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~~The word~~ '''map''' ~~could refer~~ to:

A '''map''' or '''mapping'''<ref group="note">It is possible to distinguish the "mapping", which is the function, from the "map", which is its image. However, both terms are used interchangeably in most contexts.</ref> is an association of elements from a set to elements of another set. In mathematics, a map is a generalized function.

* In general mathematics, a "map" is any function from one set to ~~another. For more information, see [[Wikipedia:Map_(mathematics)|Wikipedia:Map (mathematics)]]~~.

Various kinds of maps relevant to microtonal music theory are introduced below.

* In [[regular temperament theory]], the term "map~~" is used in the more specific sense of a [[Wikipedia:Linear_map|~~''~~linear~~'' map]], which, informally, can be thought of as a function that can be represented by a matrix. Examples include [[tuning map]]s, [[projection matrix|projection map]]s (sometimes called projection matrices), and [[temperament mapping matrix|temperament map]]s (usually called "temperament mapping matrices", or "mapping ~~matrices" or even just "mappings" for short). In~~ the past, the terms "M-map" and "V-map" were also sometimes used to refer to temperament mappings and [[subgroup basis matrices]], although the terminology has since changed on this wiki. A rank-1 temperament mapping is also called a ~~[[val]]~~.

== Keyboard map ==

A '''keyboard map''' or '''keyboard mapping''' specifies the actions associated with each key of a keyboard.

* [[~~Douglas Blumeyer~~]] ~~and~~ [[~~Dave Keenan~~]] ~~recommend reserving~~ the ~~word "map" for a mapping with one row, so that all maps~~ are mappings ~~but not all~~ mappings ~~are maps; a simple tip to remember this usage is that the shorter word refers to the simpler object~~.

=== Musical instruments ===

A [[keyboard]] mapping for a musical [[instrument]] ([[List of microtonal software plugins|software]] or [[List of microtonal hardware synths|hardware]]) associates notes or pitches from a scale file to the instrument's keys. Notable examples are [[Scala]] keyboard mappings (.kbm files) and [[Lumatone]] keyboard mappings (.ltn files).

=== Computer keyboard bindings ===

A keyboard map for a computer keyboard associates characters to certain keys or combinations of keys. It can be used to type special characters more quickly. An example is provided in [[Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations#WinCompose]].

== Maps in regular temperament theory ==

[[File:Temperament and tuning matrices 2.png|thumb|alt=An overview of maps in regular temperament theory.|An overview of maps in regular temperament theory.]]

In [[regular temperament theory]] (RTT), maps are generally assumed to be [[Wikipedia:Linear map|''linear'' maps]]<ref group="note">Since regular temperaments are defined on ℤ-modules, not on vector spaces, the term "linear map" is not quite correct and it would be preferable to say "[[Wikipedia:Additive map|''additive'' map]]", but "linear map" is often used by analogy since linear algebra terminology is more common in mathematics overall.</ref>, which, informally, can be thought of as a function that can be represented by a {{w|Matrix (mathematics)|matrix}}.

The most common maps in RTT are discussed below.

=== Temperament map ===

A temperament map or temperament mapping<ref group="note">[[Douglas Blumeyer]] and [[Dave Keenan]] recommend reserving the word "map" for a mapping with one row, so that all maps are mappings but not all mappings are maps; a simple tip to remember this usage is that the shorter word refers to the simpler object. In all occurrences at present on this wiki, as well as in Graham Breed's temperament finder, the term "map" (and not "mapping") consistently refers to a single-row mapping, so following this suggestion would be seamless moving forward. A "tuning map", which maps from generators to cents, is a map in "tuning space"; by analogy, a val is a map in "temperament space", and so it would be perfectly consistent with existing terminology to refer to a val as a "temperament map" as opposed to a "temperament mapping", and then when it is clear from the context that it is a ''temperament'' map, the qualifier "temperament" can be dropped, as is done with "temperament mapping matrix" being abbreviated to "mapping matrix". So the suggestion is equivalent to unqualified occurrences of "map" being assumed to be temperament maps, or in other words, synonymous with vals (except for the integer entry requirement), not tuning maps. Dave and Douglas recommend using "map" rather than "val", for two reasons. First, "map" is a basic linear algebra term with wide familiarity (being specialized for this purpose) while "val" is unnecessary jargon that creates a barrier to understanding by newcomers. Second, the coinage of "val" from the obscure mathematical term "valuation" is tenuous and unlikely to provide helpful insight: "p-adic valuation" is an obscure term for "prime count", which would be an element of a prime-count vector ("monzo"), not a map ("val").</ref> specifies the structure of a regular temperament. It transforms [[interval]]s, usually from a [[just intonation subgroup]] and expressed as prime-count vectors ([[monzo]]s), into [[mapped interval]]s ([[tmonzo]]s). It is often represented by a matrix, thus called a temperament mapping matrix<ref group="note">The terms "M-map" and "V-map" were also sometimes used to refer to temperament mappings and subgroup basis matrices, although these terms are rarely used nowadays.</ref>, or more simply a mapping matrix or even a mapping when the context is sufficiently clear.

