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 kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.

Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is ~~the~~ rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an ~~nxn~~ identity matrix~~, with wedgies of <1|, <<1||, <<<1|||...~~ .

Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. 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]] version is rank-3 ("classical"), and so on. The mapping for this temperament is an ''n''×''n'' identity matrix.

'''Om''' temperament 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; it could be described as [[0edo]]. The mapping for this is the 0-val, ~~<~~0 0 ... 0|. 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. 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|'''Om''' temperament]] 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; it could be described as 0edo. The mapping for this is the 0-val, {{val| 0 0 ... 0 }}. 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. 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.

== Notes and references ==

[[Category:Regular temperament theory]]

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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 kinds of trivial temperaments - [[JI]], in which nothing is tempered, and '''Om''' temperament, in which everything is tempered.
-Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. The [[2-limit]] version is the equal temperament [[1edo]]. The [[3-limit]] version is the rank-2 temperament [[pythagorean]], which has all the properties of any other rank-2 temperament except that it tempers no commas. The [[5-limit]] version is rank-3, and so on. The mapping for this temperament is an nxn identity matrix, with wedgies of &lt;1|, &lt;&lt;1||, &lt;&lt;&lt;1|||... .
+Just intonation is a codimension-0 "temperament", which means nothing is tempered. The set of commas that are made to [[vanish]] is the set {1/1}, but that's still a set, so JI is still a regular temperament. There is an infinite family of these "temperaments", one for each subgroup of JI. 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]] version is rank-3 ("classical"), and so on. The mapping for this temperament is an ''n''&times;''n'' identity matrix.
-'''Om''' temperament 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; it could be described as [[0edo]]. The mapping for this is the 0-val, &lt;0 0 ... 0|. 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. 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|'''Om''' temperament]] 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; it could be described as 0edo. The mapping for this is the 0-val, {{val| 0 0 ... 0 }}. 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. 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.
+== Notes and references ==
+<references />
 [[Category:Regular temperament theory]]