Val: Difference between revisions
Clarify generator chain |
Cmloegcmluin (talk | contribs) clarify difference from map |
||
| Line 61: | Line 61: | ||
== Vals in regular temperaments == | == Vals in regular temperaments == | ||
There is also a notion of a "tempered val" on a group of "tempered monzos", representing intervals in some [[regular temperament]]. This name is sometimes abbreviated as [[Tmonzos and Tvals|"tmonzos" and "tvals"]]. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]] temperament, we can use names like "P8" and "P5," so that the tval "P8.P5 {{val| 12 7 }}" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a [[transversal]] can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 {{val| 12 7 }}, or "(meantone) 2.3/2 {{val| 31 18 }}". | There is also a notion of a "tempered val" on a group of "tempered monzos", representing intervals in some [[regular temperament]]. This name is sometimes abbreviated as [[Tmonzos and Tvals|"tmonzos" and "tvals"]]. Typically, this is made explicit by writing the generators beforehand. When the tempered intervals have accepted names, such as in [[meantone]] temperament, we can use names like "P8" and "P5," so that the tval "P8.P5 {{val| 12 7 }}" represents the 12-edo "patent tval" in meantone temperament (given that particular basis). If the intervals don't have names, a [[transversal]] can be given instead, preceded with the temperament name, so that we have "(meantone) 2.3/2 {{val| 12 7 }}, or "(meantone) 2.3/2 {{val| 31 18 }}". | ||
== Vals vs. maps == | |||
A val is more specific than a [[map]]: | |||
# It is a specific type of [[Wikipedia:Linear_map|(linear) map]], 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 1D 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. | |||
# It has only integer entries. | |||
# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sums of [[Wikipedia:P-adic_order|p-adic valuations]]. | |||
# It is [[Wikipedia:Homomorphism|homomorphic]]. A val is a map from every positive rational number to an associated integer, usually restricted to a given prime limit. | |||
In practice, most single-row maps in RTT are vals, because we usually deal with maps with integer entries, and the other specifications only mean anything to advanced mathematicians. | |||
== See also == | == See also == | ||
* [[Mapping]] | * [[Mapping]] | ||
* [[Map]] | |||
* [[Monzo]] | * [[Monzo]] | ||
* [[Monzos and interval space]] | * [[Monzos and interval space]] | ||