Trivial temperament

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 entry. It could also be called the unison temperament, 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.