Diamond tradeoff

A tuning for a rank-r p-limit regular temperament is diamond purer, or diamond strict, if it fits the following definition: we may define the diamond purer tuning range by finding the convex hull in tuning space of the set of all tunings with r eigenmonzos chosen as follows: one eigenmonzo 2 (pure octaves tunings) and the rest of the eigenmonzos any set of r - 1 members of the p-odd limit tonality diamond, whenever such a tuning is defined.

In the original work by Andrew Milne, Bill Sethares and James Plamondon — and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "nice" tuning range (this is not the same thing as what is now called diamond nice, which is the combination of diamond purer and diamond monotone).

Diamond purer tunings are always guaranteed to occur, but diamond monotone tunings are not.

For examples and other information, see the topic page Tuning ranges of regular temperaments.