Tuning map: Difference between revisions

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A '''tuning map''' represents the tuning of a [[regular temperament]]. 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 ~~cents~~ or ~~octaves~~ (or any other logarithmic pitch unit).

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).

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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From the generator tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈M</math>. 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]] <math>M^{+}</math>, as in <math>𝒈 = 𝒕M^{+}</math>. For more information, see the explanation [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_in_nonstandard_domains#9._Find_pseudoinverse|here]].

== Error map ==

An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually [[JI]]), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the [[just tuning map]].

== Example ==

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 A '''tuning map''' represents the tuning of a [[regular temperament]]. 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 cents or octaves (or any other logarithmic pitch unit).
+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).
 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]]).
@@ Line 12: / Line 12: @@
 From the generator tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈M</math>. 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]] <math>M^{+}</math>, as in <math>𝒈 = 𝒕M^{+}</math>. For more information, see the explanation [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_in_nonstandard_domains#9._Find_pseudoinverse|here]].
+== Error map ==
+An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually [[JI]]), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the [[just tuning map]].
 == Example ==