Map: Difference between revisions
Major rewrite |
→Maps in regular temperament theory: Add mapping to lattice |
||
| Line 35: | Line 35: | ||
{{Main|Projection matrix}} | {{Main|Projection matrix}} | ||
A projection matrix (or rarely a projection map) uniquely identifies a specific tuning of a specific regular temperament. | 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 == | == See also == | ||
Latest revision as of 05:23, 28 July 2026
A map or mapping[note 1] is an association of elements from a set to elements of another set. In mathematics, a map is a generalized function.
Various kinds of maps relevant to microtonal music theory are introduced below.
Keyboard map
A keyboard map or keyboard mapping specifies the actions associated with each key of a keyboard.
Musical instruments
A keyboard mapping for a musical instrument (software or 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
In regular temperament theory (RTT), maps are generally assumed to be linear maps[note 2], which, informally, can be thought of as a function that can be represented by a matrix.
The most common maps in RTT are discussed below.
Temperament map
A temperament map or temperament mapping[note 3] specifies the structure of a regular temperament. It transforms intervals, usually from a just intonation subgroup and expressed as prime-count vectors (monzos), into mapped intervals (tmonzos). It is often represented by a matrix, thus called a temperament mapping matrix[note 4], 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 sizes. 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
Notes
- ↑ 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.
- ↑ 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 "additive map", but "linear map" is often used by analogy since linear algebra terminology is more common in mathematics overall.
- ↑ 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").
- ↑ 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.