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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More comprehensive lists:

* [[Map of rank-2 temperaments]]: a visual map of temperaments based on the size of their period and generator

* [[Bird's eye view of temperaments by accuracy]] (article): temperaments the Xen Wiki contributors find most useful for approximating JI - with edo tunings and note counts for the harmonies they target, and explanations of their structure

* [[Bird's eye view of temperaments by accuracy]]: ~~a compilation of rank 2~~ temperaments ~~that people think are~~ most ~~valuable~~ for approximating JI, with edo tunings and note counts for the harmonies they target and explanations of their structure

* [[Survey of efficient temperaments by subgroup]] (table): good general-purpose temperaments, sorted by size (notes per equave) and by JI subgroup

* [[Survey of efficient temperaments by subgroup]]: ~~a visual map of~~ temperaments ~~based on~~ notes ~~needed~~ per equave and JI subgroup

* [[Map of rank-2 temperaments]] (table): temperaments (some general, some niche) sorted by the size of their period and generator

* [[Tour of regular temperaments]]: a huge gallery of the dozens of families of temperaments that have been described; not for the faint of heart

* [[Temperaments for MOS shapes]] (table): temperaments (some general, some niche) sorted by the scale shape they generate

* [[Tour of regular temperaments]] (article): huge gallery of the dozens of families of temperaments that have been described; ''very technical - not for the faint of heart''

=== Other writings on 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: 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 ==
@@ Line 79: / Line 79: @@
 More comprehensive lists:
-* [[Map of rank-2 temperaments]]: a visual map of temperaments based on the size of their period and generator
+* [[Bird's eye view of temperaments by accuracy]] (article): temperaments the Xen Wiki contributors find most useful for approximating JI - with edo tunings and note counts for the harmonies they target, and explanations of their structure
-* [[Bird's eye view of temperaments by accuracy]]: a compilation of rank 2 temperaments that people think are most valuable for approximating JI, with edo tunings and note counts for the harmonies they target and explanations of their structure
+* [[Survey of efficient temperaments by subgroup]] (table): good general-purpose temperaments, sorted by size (notes per equave) and by JI subgroup
-* [[Survey of efficient temperaments by subgroup]]: a visual map of temperaments based on notes needed per equave and JI subgroup
+* [[Map of rank-2 temperaments]] (table): temperaments (some general, some niche) sorted by the size of their period and generator
-* [[Tour of regular temperaments]]: a huge gallery of the dozens of families of temperaments that have been described; not for the faint of heart
+* [[Temperaments for MOS shapes]] (table): temperaments (some general, some niche) sorted by the scale shape they generate
+* [[Tour of regular temperaments]] (article): huge gallery of the dozens of families of temperaments that have been described; ''very technical - not for the faint of heart''
 === Other writings on temperaments ===