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 ~~tempered out~~ 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 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|||... .

'''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>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval ~~tempered out~~ is the [[unison]] (and therefore all intervals are brought together in 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.

'''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>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval made to vanish is the [[unison]] (and therefore all intervals are brought together in 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.

[[Category:Regular temperament theory]]

[[Category: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 tempered out 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 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|||... .
-'''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>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval tempered out is the [[unison]] (and therefore all intervals are brought together in 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.
+'''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>, following the common pattern whereby temperaments are named after the intervals they [[temper out]], where in this case the interval made to vanish is the [[unison]] (and therefore all intervals are brought together in 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.
 [[Category:Regular temperament theory]]
 [[Category:Theory]]