Recoverability

Definition

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.

Segments

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.

The recoverability norm

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 multivector 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.

The rank one case

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... .

If W is the patent val for N-edo and ]W∧J[ > 600 cents, it can happen that the minimum value for recoverability relative error/simple badness, defined as ]V∧J[ for any N-edo val V, does not occur for V = W. Examples are 13-limit 12f, where ]12f∧J[ = 1217.949 is much smaller than ]12∧J[ = 3844.172, 5-limit 17c, ]17c∧J[ = 847.730 compared to ]17∧J[ = 1054.225, and 11-limit 27e, where ]27e∧J[ = 1169.472 is less than ]27∧J[ = 2405.855. Of course two possibilities can be close enough that both are plausible, as with 7-limit 34d at 1169.472 cents versus 34 at 1437.444 cents.

Complete searches for temperaments

By a complete search for regular temperaments is meant a search which is guaranteed to find all temperaments meeting certain specified conditions. Recoverability conditions provide one approach to these. The first segment of W∨2 consists of C(n-1, r-1) zeros, and the second segment of C(n-1, r) integers identical to the initial, 2 containing, segment of W. By beginning with such a multivector of integer coefficients, wedging with J, and rounding, we obtain a multivector which is a candidate for a p-limit rank r wedgie, defining a regular temperament. It will not in general be a wedgie, but all recoverable wedgies can be obtained in this way. Hence all that remains to do, as discussed in The wedgie, is to test if the multivector in question is actually a wedgie, and also if it passes any further conditions on complexity, error, or badness we wish to place on our list of wedgies.