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 | 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). | ||
== Identity temperament == | |||
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. | |||
[[ | 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. | ||
== Om temperament == | |||
'''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 zero. 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. | |||
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 04:41, 6 October 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 types of trivial temperaments: identity temperaments (which leave all intervals untempered) and Om temperaments (which temper out all intervals).
Identity temperament
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.
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.
Om temperament
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, ⟨0 0 ... 0], and its multival is a single zero. It could also be called the unison temperament[1], as all intervals are equated to the unison.
As with identity temperaments, there is technically an Om temperament for every subgroup.
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.