Expansion and retraction

An expansion of a regular temperament from a JI subgroup to an expanded subgroup is a new temperament that contains the intervals and structure of the original temperament, with extra generators added to reach the new JI representations in the larger subgroup. Because new generators are added, expanding a temperament always increases its rank. For example, didacus is the rank-2 temperament that tempers 3136/3125 in the 2.5.7 subgroup. It can be expanded to the full 7-limit by adding a generator representing 3, resulting in the rank-3 temperament hemimean. The opposite of expansion is retraction.

For temperaments with many reasonable extensions to reach certain primes, it may be useful to consider the expansion of that temperament that includes the desired intervals and tempers any new commans shared between the extensions. Then, the extensions of the original temperament can be thought of as ways to temper the expanded temperament back to the original rank (especially when some new commas are shared between the extensions, which is common). For example, the rank-2 2.3.7-subgroup temperament slendric has several extensions to the full 7-limit, as described on that page, but all of those extensions can be further extended to the 11-limit by tempering 385/384~441/440. Therefore it is logical to first expand slendric to the 11-limit by adding one more generator and tempering 385/384~441/440, resulting in the rank-3 temperament portent. The 11-limit slendric extensions then represent methods of tempering portent back into a rank-2 temperament.

See also

• Subgroup temperament families, relationships, and genes – formal definitions
• Extension and restriction