Mapping: Difference between revisions
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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. | ||
== 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]]. | ||
== 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. | |||
== See also == | == See also == | ||