Constrained tuning: Difference between revisions

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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.

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 give similar results. 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.

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 give similar results. 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 ===

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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.
-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 give similar results. 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.
+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 give similar results. 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 ===