Mapping: Difference between revisions
Move correspondence with mathematical terminology to the bottom, beginners don't need to know about matrices for this |
Undo revision 187782 by VectorGraphics (talk) The term "matrix" is present throughout the page, so it is appropriate to introduce it at the beginning, alongside the rest of the note which is also useful to disambiguate from other meanings of the term "mapping". Tag: Undo |
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Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval. | Naively, one might think that a simple rounding function might be suitable for a mapping: let the "tempered version" of each JI pitch simply be the tempered pitch that is closest to it. However, this (usually) does not result in a regular temperament at all! The reason is that, although this mapping assigns a tempered pitch to each JI pitch, it does not do so in a ''consistent'' way—some instances of the same JI interval are represented by different tempered intervals if they occur in different places. A regular temperament mapping always represents each JI interval by the ''same'' tempered interval, even if that tempered interval is not the closest tempered interval to the JI interval. | ||
== A note on mathematical terminology == | |||
In mathematics generally, "mapping" is synonymous with "map" and "function". In RTT, "mapping" has the more specific meaning of a {{w|Linear map|''linear'' mapping}}, which is a function that can be represented by a matrix. | |||
== Equal temperament mappings == | == Equal temperament mappings == | ||
An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of | An equal temperament, also known as a rank-1 temperament (see below for a discussion of rank), is not merely a set of equally spaced pitches. An equal temperament consists of | ||
# A JI subgroup that is being represented, such as "5-limit JI", and | # A JI subgroup that is being represented, such as "5-limit JI", and | ||
# A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer). | # A mapping that assigns every pitch of this JI subgroup to a note of the equal temperament (which can be represented as an integer). | ||
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For example: | For example: | ||
# An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator. | # An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator. | ||
# Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains. | # Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank-2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains. | ||
# Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like). | # Temperaments which consist of three generators, or more commonly a period and two generators, are rank-3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like). | ||
# 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc. | # 7-limit JI would be rank-4, 11-limit JI would be rank-5, 13-limit JI would be rank-6, etc. | ||
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== Units == | == Units == | ||
It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]]. | It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]]. | ||
== See also == | == See also == | ||