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 set of all tunings with 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. This is the nice tuning range.
• We may define another tuning range by requiring that 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). This is 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]. 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 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.

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

Example: 5-limit meantone

To illustrate the above definitions, let's consider 5-limit meantone. This is a 5-limit temperament so the tonality diamond is {1, 3, 5, 1/3, 5/3, 1/5, 3/5}, or octave reduced, {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. It is also a rank-2 temperament, so any particular tuning can be specified by tunings of the two generators, which we take to be the tempered 2/1 and 4/3.

To find the range of "nice" tunings, we fix one eigenmonzo as 2/1 and iterate through the 5-limit tonality diamond for what to use for the other eigenmonzo. (If the rank were more than 2, we would be iterating over subsets of the tonality diamond of size r - 1, but since r = 2 we are iterating over single ratios.)

• 1/1 is the 0-vector monzo and so it is always a (trivial) eigenmonzo of any tuning. We have to use a non-1/1 interval as the other eigenmonzo besides 2/1 to define a tuning.
• If 4/3 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean.
• If 3/2 is the eigenmonzo, that is equivalent to 4/3 being an eigenmonzo because if 2/1 and 4/3 are tuned pure, then 3/2 is automatically tuned pure also.
• If 5/4 is the eigenmonzo, then 5/1 is also tuned pure so the fifth is 5^(1/4) and the generator is 2/5^(1/4). Therefore the tuning is [2/1, 2/5^(1/4)], or quarter-comma meantone.
• If 8/5 is the eigenmonzo, that's equivalent to 5/4 being the eigenmonzo and leads to the same tuning.
• If 6/5 is the eigenmonzo, then 12/5 is also tuned pure so the fourth (the generator) is (12/5)^(1/3). Therefore the tuning is [2/1, (12/5)^(1/3)], or third-comma meantone.
• If 5/3 is the eigenmonzo, that's equivalent to 6/3 being the eigenmonzo.

These lead to three distinct tunings:

• [2/1, 4/3] or Pythagorean - 4/3 and 3/2 are pure
• [2/1, 2/5^(1/4)], or quarter-comma meantone - 5/4 and 8/5 are pure
• [2/1, (12/5)^(1/3)], or third-comma meantone - 6/5 and 5/3 are pure

These three are the possible extreme points of the "nice" tuning range, so to describe the whole range, we must take their convex hull. In this case it is easy because they are all collinear, and Pythagorean and third-comma are the two endpoints of the line segment. So the "nice" tuning range of 5-limit meantone consists of exactly those tunings with a pure 2/1 as the period and anywhere from 4/3 to (12/5)^(1/3) (498.045 to 505.214 in cents) as the generator.

To find the range of "valid" tunings, we need all the steps in between consecutive members of the tonality diamond to be positive. So 6/5, 25/24, 16/15, and 9/8 must all be positive for the tuning to be "valid". If we denote the octave period by p and the perfect fourth generator by g, this yields the equations:

• tempered 6/5 = 3g - p > 0
• tempered 25/24 = 3p - 7g > 0
• tempered 16/15 = 5g - 2p > 0
• tempered 9/8 = p - 2g > 0

These are all homogenous equations, so we can divide through by p and rearrange to get restrictions on the ratio g/p:

• g/p > 1/3
• g/p < 3/7
• g/p > 2/5
• g/p < 1/2

Of these it can be seen that the first and last are redundant, and the overall "valid" tuning range can be summarized as 2/5 < g/p < 3/7, in other words all meantone tunings between 5edo and 7edo (480 to 514.286 cents).

Note that, since the definition of "valid" only depends on the ordering of intervals and not on their absolute size, in theory any amount of stretching or compression is allowed. For example, p = 12 cents and g = 5 cents is technically a "valid" meantone tuning, as is p = 12000 cents and g = 5000 cents.

The "strict" tuning range includes those that are both "nice" and "valid", but in this particular case all of the "nice" tunings are also "valid", so the "strict" range is identical to the "nice" range.