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 also consists of two segments, where 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. Also, W∧J contains three segments, the first consisting of basis elements with 2 in the product, the second consisting of basis elements which are products of odd primes where log₂(1) was involved in the calculation of the coefficients, and the third also with odd primes, where only logs of odd primes appear in the computation. Finally Ƹ = W - R also has two segments, where the first segment consists of zeros, and the second segment corresponds to the second segment of W∧J where the second segment of Ƹ has coefficients identical, up to sign, to the second segment 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 up to sign 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). When the rank r is greater than one, 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 rank 2 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. The rank 3 case, like the rank 2 case in the 5-limit, involves a single comma where the relative error is the size in cents of the comma; again, not a serious restriction. In the case of rank one, if W is an N-edo val, ie whose first coefficient is N. then W∨2 = N and (W∨2)∧J = NJ, so that rounding it gives the patent val for N. Hence, W is recoverable if and only if it is a patent val. However, ]W∧J[ < 600 cents is now a stringent condition for p>5, especially in higher prime limits. In the 5-limit we have 4, 5, 7, 8, 12, 15, 19, 22, 23, 26, 27, 31, 34 ... . The 7-limit grows somewhat restrictive: 19, 27, 31, 41, 49, 60, 68, 72, 80, 91, 99... . In the 11-limit we have 49, 72, 103, 239, 270, 342, 391, 414, 445, 494, 552, 612... . The 13-limit is already quite restrictive: 552, 954, 1133, 1236, 1506..., and the 17-limit starts off 4452, 5527, 7033, 7315, 9896... .
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 also consists of two segments, where 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. Also, W∧J contains three segments, the first consisting of basis elements with 2 in the product, the second consisting of basis elements which are products of odd primes where log₂(1) was involved in the calculation of the coefficients, and the third also with odd primes, where only logs of odd primes appear in the computation. Finally Ƹ = W - R also has two segments, where the first segment consists of zeros, and the second segment corresponds to the second segment of W∧J where the second segment of Ƹ has coefficients identical, up to sign, to the second segment 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 up to sign 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). When the rank r is greater than one, 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 rank 2 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. The rank 3 case, like the rank 2 case in the 5-limit, involves a single comma where the relative error is the size in cents of the comma; again, not a serious restriction.<br /> <br /> In the case of rank one, if W is an N-edo val, ie whose first coefficient is N. then W∨2 = N and (W∨2)∧J = NJ, so that rounding it gives the patent val for N. Hence, W is recoverable if and only if it is a patent val. However, ]W∧J[ < 600 cents is now a stringent condition for p>5, especially in higher prime limits. In the 5-limit we have 4, 5, 7, 8, 12, 15, 19, 22, 23, 26, 27, 31, 34 ... . The 7-limit grows somewhat restrictive: 19, 27, 31, 41, 49, 60, 68, 72, 80, 91, 99... . In the 11-limit we have 49, 72, 103, 239, 270, 342, 391, 414, 445, 494, 552, 612... . The 13-limit is already quite restrictive: 552, 954, 1133, 1236, 1506..., and the 17-limit starts off 4452, 5527, 7033, 7315, 9896... .</body></html>