Regular temperament: Difference between revisions

Line 8:

A '''regular temperament''' is an abstract [[tuning 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''' ('''RT''') is an abstract [[tuning 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 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.

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 ~~[[Wikipedia:~~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]]''.

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]]''.

== History ==

Line 28:

* [[George Secor]] (1943–2020): miracle temperament

A significant amount of this theory's early development occurred online via the ~~[[Wikipedia:~~ Yahoo! Groups~~|Yahoo! Groups]]~~ service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and [[Paul Hahn]] in the late 1990's.

A significant amount of this theory's early development occurred online via the {{Yahoo! Groups}} service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and [[Paul Hahn]] in the late 1990's.

In 2001 [[Gene Ward Smith]] joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as [[Mike Battaglia]] also did much of the work to document RTT on this wiki.

@@ Line 8: / Line 8: @@
 {{Wikipedia}}
-A '''regular temperament''' is an abstract [[tuning 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''' ('''RT''') is an abstract [[tuning 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 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.
-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 [[Wikipedia: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]]''.
+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]]''.
 == History ==
@@ Line 28: / Line 28: @@
 * [[George Secor]] (1943–2020): miracle temperament
-A significant amount of this theory's early development occurred online via the [[Wikipedia: Yahoo! Groups|Yahoo! Groups]] service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and [[Paul Hahn]] in the late 1990's.
+A significant amount of this theory's early development occurred online via the {{Yahoo! Groups}} service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and [[Paul Hahn]] in the late 1990's.
 In 2001 [[Gene Ward Smith]] joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as [[Mike Battaglia]] also did much of the work to document RTT on this wiki.