Mathematical theory of regular temperaments: Difference between revisions

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To use a concrete example, if [[Meantone family #Septimal meantone|7-limit meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 ([[syntonic comma]]), is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".

A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of optimality that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament ''regular''.

A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of optimality that disagree with each other, so in general we can consider a regular temperament as having a [[Tuning ranges of regular temperaments|range of possible tunings of the generators]]. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament ''regular''.

== Dimensionality, or rank ==

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 To use a concrete example, if [[Meantone family #Septimal meantone|7-limit meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 ([[syntonic comma]]), is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
-A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of optimality that disagree with each other, so in general we can consider a regular temperament as having a range of possible tunings of the generators. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament ''regular''.
+A regular temperament is abstract, and has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of optimality that disagree with each other, so in general we can consider a regular temperament as having a [[Tuning ranges of regular temperaments|range of possible tunings of the generators]]. Once a tuning of each generator is provided the tuning of any interval can be computed as an integer linear combination of generator tunings. This property that all intervals are linear combinations of the generators is in fact what makes a temperament ''regular''.
 == Dimensionality, or rank ==