Val: Difference between revisions

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== Vals vs. mappings ==

A val is more specific than a [[mapping]]. To be precise:

A val is more specific than a general linear map, or even an arbitrary [[mapping|temperament mapping]]. To be precise:

# A val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector)~~. Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row.~~

# A val can be thought of as a temperament mapping matrix with one row. Put another way, the rows of integer mapping matrices are vals, so that mapping matrices can be thought of as being built up from vals.

~~# To be precise, the rows of integer mapping matrices are vals, so that mapping matrices can be thought of as being built up from vals.~~

# Vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis).

~~# It has only integer entries~~.

# To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).

# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear ~~sums~~ of [[Wikipedia:P-adic_order|p-adic valuations]].

# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sum of [[Wikipedia:P-adic_order|p-adic valuations]].

@@ Line 64: / Line 64: @@
 == Vals vs. mappings ==
-A val is more specific than a [[mapping]]. To be precise:
+A val is more specific than a general linear map, or even an arbitrary [[mapping|temperament mapping]]. To be precise:
-# A val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector). Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row.
+# A val can be thought of as a temperament mapping matrix with one row. Put another way, the rows of integer mapping matrices are vals, so that mapping matrices can be thought of as being built up from vals.
-# To be precise, the rows of integer mapping matrices are vals, so that mapping matrices can be thought of as being built up from vals.
+# Vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis).
-# It has only integer entries.
+# To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).
-# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sums of [[Wikipedia:P-adic_order|p-adic valuations]].
+# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sum of [[Wikipedia:P-adic_order|p-adic valuations]].