Regular temperament: Difference between revisions

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This article focuses on the mathematical theories, particularly the mathematical tools to describe a regular temperament. For an introduction to regular temperaments, see [[Tour of Regular Temperaments]].

A '''regular temperament''' is a sort of mathematical object that maps [[just intonation|JI]] intervals (aka rational numbers) to tempered intervals. Thus it is a function, whose domain is JI (or in general any "target tuning" we wish to approximate, but very commonly assumed to be JI), and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are "tempered together", or that the difference between them is a "comma" that is "tempered out" in the temperament.

As an example, if [[meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".

An abstract regular temperament has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" 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".)

== Characterizing a regular temperament ==

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 This article focuses on the mathematical theories, particularly the mathematical tools to describe a regular temperament. For an introduction to regular temperaments, see [[Tour of Regular Temperaments]].
+A '''regular temperament''' is a sort of mathematical object that maps [[just intonation|JI]] intervals (aka rational numbers) to tempered intervals. Thus it is a function, whose domain is JI (or in general any "target tuning" we wish to approximate, but very commonly assumed to be JI), and its range is the intervals of the temperament. In general, this mapping is many-to-one, and two different rational numbers may be mapped to the same tempered interval — in this case we say that the two JI intervals are "tempered together", or that the difference between them is a "comma" that is "tempered out" in the temperament.
+As an example, if [[meantone temperament]] is a function M, then M(6/5) = M(32/27) = "minor third". The difference between these, 81/80 or the "[[syntonic comma]]", is tempered out in meantone temperament. M(81/80) = M(1/1) = "unison".
+An abstract regular temperament has no preferred exact tuning. There are ways to compute an optimal tuning for any given temperament, but there are multiple definitions of "optimal" 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".)
 == Characterizing a regular temperament ==