Each row of a temperament mapping matrix, or the only row of a rank-1 temperament mapping matrix, is often called a [[val]].

A [[subgroup basis matrix]] is the dual of a temperament mapping matrix.

=== Tuning map ===

A tuning map (or rarely a tuning mapping) specifies the tuning of a regular temperament. A (tempered-prime) tuning map transforms intervals into tempered [[interval size]]s. A generator tuning map transforms mapped intervals into interval sizes.

=== Projection map ===

A projection matrix (or rarely a projection map) uniquely identifies a specific tuning of a specific regular temperament.

=== Lattice map ===

A lattice map or mapping to lattice transforms intervals into their coordinates in a regularly tempered [[lattice]].

== See also ==

* [[Harmonic template]]

== Notes ==

~~[[Category:Temperament]]~~

[[Category:Regular temperament theory]]

~~[[Category:Theory]]~~

[[Category:Terms]]

[[Category:Math]]

[[Category:Val]]

[[Category:Mapping]]

~~[[Category:Tuning]]~~

@@ Line 1: / Line 1: @@
-The word '''map''' could refer to:
+{{Wikipedia|Map (mathematics)}}
+A '''map''' or '''mapping'''<ref group="note">It is possible to distinguish the "mapping", which is the function, from the "map", which is its image. However, both terms are used interchangeably in most contexts.</ref> is an association of elements from a set to elements of another set. In mathematics, a map is a generalized function.
-* In general mathematics, a "map" is any function from one set to another. For more information, see [[Wikipedia:Map_(mathematics)|Wikipedia:Map (mathematics)]].
+Various kinds of maps relevant to microtonal music theory are introduced below.
-* In [[regular temperament theory]], the term "map" is used in the more specific sense of a [[Wikipedia:Linear_map|''linear'' map]], which, informally, can be thought of as a function that can be represented by a matrix. Examples include [[tuning map]]s, [[projection matrix|projection map]]s (sometimes called projection matrices), and [[temperament mapping matrix|temperament map]]s (usually called "temperament mapping matrices", or "mapping matrices" or even just "mappings" for short). In the past, the terms "M-map"  and "V-map" were also sometimes used to refer to temperament mappings and [[subgroup basis matrices]], although the terminology has since changed on this wiki. A rank-1 temperament mapping is also called a [[val]].
+== Keyboard map ==
+A '''keyboard map''' or '''keyboard mapping''' specifies the actions associated with each key of a keyboard.
-* [[Douglas Blumeyer]] and [[Dave Keenan]] recommend reserving the word "map" for a mapping with one row, so that all maps are mappings but not all mappings are maps; a simple tip to remember this usage is that the shorter word refers to the simpler object.
+=== Musical instruments ===
+{{See also|Scale design software#Keyboard mapping}}
+A [[keyboard]] mapping for a musical [[instrument]] ([[List of microtonal software plugins|software]] or [[List of microtonal hardware synths|hardware]]) associates notes or pitches from a scale file to the instrument's keys. Notable examples are [[Scala]] keyboard mappings (.kbm files) and [[Lumatone]] keyboard mappings (.ltn files).
+=== Computer keyboard bindings ===
+A keyboard map for a computer keyboard associates characters to certain keys or combinations of keys. It can be used to type special characters more quickly. An example is provided in [[Dave Keenan & Douglas Blumeyer's guide to RTT/Conventions for names, variables, units, and notations#WinCompose]].
+== Maps in regular temperament theory ==
+[[File:Temperament and tuning matrices 2.png|thumb|alt=An overview of maps in regular temperament theory.|An overview of maps in regular temperament theory.]]
