Revision as of 22:04, 15 August 2024 by Fredg999(talk | contribs)(Created page with "{{Beginner}} Regular temperaments are often described with several mathematical properties. This information can be condensed in the form of ''temperament data'' tables, w...")
Not all temperament tables provide the same information, nor do they all provide it in exactly the same way, but the following properties should cover most needs.
The subgroup (or domain basis) of a regular temperament is the set of all intervals which are considered to be approximated by the temperament. For example, it is common to consider that 3/2 is approximated by 12-tone equal temperament, therefore 3/2 would be included in this set, but other intervals like 11/8 could be excluded. Most of the time, a subgroup exclusively contains just intonation intervals.
In a subgroup, all intervals must be reachable by stacking (up and down) copies of a few "generating intervals", or generators. Continuing the previous example, if 3/2 is taken as a generator of the subgroup, then 9/4 is also included in the subgroup. If 2/1 is added to the list of subgroup generators, then intervals like 4/3 can be reached by combining a 3/2 down with a 2/1 up (i.e. 2/3 × 2/1 = 4/3).
A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of 2/1, 3/1 and 5/1. The 2.3.5 subgroup is equivalent to the 5-limit, because it contains all prime harmonics up to 5, but temperament data tables typically prefer the first notation.
