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.

The diamond purer tuning range marks tuning boundaries inside of which the temperament's approximations to simple low-ratio frequency ratios can be "traded" against each other - i.e., if I make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, you will improve the tunings of *all* such intervals by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.

However, it is quite clear that tunings outside of this purer range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the purer tuning range.

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