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. For example, 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; if so, the temperament is irregular.

An "abstract tuning system" doesn't refer to a concrete tuning, but to a set of "characteristics" or "rules" that a concrete tuning can have. For example, any regular tuning that follows the rule that ~3/2^4 = ~5/1 is a tuning of [[meantone]] temperament; if you change the rule to ~3/2^4 = ~36/7, then the same tuning can be treated as a tuning of [[archy]] temperament.

One particularly simple kind of regular temperaments is the equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a {{w|Triviality (mathematics)|trivial}} temperament where no tempering is happening: no [[comma]]s 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]]''.

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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. For example, 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; if so, the temperament is irregular.
-An "abstract tuning system" doesn't refer to a concrete tuning, but to a set of "characteristics" or "rules" that a concrete tuning can have. For example, any regular tuning that follows the rule that ~3/2^4 = ~5/1 is a tuning of [[meantone]] temperament; if you change the rule to ~3/2^4 = ~36/7, then the same tuning can be treated as a tuning of [[archy]] temperament.
 One particularly simple kind of regular temperaments is the equal temperaments, which represent all intervals by multiples of a single smallest step. At the other extreme, JI itself can be considered a {{w|Triviality (mathematics)|trivial}} temperament where no tempering is happening: no [[comma]]s 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]]''.