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.

This tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4, which will be smaller than the interval representing 4/3. (As with the diamond tradeoff range, the precise boundary tunings depend on the intervals we wish to privilege—privileging those in the p-odd-limit tonality diamond is an arguably reasonable choice).

The "empirical" range is likely to fall somewhere between diamond monotone and diamond tradeoff. Though, when one is using tempered spectra to match the tuning, it is possible the empirical range can be made wider.

Temperaments without diamond monotone tunings

While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not.

Let's look at an example: the rank-2 temperament with mapping [⟨1 0 5], ⟨0 1 -2]], tempering out 45/32.

All of this temperaments tunings are some linear combination of these two mapping rows. We could express that idea in the form ⟨1 0 5] + a⟨0 1 -2] = ⟨1 a 5-2a]. So one example tuning would be if this a variable was 7/5, which would give us the map ⟨1 7/5 5−2(7/5)] = ⟨1 7/5 25/5−14/5] = ⟨5 7 11]. Another example tuning would be 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. Let us work through how to establish that:

1. The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3].
2. The intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5].
3. We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8].
4. In vector form those are [[1 1 -1⟩, [-3 -1 2⟩, [4 -1 -1⟩, [-3 2 0⟩], respectively.
5. If we map those using ⟨1 a 5−2a] we obtain the tempered sizes [3a − 4, 7 − 5a, a − 1, 2a− 3].
6. 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.

We can see that these inequalities have no solution: there is 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; any tuning must necessarily either map 6/5 wider than 5/4 or 4/3 wider than 3/2.

See also

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