Trivial temperament: Difference between revisions
Wikispaces>keenanpepper **Imported revision 300788560 - Original comment: ** |
Use common terms rather than unattested/rare idiosyncratic terms (including "Om temperament"!) |
||
| (25 intermediate revisions by 12 users not shown) | |||
| Line 1: | Line 1: | ||
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. | |||
Just intonation | == Just intonation == | ||
{{Main| Just intonation }} | |||
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. | |||
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. | |||
[[User:VectorGraphics|Vector]] proposes the name ''identity temperament''{{idio}} for this family of temperaments. | |||
== 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 == | |||
<references /> | |||
[[Category:Regular temperament theory]] | |||
Latest revision as of 14:19, 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: just intonation, which leaves all intervals untempered, and single-pitch tuning, which tempers out all intervals.
Just intonation
The mapping for a just intonation subgroup of rank n is an n×n 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.
There is an infinite family of these temperaments, one for each subgroup of JI. The 2-limit version is equivalent to 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.
Vector proposes the name identity temperament[idiosyncratic term] for this family of temperaments.
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, ⟨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[1], as all intervals are equated to the unison. Keenan Pepper proposes the name Om temperament[idiosyncratic term]. 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.