Tuning ranges of regular temperaments

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

Diamond tuning ranges

Andrew Milne, Bill Sethares and James Plamondon defined some important tuning ranges. Their "valid" range was defined in Tuning Continua and Keyboard Layouts in the Journal of Mathematics and Music[1]; according to Milne, this tuning range was Sethares's contribution. Their "purer" range was discussed in the technical report X_System in the Open University’s repository.

In 2014, these tuning ranges were discussed by Milne, Gene Ward Smith, and others. The "valid" range was proposed names such as "lax", "monotone", and "normal", while the "purer" range was proposed to be named "strict", and the combination proposed to be named "nice". After some churn (in the edit history of this page), valid/lax/monotone/normal became "valid", while purer/strict became "nice", and the combination became "strict". In other words, "nice" and "strict" for some reason got switched, and for some time and to some extent on the wiki and in the regular temperament community that's how they stuck.

In May 2021 Milne agreed with a community effort to revise the names to be more specific, descriptive, and closer to their original meaning. So the original "valid" became "diamond monotone", the original "purer" become "diamond tradeoff", and the combination of these two was left unnamed.

Examples

5-limit meantone

To illustrate the diamond tuning ranges, 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.

Diamond tradeoff

To find the range of diamond tradeoff 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 diamond tradeoff 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 diamond tradeoff 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.

Diamond monotone

To find the range of diamond monotone 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 diamond monotone. 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 diamond monotone 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 diamond monotone 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 diamond monotone meantone tuning, as is p = 12000 cents and g = 5000 cents.

Diamond tradeoff and diamond monotone

In this particular case all of the diamond tradeoff tunings are also diamond monotone, so the diamond tradeoff range is entirely inside the diamond monotone range.

11-limit marvel

Diamond monotone

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 diamond monotone range.

Diamond tradeoff

The diamond tradeoff 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]].

Diamond tradeoff and diamond monotone

The three vertices with entirely rational number values for the approximations of 3 and 5 are not in the diamond monotone range, so only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is both diamond tradeoff and diamond monotone. Other examples of tunings that are both diamond tradeoff and diamond monotone are 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the diamond tradeoff range.

Other tuning ranges

The diamond tuning ranges, though they have historical momentum, do not preclude definition of other validity ranges for the tuning of temperaments. The topic of tuning ranges is relatively subjective. Milne himself has described the diamond tuning ranges as "convenient mathematical fictions", and proposed that the reality would be to define some sort of empirically obtained range of tunings over which a sample of participants can correctly identify that tuning's intervals in the way prescribed by the mapping. But realistically, that is an almost impossible question to even ask of participants, and relies upon all sorts of a priori assumptions about categorizations of intervals by their ratio, which is quite possibly an entirely bogus notion.

Paul Erlich has proposed other tuning ranges, such as the set of regular tunings in which the temperament has up to 2×, 5×, 10× etc. its optimal damage under some metric (such as KE), or a set of absolute cutoffs on damage applied across all temperaments, though there could be no objective value for such cutoffs that would be both amenable to the entire community as well as useful for the entire set of regular temperaments (including extreme cases like macrotemperaments and microtemperaments).

Dave Keenan has proposed the range over which there is not a "better" temperament that maps the generators differently, for some definition of "better", likely taking into account both error and complexity.

Others have proposed the step ratio spectrum as a helpful way of thinking about tuning ranges.

References

  1. Andrew Milne, William Sethares & James Plamondon (2008) Tuning continua and keyboard layouts, Journal of Mathematics and Music, 2:1, 1-19, DOI: 10.1080/17459730701828677