Constrained tuning: Difference between revisions
→CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning: I'll try to find out the thread that suits a reference for the claim. For now I'm labeling it unsourced; style |
→CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning: revise the tone and note the assumptions in the hope to be more objective |
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== CTE tuning vs POTE tuning vs CWE tuning vs CTWE tuning == | == CTE tuning vs POTE tuning vs 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.{{citation needed}} One good example is | 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.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0-386-720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle. | ||
This sort of thing was important historically when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property, | This sort of thing was important historically when looking at optimal tunings for meantone, and is ultimately the motivation for advanced tuning methods such as TOP, TE, etc to begin with. Thus, if our goal is to extend this principle in an elegant way to all intervals (and hopefully, triads and large chords), it would seem to defeat the purpose if we use a tuning optimization that doesn't also have this property, | ||
As a result of this | As a result of this, historically, the POTE tuning was used instead, which tunes it to a result that is approximately [[delta-rational]] 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 math involved and the historical reasoning behind this. | ||
=== The problem === | === 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. | 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 | One criticism that has sometimes been brought up is to note the interval 13/1 nearly four octaves wide, which is considered very large in certain genres of music. In these cases, 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. | ||
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 | The phenomenon becomes much more evident 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 compact triads out of these close-voiced intervals, they tend to be tuned in a very lopsided fashion. | ||
This is one of the reasons why the tuning of 4 | This is one of the reasons why the tuning of 1-5/4-3/2 is so skewed in CTE blackwood. 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 1-5/4-3/2 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 compact triads has not quite translated to CTE. Another way to look at this situation is that with CTE, 5/4 is prioritized more strongly than 6/5, and also 1-3-5 is tuned as nicely as possible, instead of 1-5/4-3/2. | ||
The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 4 | === Using destretch === | ||
The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 1-5/4-3/2 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. | 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. | ||
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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 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. | 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. | 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. | ||
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<math>\langle \begin{matrix} 1200.000 & 1920.000 & 2786.314 \end{matrix} ]</math> | <math>\langle \begin{matrix} 1200.000 & 1920.000 & 2786.314 \end{matrix} ]</math> | ||
Since prime 5 is not involved in the comma to begin with, it is understandable that it is tuned pure as in 5-limit JI. This, as mentioned above, leads to very lopsided behavior for compact chords like 1-5/4-3/2. Note that the tunings for KE and POTE distribute the error between 5/4 and 6/5 relatively evenly; both are very close to the delta-rational 0-397-720. The CTE tuning, on the other hand, has that chord tuned to 0-386-720, so that all of the error is on the 6/5 at about 18 cents sharp. | |||
== Special constraint == | == Special constraint == | ||