Constrained tuning: Difference between revisions
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'''Constrained tunings''' are tuning [[optimization]] techniques using the constraint of some purely tuned intervals (i.e. [[eigenmonzo|unit eigenmonzos, or unchanged-intervals]]). '''CTE tuning''' ('''constrained Tenney-Euclidean tuning''') is the most typical instance. It has a more sophisticated variant, ''' | '''Constrained tunings''' are tuning [[optimization]] techniques using the constraint of some purely tuned intervals (i.e. [[eigenmonzo|unit eigenmonzos, or unchanged-intervals]]). '''CTE tuning''' ('''constrained Tenney-Euclidean tuning''') is the most typical instance. It has a more sophisticated variant, '''CWE tuning''' ('''constrained Weil-Euclidean tuning'''), a.k.a. '''KE tuning''' ('''Kees-Euclidean tuning'''). Both of these are special cases of the ''CTWE tuning''' ('''constrained Tenney-Weil-Euclidean tuning'''). These two tunings will be the focus of this article. Otherwise normed tunings can be defined and computed analogously. | ||
All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a {{w|least squares|least squares problem}}, CTE tuning can be viewed as an equality-constrained least squares problem. | All constrained tunings are standard temperament optimization problems. Specifically, as [[TE tuning]] can be viewed as a {{w|least squares|least squares problem}}, CTE tuning can be viewed as an equality-constrained least squares problem. | ||
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which is almost an analytical solution. Notice we introduced the vector of lagrange multipliers ''Λ'', with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be discarded. | which is almost an analytical solution. Notice we introduced the vector of lagrange multipliers ''Λ'', with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be discarded. | ||
== CTE tuning vs CTWE tuning == | == CTE tuning vs POTE tuning CWE tuning vs CTWE tuning == | ||
People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results. One good example is Blackwood, where the 4:5:6 chord is tuned to 0-386-720 cents, so that the error is no longer evenly divided between the 5/4, 6/5, and 3/2 in a way that was important historically - and which tunings like TE, TOP, etc seem to do for you automatically. The reasons for this are subtle, but as a result of this, as a result, historically, the POTE tuning was used instead, which tunes it to the much less lopsided (and approximately isoharmonic) 0-400-720 cents. People have also suggested using the Kees-Euclidean or KE tuning, also known as the constrained-Weil-Euclidean or CWE tuning. Here is a summary of the historical reasoning behind this. | |||
=== The Problem === | |||
The CTE tuning can be thought of as a modified TE tuning in which the weighting (in monzo space) on the 2/1 coordinate has been changed to 0, making it a kind of seminorm rather than a norm. As a result, all elements in the same octave-equivalence class are weighted identically: they are all given complexity equal to the '''representative''' in each equivalence class in which all factors of 2 have been removed. Thus 5/4 is given the same complexity as 5/1, 13/8 as 13/1, and so on. | |||
One criticism that has sometimes been brought up is to note the interval 13/1 is huge: it's four octaves large. We may not really care about 13/1 more than 13/8, or 15/1 more than 15/8, and so on. Instead, we often care most about intervals which are maybe within an octave or two at most in span. This can be viewed as a criticism of Tenney-weighting in general, perhaps, but it is has often been noted that the situation makes little difference there, | |||
as many of the alternative metrics suggested give similar results. For this reason, and because the Tenney and TE norms are easy to use, and also because not everyone has been convinced by this criticism to begin with, the use of Tenney-weighting has remained the standard. | |||
The problem becomes much more severe when we essentially assign these weightings to entire equivalence classes. In this situation, all of these small, intra-octave intervals essentially have their weightings scrambled relative to one another. Small intervals which used to be similarly-calibrated are now totally different: 16/13 is weighted much more strongly than 6/5; 15/8 and 6/5 are given equal weight; 5/4 is now much more important than 6/5, and so on. This is because some of these intervals happen to be octave-equivalent to prime numbers 3-4 octaves up, and others aren't. As a result, when we build simple triads out of these manageably-sized intervals, they tend to be tuned in a very lopsided fashion. | |||
This is one of the reasons why the tuning of 4:5:6 is so skewed in CTE Blackwood: 5/4 is prioritized more strongly than 6/5. This problem doesn't happen with the TE tuning: the extra degree of freedom in adjusting the octave, and different weights, tend to even this kind of thing out. TE Blackwood has 4:5:6 tuned to approximately 0-398-717 cents, which does seem to evenly split the error between the 5/4 and 6/5. We can see that something good about the way that TE tunes important triads has not quite translated to CTE. What has basically happened is that 1:3:5 is tuned as nicely as possible, at the cost of 4:5:6. | |||
