# 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 quadrilateral, 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 vertices 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.

Milne, Sethares and Plamodon define valid tunings in * Tuning Continua and Keyboard Layouts* in the premiere issue of

*Journal of Mathematics and Music*; they discuss nice tunings in

*in the Open University’s repository.*

**X_System**