Constrained tuning: Difference between revisions

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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 ~~be good enough~~. 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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 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 be good enough. 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 ===