Constrained tuning: Difference between revisions

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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, 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.

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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 defense ===

Anyone who performs tuning optimization has [[octave equivalence]] to unlearn. It is tempting to optimize for close-voiced chords such as 1-5/4-3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the original version being part of the [[chord of nature]]. It is thus important to recognize that 1-3-5 is the true basis of harmony and that 1-5/4-3/2 is but a simplification.

Real music is all about various open voicings, except for the archaic {{w|Alberti bass}} and the few other similar patterns. It should be noted that 13/1, dismissed as too wide in the section above, is still within the range of a full choir, not to mention a {{w|rock band}}, {{w|concert band}} or {{w|orchestra}}. {{w|Ludwig van Beethoven|Beethoven}}'s {{w|Symphony No. 3 (Beethoven)|''Symphony No. 3''}} opens with 1-2-5/2-4-5-6-8-10-12-16. Such a chord will be much overtempered, its tuning profile unreasonably squeezed and strained, if we set 1-5/4-3/2 as our target.

CTE blackwood does not try to approximate a delta-rational 1-5/4-3/2, and not even a delta-rational 1-3-5. This is also justifiable: since prime 5 is never involved in the comma that is tempered out, it only makes sense that it is tuned pure. And furthermore, any new prime added to the temperament is automatically tuned pure, as in JI. The dent in prime 3 does not spread to what it does not have to, unlike the schemes introduced below.

=== 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).

The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 1-5/4-3/2 to 0-400-720, translates the near-delta-rational 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.

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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 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.

@@ Line 334: / Line 334: @@
 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, 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.
@@ Line 346: / Line 346: @@
 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 defense ===
+Anyone who performs tuning optimization has [[octave equivalence]] to unlearn. It is tempting to optimize for close-voiced chords such as 1-5/4-3/2 without much consideration, since textbooks often present harmony in this way. The close-voiced chord, 1-5/4-3/2, is an octave-reduced version of 1-3-5, with the original version being part of the [[chord of nature]]. It is thus important to recognize that 1-3-5 is the true basis of harmony and that 1-5/4-3/2 is but a simplification.
+Real music is all about various open voicings, except for the archaic {{w|Alberti bass}} and the few other similar patterns. It should be noted that 13/1, dismissed as too wide in the section above, is still within the range of a full choir, not to mention a {{w|rock band}}, {{w|concert band}} or {{w|orchestra}}. {{w|Ludwig van Beethoven|Beethoven}}'s {{w|Symphony No. 3 (Beethoven)|''Symphony No. 3''}} opens with 1-2-5/2-4-5-6-8-10-12-16. Such a chord will be much overtempered, its tuning profile unreasonably squeezed and strained, if we set 1-5/4-3/2 as our target.
+CTE blackwood does not try to approximate a delta-rational 1-5/4-3/2, and not even a delta-rational 1-3-5. This is also justifiable: since prime 5 is never involved in the comma that is tempered out, it only makes sense that it is tuned pure. And furthermore, any new prime added to the temperament is automatically tuned pure, as in JI. The dent in prime 3 does not spread to what it does not have to, unlike the schemes introduced below.
 === 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).
+The POTE tuning, which simply "de-stretches" the TE tuning and tunes the 1-5/4-3/2 to 0-400-720, translates the near-delta-rational 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.
@@ Line 360: / Line 367: @@
 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.