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Trivial temperament: Difference between revisions

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A '''trivial temperament''' is something that fits the mathematical definition of "regular temperament", but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two types of trivial temperaments: identity temperaments (which leave all intervals untempered) and Om temperaments (which temper out all intervals).
A '''trivial temperament''' is something that fits the mathematical definition of [[regular temperament]], but is a unique, extreme case that people might be uncomfortable calling a "[[temperament]]". There are two types of trivial temperaments: [[just intonation]], which leaves all intervals [[tempering|untempered]], and [[single-pitch tuning]], which [[tempering out|tempers out]] all intervals.


== Identity temperament ==
== Just intonation ==
The '''identity temperament''' for a subgroup of rank ''n'', so called because a possible mapping is an ''n''×''n'' identity matrix, transforms said subgroup to itself. In musical terms, this means that nothing is tempered. The set of commas that are [[tempered out]] is {1/1}, but that's still a valid set, so identity temperaments are still valid regular temperaments. An identity temperament exists for each subgroup of JI, and there is an identity extension for any given temperament.
{{Main| Just intonation }}


The 2-limit version is the equal temperament [[1edo]]. The [[3-limit]] version is a rank-2 temperament ("[[pythagorean]]"), which has all the properties of any other rank-2 temperament except that it tempers no commas. The 5-limit identity temperament is rank-3 ("classical" - though note that this might be confused with [[meantone]]), the 7-limit identity temperament is rank-4 ("septimal"), etc.
The [[mapping]] for a [[just intonation subgroup]] of rank ''n'' is an ''n''×''n'' {{w|identity matrix}}, and transforms said subgroup to itself. In musical terms, this means that nothing is tempered. The set of commas that are tempered out is {1/1}, but that is still a valid set, so just intonation still counts as valid regular temperaments.  


== Om temperament ==
There is an infinite family of these temperaments, one for each subgroup of JI. The 2-limit version is equivalent to [[1edo|1et]]. The [[3-limit]] version, or [[pythagorean tuning]], is a rank-2 temperament, which has all the properties of any other rank-2 temperament except that it tempers out no commas. 5-limit JI is rank-3, 7-limit JI is rank-4, etc.
'''Om''' temperament is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note, leading to [[single-pitch tuning]]. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, {{val| 0 0 ... 0 }}, and its multival is a single entry. It could also be called the ''unison temperament''<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison.  


As with identity temperaments, there is technically an Om temperament for every subgroup.
[[User:VectorGraphics|Vector]] proposes the name ''identity temperament''{{idio}} for this family of temperaments.  


The name "Om" is a reference to [[Wikipedia:Om|that syllable's use in Hindu meditation practices]]; [[Keenan Pepper]] gave it this name because there's only one temperament-distinct pitch in the whole system, in the same way that "Om" in the meditation sense is the only word you need to create the whole universe.
== Single-pitch tuning ==
{{Main| Single-pitch tuning }}
 
The single-pitch tuning is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, {{val| 0 0 … 0 }}, and its [[wedgie]] is a single entry.
 
As with JI, there is technically a temperament of a single pitch for every subgroup.
 
[[Gene Ward Smith]] proposes the name ''unison temperament'' for this family of temperaments<ref>http://www.robertinventor.com/tuning-math/s__12/msg_11050-11074.html</ref>, as all intervals are equated to the unison. [[Keenan Pepper]] proposes the name ''Om temperament''{{idio}}. [[Wikipedia:Om|''Om'']] is a reference to that syllable's use in Hindu meditation practices, for there is only one temperament-distinct pitch in the whole system, in the same way that ''Om'' in the meditation sense is the only word you need to create the whole universe.


== Notes and references ==
== Notes and references ==
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