Extension and restriction
An extension of a regular temperament on a JI subgroup G to a larger subgroup G' is a new temperament that contains the same intervals as the original temperament, and whose rank remains the same, with the same JI interpretations in the subgroup G, but gives them new JI interpretations not in the original subgroup (but are in the larger subgroup). The opposite of extension is restriction.
For example, septimal meantone and flattone are both extensions of 5-limit (2.3.5) meantone to the 7-limit (22.214.171.124), because C-E (4 fifths) represents 5/4 in both. They are different extensions, because in septimal meantone, 7/4 is C-A# (10 fifths), while in flattone, 7/4 is C-Bbb (-9 fifths).
A strong extension is an extension in which the generators are not split compared to the original temperament. The structure of a strong extension is not changed, and no new intervals are introduced. A weak extension is one in which the generators are split.
For example, both septimal meantone and flattone are strong extensions of 5-limit meantone since they all share the same period and generator. Godzilla is a weak extension of meantone but a strong extension of semaphore.
- Subgroup Temperament Families, Relationships, and Genes – formal definitions