Val: Difference between revisions

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== Vals vs. mappings ==

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

A val is more specific than a mapping, both as in the general mathematical sense as well as [[mapping|the regular temperament sense]]:

# 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~~.

# A val can be thought of as a mapping with one row. Put another way, the rows of mappings are vals. 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"]], which 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).

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

# 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 sum of [[Wikipedia:P-adic_order|p-adic valuations]].

In practice, most single-row mappings in RTT are vals, because we usually deal with integer entries, and the other specifications only mean anything to advanced mathematicians.

@@ Line 64: / Line 64: @@
 == Vals vs. mappings ==
-A val is more specific than a general linear map, or even an arbitrary [[mapping|temperament mapping]]. To be precise:
+A val is more specific than a mapping, both as in the general mathematical sense as well as [[mapping|the regular temperament sense]]:
-# 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.
+# A val can be thought of as a mapping with one row. Put another way, the rows of mappings are vals. 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"]], which 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).
 # Vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis).
-# 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 sum of [[Wikipedia:P-adic_order|p-adic valuations]].
 In practice, most single-row mappings in RTT are vals, because we usually deal with integer entries, and the other specifications only mean anything to advanced mathematicians.