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Some tunings make no attempt to approximate JI, and so should not be considered temperaments.
Breed's last sentence in the definition above is exactly the reason for coining the term EDO. However, "EDO" is by now (May 2020) often used to represent any equal-division of the octave, including the instances when they are being used as a temperament.
The most familiar types of temperament today (1998) are the equal temperaments, but historically many temperaments have also been unequal; all well-temperaments, and all those belonging to the meantone family which are not EDOs (i.e., the numerous fraction-of-a-comma varieties, LucyTuning, golden-meantone) fall into the latter category.
I would say a regular temperament is determined by a homomorphic mapping from the p-limit, or possibly another finitely-generated subgroup of the positive rationals, to an abstract free group of smaller rank. This can be specified (uniquely determined) by giving a wedgie, a kernel, or an explicit mapping.
Note I do not include the tuning map as part of the definition, so this is an abstract definition of what a regular temperament is. However, this is how the word is most commonly used; people may object to it but the same people talk of 1/4-comma meantone or 2/7-comma meantone as if they were both meantone. My definition also says that even though 31-et meantone is tuned to a group of rank one, it is still qua meantone a group of rank two, and the tuning mapping is another issue.
You might also note that this definition, which says a temperament is a morphism, makes no sense unless you get the category right, which connects to the thread about spaces.
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