Tuning Ranges of Regular Temperaments

There are various methods which have been suggested for defining tuning ranges appropriate to a given regular temperament.

Given a rank r p-limit regular temperament, we may define a tuning range by finding the convex hull in tuning space of the tunings with one eigenmonzo 2 (pure octaves tunings) and the rest a set of r-1 members of the p-limit tonality diamond, when this tuning is defined. This is the nice tuning range. We may define another tuning range by requiring that the tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also monotone weakly increasing. This we may call the valid tuning range. A tuning which is both nice and valid is a strict tuning and this defines the strict tuning range.

While nice tunings are always guaranteed to occur, valid 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|. Applying this to the list of steps between the notes of the 5-limit tonality diamond, [6/5, 25/24, 16/15, 9/8], we obtain [3a-4, 7-5a, a-1, 2a-3] from which it follows that a≥4/3, a≤7/5, a≥1, a≥3/2, the solution set of which is empty. Hence there are no valid tunings of this temperament.

For a more typical example, consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form <1 a b 2a+ab-5 12-a-3b|. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ a ≤ 49/31, 2+a/5 ≤ b ≤ 4a-4} with {49/31 ≤ a ≤ 35/22, 2+a/5 ≤ b ≤ 3-3a/7}, which is the triangular region bounded by the tunings for 19, 22, and 31. This is the valid range. The nice tuning range is a quadralateral, with vertices, given in terms of frequency ratios rather than log base 2 or cents, [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)], [2, 3, 5, 225/32, 4096/375]]. The three verticies with all rational number values for the approximate 3 and 5 are not in the valid range, so that only the [2, 1620/539, 4/3*sqrt(14), 291600/41503, 44/15*sqrt(14)] tuning is valid and hence strict. Other examples of strict tunings are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the nice range.

In Tuning Continua and Keyboard Layouts in the premiere issue of Journal of Mathematics and Music, Milne, Sethares and Plamodon define valid tunings; they discuss nice tunings in X_System in the Open University’s repository.