Tuning map: Difference between revisions

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A '''tuning map''' represents the tuning of a [[regular temperament]]. It is similar to a standard [[val]], but it specifies the tuning of a temperament in terms of logarithmic [[interval size unit]]s (such as [[cent]]s) rather than scale steps. Some people consider it a type of val. It can take a vector representation of an interval ([[monzo]]) as input and outputs its pitch, usually measured in cents or octaves.

A tuning map has one entry for each [[basis element]] of the temperament, giving its size in [[cent]]s or [[octave]]s (or any other logarithmic pitch unit).

There are two kinds of tuning maps: a '''subgroup tuning map'''{{Idiosyncratic}} and a '''generator tuning map.''' A subgroup tuning map represents a tuning of the "formal primes" of the JI subgroup being tempered (e.g. 2, 9, 5 in a 2.9.5 temperament), and takes a monzo representing a JI interval. A generator tuning map directly represents a tuning of the temperament's generators (given a set of generators), and takes a monzo in tempered interval space (a "[[Tmonzos and tvals|tmonzo"]]).

== Subgroup tuning map ==

A subgroup tuning map has one entry for each [[basis element]] of the temperament's JI subgroup, giving its size in [[cent]]s or [[octave]]s (or any other logarithmic pitch unit).

It may be helpful, then, to think of the units of each entry of a tuning map as <math>{\large\mathsf{¢}}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime (for more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]]).

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It may be helpful, then, to think of the units of each entry of a generator tuning map as <math>{\large\mathsf{¢}}\small /𝗴</math> (read "cents per generator"), <math>\small \mathsf{oct}/𝗴</math> (read "octaves per generator"), or any other logarithmic pitch unit per generator.

From the generator tuning map ''G'' and the mapping ''M'', we can obtain the tuning map ''T'' as

From the generator tuning map ''G'' and the mapping ''M'', we can obtain the subgroup tuning map ''T'' as

$$T = GM$$

To go the other way – that is, to find the generator tuning map from the ~~(primes)~~ tuning map – we can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in

To go the other way – that is, to find the generator tuning map from the subgroup tuning map – we can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in

$$G = TM^{+}$$

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[[JI]] can be conceptualized as the temperament where no intervals are made to [[vanish]], and as such, the untempered primes can be thought of as its generators~~, or of course its basis elements~~. So, JI subgroups have generator tuning maps and tuning maps too; the generator tuning maps and tuning maps are always the same thing as each other, and they are all subsets of the entries of the [[JIP]].

[[JI]] can be conceptualized as the temperament where no intervals are made to [[vanish]], and as such, the untempered primes can be thought of as its generators. So, JI subgroups have generator tuning maps and subgroup tuning maps too; the generator tuning maps and subgroup tuning maps are always the same thing as each other, and they are all subsets of the entries of the [[JIP]].

== Error map ==

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 A '''tuning map''' represents the tuning of a [[regular temperament]]. It is similar to a standard [[val]], but it specifies the tuning of a temperament in terms of logarithmic [[interval size unit]]s (such as [[cent]]s) rather than scale steps. Some people consider it a type of val. It can take a vector representation of an interval ([[monzo]]) as input and outputs its pitch, usually measured in cents or octaves.
-A tuning map has one entry for each [[basis element]] of the temperament, giving its size in [[cent]]s or [[octave]]s (or any other logarithmic pitch unit).
+There are two kinds of tuning maps: a '''subgroup tuning map'''{{Idiosyncratic}} and a '''generator tuning map.''' A subgroup tuning map represents a tuning of the "formal primes" of the JI subgroup being tempered (e.g. 2, 9, 5 in a 2.9.5 temperament), and takes a monzo representing a JI interval. A generator tuning map directly represents a tuning of the temperament's generators (given a set of generators), and takes a monzo in tempered interval space (a "[[Tmonzos and tvals|tmonzo"]]).
+== Subgroup tuning map ==
+A subgroup tuning map has one entry for each [[basis element]] of the temperament's JI subgroup, giving its size in [[cent]]s or [[octave]]s (or any other logarithmic pitch unit).
 It may be helpful, then, to think of the units of each entry of a tuning map as <math>{\large\mathsf{¢}}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime (for more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]]).
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 It may be helpful, then, to think of the units of each entry of a generator tuning map as <math>{\large\mathsf{¢}}\small /𝗴</math> (read "cents per generator"), <math>\small \mathsf{oct}/𝗴</math> (read "octaves per generator"), or any other logarithmic pitch unit per generator.
-From the generator tuning map ''G'' and the mapping ''M'', we can obtain the tuning map ''T'' as
+From the generator tuning map ''G'' and the mapping ''M'', we can obtain the subgroup tuning map ''T'' as
 $$T = GM$$
-To go the other way – that is, to find the generator tuning map from the (primes) tuning map – we can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in
+To go the other way – that is, to find the generator tuning map from the subgroup tuning map – we can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in
 $$G = TM^{+}$$
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 {{Main| JIP }}
-[[JI]] can be conceptualized as the temperament where no intervals are made to [[vanish]], and as such, the untempered primes can be thought of as its generators, or of course its basis elements. So, JI subgroups have generator tuning maps and tuning maps too; the generator tuning maps and tuning maps are always the same thing as each other, and they are all subsets of the entries of the [[JIP]].
+[[JI]] can be conceptualized as the temperament where no intervals are made to [[vanish]], and as such, the untempered primes can be thought of as its generators. So, JI subgroups have generator tuning maps and subgroup tuning maps too; the generator tuning maps and subgroup tuning maps are always the same thing as each other, and they are all subsets of the entries of the [[JIP]].
 == Error map ==