Constrained tuning: Difference between revisions

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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 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.

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 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 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.