Regular temperament: Difference between revisions

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This article gives an introduction to regular temperaments. For a formal mathematical discussion, see [[Mathematical theory of regular temperaments]]. For an organized list of regular temperaments, see [[Tour of Regular Temperaments]].

A '''regular temperament''' is a kind of abstract musical system that looks the same no matter which pitch you start from (or consider the "tonic"). In other words, unlimited free modulation is possible - any interval can be stacked as many times as you like. Regular temperaments generally have an infinite number of notes; and other than [[equal temperament]]s, every regular temperament actually has an infinite number of notes in between ''any two other notes''!

A '''regular temperament''' is a kind of abstract musical system that looks the same no matter which pitch you start from (or consider the tonic). In other words, unlimited free modulation is possible – any interval can be stacked as many times as you like. Regular temperaments generally have an infinite number of notes; and other than [[equal temperament]]s, every regular temperament actually has an infinite number of notes in between ''any two other notes''.

In addition to unlimited modulation, regular temperaments are usually thought of as being "tempered versions" of some more complicated system of "pure" or "target" intervals, very often [[just intonation]] (JI). A temperament only qualifies as a regular temperament if this approximation 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~~'re "~~tempered together"), but a single pure interval must never be represented by different tempered intervals - if so, the temperament is irregular.

In addition to unlimited modulation, regular temperaments are usually thought of as being tempered versions of some more complicated system of pure or target intervals, very often [[just intonation]] (JI). A temperament only qualifies as a regular temperament if this approximation 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.

One particularly simple kind of regular temperaments are the equal temperaments, which represent all intervals by multiples of a single smallest step.

At the other extreme, JI itself can be considered a kind of "temperament" where no tempering is happening (no [[comma]]s are tempered out but all are preserved as small pitch differences).

At the other extreme, JI itself can be considered a kind of 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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 This article gives an introduction to regular temperaments. For a formal mathematical discussion, see [[Mathematical theory of regular temperaments]]. For an organized list of regular temperaments, see [[Tour of Regular Temperaments]].
-A '''regular temperament''' is a kind of abstract musical system that looks the same no matter which pitch you start from (or consider the "tonic"). In other words, unlimited free modulation is possible - any interval can be stacked as many times as you like. Regular temperaments generally have an infinite number of notes; and other than [[equal temperament]]s, every regular temperament actually has an infinite number of notes in between ''any two other notes''!
+A '''regular temperament''' is a kind of abstract musical system that looks the same no matter which pitch you start from (or consider the tonic). In other words, unlimited free modulation is possible – any interval can be stacked as many times as you like. Regular temperaments generally have an infinite number of notes; and other than [[equal temperament]]s, every regular temperament actually has an infinite number of notes in between ''any two other notes''.
-In addition to unlimited modulation, regular temperaments are usually thought of as being "tempered versions" of some more complicated system of "pure" or "target" intervals, very often [[just intonation]] (JI). A temperament only qualifies as a regular temperament if this approximation 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're "tempered together"), but a single pure interval must never be represented by different tempered intervals - if so, the temperament is irregular.
+In addition to unlimited modulation, regular temperaments are usually thought of as being tempered versions of some more complicated system of pure or target intervals, very often [[just intonation]] (JI). A temperament only qualifies as a regular temperament if this approximation 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.
 One particularly simple kind of regular temperaments are the equal temperaments, which represent all intervals by multiples of a single smallest step.
-At the other extreme, JI itself can be considered a kind of "temperament" where no tempering is happening (no [[comma]]s are tempered out but all are preserved as small pitch differences).
+At the other extreme, JI itself can be considered a kind of 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]]''.