Regular temperament: Difference between revisions

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In addition to unlimited modulation, regular temperaments by definition are thought of as being approximations of some more complicated system of pure or target intervals, very often a [[just intonation]] (JI) [[subgroup]]. Each abstract interval is interpreted as a tempered, or detuned, version of the target interval (more accurately, a set of target intervals). A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: the sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. Multiple pure intervals may be represented by the same tempered interval (so they are tempered together), but a single pure interval must never be represented by different tempered intervals. Certain intervals are tempered to the [[1/1|unison]]; in a regular temperament, these intervals are known as [[comma]]s. Any two just intervals seperated by a comma of a temperament are mapped to the same tempered interval in that temperament. For example, [[meantone]] temperament, the [[5-limit]] temperament tempering out [[81/80]], maps [[81/64]] and [[5/4]] to the same interval, as 81/64 and 5/4 differ by 81/80 in JI.

One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a [[trivial temperament]] where no tempering is happening: no commas are tempered out, but all are preserved as small pitch differences. In between lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].

One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a [[trivial temperament]] where no tempering is happening: no commas are tempered out, but all are preserved as small pitch differences. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and there is only a single pitch. In between lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].

== History ==

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 In addition to unlimited modulation, regular temperaments by definition are thought of as being approximations of some more complicated system of pure or target intervals, very often a [[just intonation]] (JI) [[subgroup]]. Each abstract interval is interpreted as a tempered, or detuned, version of the target interval (more accurately, a set of target intervals). A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: the sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. Multiple pure intervals may be represented by the same tempered interval (so they are tempered together), but a single pure interval must never be represented by different tempered intervals. Certain intervals are tempered to the [[1/1|unison]]; in a regular temperament, these intervals are known as [[comma]]s. Any two just intervals seperated by a comma of a temperament are mapped to the same tempered interval in that temperament. For example, [[meantone]] temperament, the [[5-limit]] temperament tempering out [[81/80]], maps [[81/64]] and [[5/4]] to the same interval, as 81/64 and 5/4 differ by 81/80 in JI.
-One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a [[trivial temperament]] where no tempering is happening: no commas are tempered out, but all are preserved as small pitch differences. In between lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].
+One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a [[trivial temperament]] where no tempering is happening: no commas are tempered out, but all are preserved as small pitch differences. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and there is only a single pitch. In between lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].
 == History ==