the trailing entry (the final non-zero entry) must be positive. For a covariant vector, such as the kind that represent mapping-rows, canonical form means that the leading entry (the first non-zero entry) must be positive.
-Sometimes the canonical form of a vector is not the most popular form. For instance, the meantone comma is usually expressed in positive form, that is, with its numerator greater than its denominator, so that its cents value is positive, or in other words, it's the meantone comma upwards in pitch, not downwards. But the prime count vector for that form, 81/80, is {{vector|-4 4 -1}}, and as we can see, its trailing entry -1 is negative. So the canonical form of meantone is actually {{vector|4 -4 1}}.
+Sometimes the canonical form of a vector is not the most popular form. For instance, the meantone comma is usually expressed in positive form, that is, with its numerator greater than its denominator, so that its cents value is positive, or in other words, it's the meantone comma upwards in pitch, not downwards. But the prime-count vector for that form, 81/80, is {{vector|-4 4 -1}}, and as we can see, its trailing entry -1 is negative. So the canonical form of meantone is actually {{vector|4 -4 1}}.
 =<math>g_{\text{min}}>1</math>=