Trivial temperament: Difference between revisions
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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 ''n''×''n'' identity matrix, with wedgies of {{val| 1 }}, {{multival|rank=2| 1 }}, {{multival|rank=3| 1 }}, etc. | 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 ''n''×''n'' identity matrix, with wedgies of {{val| 1 }}, {{multival|rank=2| 1 }}, {{multival|rank=3| 1 }}, etc. | ||
'''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 | [[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 == | == Notes and references == | ||
Revision as of 06:15, 13 February 2025
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 n×n identity matrix, with wedgies of ⟨1], ⟨⟨ 1 ]], ⟨⟨⟨ 1 ]]], etc.
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[1], as all intervals are equated to the unison. The name "Om" is a reference to 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.