Recoverability
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Original Wikitext content:
Suppose W is a p-limit rank r wedgie, and so of dimension n = π(p). Let R be the p-limit rank r multivector (W∨2)∧J, where J = <1 log₂(3) log₂(5) ... log₂(p)|. Then if rounding the coefficients of R to the nearest integer gives W, W is //recoverable//. W∨2 consists of two segments. The first segment consists of coefficients for basis elements (which are (r-1)-fold wedge products of primes) containing 2 in the product. Since we are examining W∨2, these coefficients are all identically 0. The second segment consists of the initial segment of W itself, being all coefficients for basis elements in W containing 2. Then R = (W∨2)∧J consists of three segments. The first segment consists of coefficients for basis elements with 2 in the product. Because of the zeros in the initial segment of W∨2, these coefficients are identical to the initial segment of W, meaning the coefficients of those basis elements with 2 in the product. There is then a segment where log₂(2) plays a role, and a final segment where only logs of odd primes are involved in the calculation. It follows that Ƹ = W - R also has three segments. The first segment consists of zeros, the second segment corresponds to the second segment of R, and the third segment to the third segment of R. The second segment of Ƹ has coefficients identical to the coefficients of W∧J. Assuming as we do throughout this article that multivals are expressed in unweighted coordinates, we may define a norm on them by the L∞ norm, so that ]M[ equals the maximum of the absolute values of the coefficients of the multival M in unweighted coordinates. Then W is recoverable precisely when ]Ƹ[ < 1/2. Since the coefficients of Ƹ are either 0 or identical to some of the coefficients of W∧J, it follows that ]Ƹ[ ≤ ]W∧J[ . But ]W∧J[ is a measure of relative error, hence W is recoverable if relative error is less than 1/2, or 600 cents if we renormalize by multiplying by 1200. This is a very loose restriction on temperaments, and it can be argued it includes all temperaments of any interest, the only debatable cases being very marginal temperaments of high error. Examples of relative error are, in the 5-limit, father (tempering out 16/15) at 111.731 cents, and bug (tempering out 27/25) at 133.238 cents, but even tempering out 4/3 (498.0450 cents) is recoverable. In the 7-limit, even ternary (<<0 0 3 0 5 7||) at 617.884 cents is nonetheless recoverable, and brutus (<<1 2 4 1 4 4||) at 561.006 cents of course presents no problem.
Original HTML content:
<html><head><title>Recoverability</title></head><body>Suppose W is a p-limit rank r wedgie, and so of dimension n = π(p). Let R be the p-limit rank r multivector (W∨2)∧J, where J = <1 log₂(3) log₂(5) ... log₂(p)|. Then if rounding the coefficients of R to the nearest integer gives W, W is <em>recoverable</em>. <br /> <br /> W∨2 consists of two segments. The first segment consists of coefficients for basis elements (which are (r-1)-fold wedge products of primes) containing 2 in the product. Since we are examining W∨2, these coefficients are all identically 0. The second segment consists of the initial segment of W itself, being all coefficients for basis elements in W containing 2. Then R = (W∨2)∧J consists of three segments. The first segment consists of coefficients for basis elements with 2 in the product. Because of the zeros in the initial segment of W∨2, these coefficients are identical to the initial segment of W, meaning the coefficients of those basis elements with 2 in the product. There is then a segment where log₂(2) plays a role, and a final segment where only logs of odd primes are involved in the calculation. It follows that Ƹ = W - R also has three segments. The first segment consists of zeros, the second segment corresponds to the second segment of R, and the third segment to the third segment of R. The second segment of Ƹ has coefficients identical to the coefficients of W∧J.<br /> <br /> Assuming as we do throughout this article that multivals are expressed in unweighted coordinates, we may define a norm on them by the L∞ norm, so that ]M[ equals the maximum of the absolute values of the coefficients of the multival M in unweighted coordinates. Then W is recoverable precisely when ]Ƹ[ < 1/2. Since the coefficients of Ƹ are either 0 or identical to some of the coefficients of W∧J, it follows that ]Ƹ[ ≤ ]W∧J[ . But ]W∧J[ is a measure of relative error, hence W is recoverable if relative error is less than 1/2, or 600 cents if we renormalize by multiplying by 1200. This is a very loose restriction on temperaments, and it can be argued it includes all temperaments of any interest, the only debatable cases being very marginal temperaments of high error. Examples of relative error are, in the 5-limit, father (tempering out 16/15) at 111.731 cents, and bug (tempering out 27/25) at 133.238 cents, but even tempering out 4/3 (498.0450 cents) is recoverable. In the 7-limit, even ternary (<<0 0 3 0 5 7||) at 617.884 cents is nonetheless recoverable, and brutus (<<1 2 4 1 4 4||) at 561.006 cents of course presents no problem.</body></html>