Diamond monotone

A tuning for a rank-r p-limit regular temperament is diamond monotone, or diamond valid, if it satisfies the following condition: the p-odd limit tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also monotone increasing (i.e. nondecreasing).

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 "valid" tuning range.

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

Temperaments without diamond monotone tunings

While diamond pure tunings are always guaranteed to occur, diamond monotone tunings are not. For instance, from the tuning map [⟨1 0 5], ⟨0 1 -2]] for the temperament tempering out 45/32 we find that all tunings are of the form ⟨1 0 5] + a⟨0 1 -2] = ⟨1 a 5-2a]. For example, if a was 7/5, then the map would be ⟨1 (7/5) 5-2(7/5)] = ⟨1 7/5 25/5-14/5] = ⟨5 7 11], and if a was 4/3 then the map would be ⟨1 (4/3) 5-2(4/3)] = ⟨1 4/3 15/3-8/3] = ⟨3 4 7]. One way to think about preserving the sorting order of the p-odd limit tonality diamond would be to ensure that none of the intervals between its pitches become negative under this temperament. The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3], and the intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5]. We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8]. In vector form those are [[1 1 -1⟩, [-3 -1 2⟩, [4 -1 -1⟩, [-3 2 0⟩]. If we map those using ⟨1 a 5-2a] we obtain the tempered sizes [3a - 4, 7 - 5a, a - 1, 2a - 3]. Now we need to make sure each of those are not negative, so we get a set of inequalities: a ≥ 4/3, a ≤ 7/5, a ≥ 1, a ≥ 3/2. These inequalities have no solution: there's no way a can be both greater or equal to 1.5 and less than or equal to 1.4. Hence there are no diamond monotone tunings of this temperament.