The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 4:5:6 to 0-400-720, does much better in translating this useful property of TE to a pure-octave tuning: the relative sizes of all intervals are preserved. Thus, we retain these nice properties for small intervals, as well as small triads, etc. In fact, one notes that if one were to actually measure the tuning error on all triads, tetrads, etc, as well as dyads, we may very well get something closer to the POTE tuning than the CTE tuning. One also notes that "de-stretching" the POTE tuning is, to first order, approximately the same as stretching all chords in it "linearly," so that "isoharmonic," "proportional," "delta-rational" chords remain so after the transformation (approximately). | |||
Another way to think of it is that as POTE destretches the equave, it keeps the angle in the tuning space unchanged, and thus can be thought of as sacrificing multiplicative (typically very large) ratios for divisive (typically very small) ratios, whereas CTE sticks to the original design book of TE-optimality without worrying about that. | |||
Historically, there was also an observation that the POTE tuning can be thought of as an approximation to the CWE/KE tuning, which we will talk about below. | |||
=== Using the Weil Norm/Kees Expressibility === | |||
Another way to solve this problem is to actually go back to the original objection that we perhaps don't care about 13/1 as much as 13/8 - or at least, that we don't care about it that much if we have to assign it to the entire equivalence class. So, we can take this objection seriously and use a different norm to begin with. | |||
The Weil norm of <math>\max(n,d)</math> can be thought of as the average of Tenney Height and the interval's span, and so inherently does this: 5/1 and 5/4 are weighted equally, so that the psychoacoustic importance of the former and small manageable size of the latter balance out. We can then do a constrained optimization using the Weil norm, and if we are using the Weil-Euclidean norm, we get the **constrained Weil-Euclidean** or **CWE tuning**. The term **Kees-Euclidean** has also sometimes been used for this (although the term has occasionally been used to refer to a de-stretched Weil-Euclidean tuning instead). | |||
The term "Kees" is from Kees Van Prooijen, whos ''Kees-expressibility'' is essentially what you get if you remove factors of 2 from an interval and take its Weil norm. This is essentially the "odd-limit" associated to the interval. So, this view essentially states that, at least when we are in these sort of octave-equivalence-class situations, we ought to use the Weil norm/Kees expressibility rather than Tenney Height. | |||
Historically, there was a sort of convoluted line of reasoning leading to this same basic idea. The thought was that we could keep Tenney height, but instead choose something like the reduced-octave version of each interval as representatives, such as 5/4, 7/4, 13/8, etc. In essence, we want the version of each interval with minimal span, so we note that we basically want to set the 2's coordinate to whatever makes the span is as small as possible. Unfortunately, doing this leads to somewhat messy nonlinear behavior, as the set of representatives no longer forms a lattice, linear subspace, etc. | |||
The "good-enough" solution suggested on the tuning list is to instead just choose an '''idealized''' 2's coordinate, which is a real number, which instead makes the span equal to zero, which ought to be good enough. This is equivalent to placing the entire span, or rather its negation, into the 2's coordinate. This happens to be the same thing as just using the Weil-norm, which can be thought of as the L1 norm in an "augmented space" where we add the span as an extra coordinate. Regardless of if we remove factors of 2 and add a coordinate for the span, or put the span in the 2's coordinate, we clearly get the same thing. | |||
=== CTWE tuning == | |||
As mentioned above, if we constrain the equave to be pure, and look for the tuning map that is closest to the JIP using the WE norm, we get the CWE tuning, a.k.a. KE tuning. | |||
It has sometimes been noted that the Weil norm can give less-than-perfect results in other ways - for instance, it weights 13/8, 13/9, 13/10, 13/11, and 13/12 all equally. This doesn't seem to cause quite as much of a problem with the WE or KE tunings, or even the minimax Kees tuning, as it does with the minimax Weil tuning. So, one simple solution is to interpolate between the two, giving the '''Tenney-Weil-Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: k=0 is the TE norm, k=1 is the WE norm, and in between we have intermediate norms. | |||
=== Comparison === | |||
These tunings can be very different from each other. Take 7-limit meantone as an example. The POTE [[tuning map]] is a little bit flatter than [[quarter-comma meantone]], with all the primes tuned flat: | These tunings can be very different from each other. Take 7-limit meantone as an example. The POTE [[tuning map]] is a little bit flatter than [[quarter-comma meantone]], with all the primes tuned flat: | ||