+In [[regular temperament theory]] (RTT), maps are generally assumed to be [[Wikipedia:Linear map|''linear'' maps]]<ref group="note">Since regular temperaments are defined on ℤ-modules, not on vector spaces, the term "linear map" is not quite correct and it would be preferable to say "[[Wikipedia:Additive map|''additive'' map]]", but "linear map" is often used by analogy since linear algebra terminology is more common in mathematics overall.</ref>, which, informally, can be thought of as a function that can be represented by a {{w|Matrix (mathematics)|matrix}}.
+The most common maps in RTT are discussed below.
+=== Temperament map ===
+{{Main|Mapping}}
+A temperament map or temperament mapping<ref group="note">[[Douglas Blumeyer]] and [[Dave Keenan]] recommend reserving the word "map" for a mapping with one row, so that all maps are mappings but not all mappings are maps; a simple tip to remember this usage is that the shorter word refers to the simpler object. In all occurrences at present on this wiki, as well as in Graham Breed's temperament finder, the term "map" (and not "mapping") consistently refers to a single-row mapping, so following this suggestion would be seamless moving forward. A "tuning map", which maps from generators to cents, is a map in "tuning space"; by analogy, a val is a map in "temperament space", and so it would be perfectly consistent with existing terminology to refer to a val as a "temperament map" as opposed to a "temperament mapping", and then when it is clear from the context that it is a ''temperament'' map, the qualifier "temperament" can be dropped, as is done with "temperament mapping matrix" being abbreviated to "mapping matrix". So the suggestion is equivalent to unqualified occurrences of "map" being assumed to be temperament maps, or in other words, synonymous with vals (except for the integer entry requirement), not tuning maps. Dave and Douglas recommend using "map" rather than "val", for two reasons. First, "map" is a basic linear algebra term with wide familiarity (being specialized for this purpose) while "val" is unnecessary jargon that creates a barrier to understanding by newcomers. Second, the coinage of "val" from the obscure mathematical term "valuation" is tenuous and unlikely to provide helpful insight: "p-adic valuation" is an obscure term for "prime count", which would be an element of a prime-count vector ("monzo"), not a map ("val").</ref> specifies the structure of a regular temperament. It transforms [[interval]]s, usually from a [[just intonation subgroup]] and expressed as prime-count vectors ([[monzo]]s), into [[mapped interval]]s ([[tmonzo]]s). It is often represented by a matrix, thus called a temperament mapping matrix<ref group="note">The terms "M-map" and "V-map" were also sometimes used to refer to temperament mappings and subgroup basis matrices, although these terms are rarely used nowadays.</ref>, or more simply a mapping matrix or even a mapping when the context is sufficiently clear.
+Each row of a temperament mapping matrix, or the only row of a rank-1 temperament mapping matrix, is often called a [[val]].
+A [[subgroup basis matrix]] is the dual of a temperament mapping matrix.
+=== Tuning map ===
+{{Main|Tuning map}}
+A tuning map (or rarely a tuning mapping) specifies the tuning of a regular temperament. A (tempered-prime) tuning map transforms intervals into tempered [[interval size]]s. A generator tuning map transforms mapped intervals into interval sizes.
+=== Projection map ===
+{{Main|Projection matrix}}
+A projection matrix (or rarely a projection map) uniquely identifies a specific tuning of a specific regular temperament.
+=== Lattice map ===
+{{Main|Mapping to lattice}}
+A lattice map or mapping to lattice transforms intervals into their coordinates in a regularly tempered [[lattice]].
+== See also ==
+* [[Harmonic template]]
+== Notes ==
+<references group="note"/>
-[[Category:Temperament]]
 [[Category:Regular temperament theory]]
-[[Category:Theory]]
 [[Category:Terms]]
 [[Category:Math]]
 [[Category:Val]]
 [[Category:Mapping]]
-[[Category:Tuning]]