Constrained tuning: Difference between revisions

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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.

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. But, this could sometimes be an issue.

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. This also gives a '''Constrained Tenney-Weil-Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE.

=== Comparison ===

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 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.
+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. But, this could sometimes be an issue.
+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. This also gives a '''Constrained Tenney-Weil-Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE.
 === Comparison ===