Interior product

Given a rank-r temperament W and a comma m not tempered out by W, the interior product of W and m is the rank-(r-1) temperament W∨m which tempers out m in addition to all the commas that are tempered out by W (thus its codimension is one dimension higher than that of W).

Definition

The interior product is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or n-map, a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).

For example, suppose W = <<6 -7 -2 -25 -20 15||, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely |1 0 0 0> and |-1 1 1 -1>, then wedging them together gives the bimonzo ||1 1 -1 0 0 0>>. The dot product with W is <<6 -7 -2 -25 -20 15||1 1 -1 0 0 0>>, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0>, |-1 1 1 1>) = 1. The fact that the result is ∓1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.

If W is a multival of rank n and m is a monzo of the same prime limit p, then form a list of (n-1) tuples of primes less than or equal to p in alphabetical order. Taking these in order, the ith element of W∨m, which we may also write W∨q where q is the rational number with monzo m, will be W(s1, s2, s3 ... s_(n-1), q), where [s1, s2, ..., s_(n-1)] is the ith tuple on the list of (n-1)-tuples of primes. This will result in W∨m, a multival of rank n-1. For instance, let Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19|||, the wedgie for 11-limit marvel temperament. To find Marv∨441/440, we form the list [[2, 3], [2, 5], [2, 7], [2, 11], [3, 5], [3, 7], [3, 11], [5, 7], [5, 11], [7, 11]]. The first element of Marv∨441/440 will be Marv(2, 3, 441/440), the second element Marv(2, 5, 441/440) and so on down to the last element, Marv(7, 11, 441/440). This gives us <<6 -7 -2 15 -25 -20 3 15 59 49||, which is the wedgie for 11-limit miracle. The interior product has added a comma to marvel to produce miracle.

If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign. We can also define the interior product of W with a multimonzo M of rank r < n, by forming a list of (n-r)-tuples of primes in alphabetical order, wedging these together with M, and taking the dot product with W to get a coefficient of W∨M.

Applications

One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if and only if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.

Another application is the use of the interior product to define the intervals of the abstract regular temperament given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r primes, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T×U (the matrix product), which can be taken to be an (r-1)-multimonzo. Then for any (r-1)-multival W∨q in the abstract regular temperament, the dot product (W∨q)∙V gives the tuning of W∨q. It should be noted that V with this property is underdetermined, so that many possible multimonzo vectors can be used to the same effect. An alternative approach is to hermite reduce the matrix [W∨2, W∨3, ... W∨p] and then solve for the linear combination which gives the desired tuning from the dot product. This makes for a simpler result; for example the dot product of Meantone∨q, where "Meantone" is the 7-limit wedgie, with |1200+300*log2(5), -1200, 0, 0> gives the value in cents of the quarter-comma meantone tuning of the interval denoted by Meantone∨q.

The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit marvel temperament. Then Marv∨45/44 = <<4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = <<-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = <<11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = <<-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = <<5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = <<6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = <<-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = <<6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨2200/2187 = <<-1 8 14 -23 15 25 -33 10, -81 -113||, garibaldi, Marv∨9801/9800 = <<-12 2 -20 6 31 2 51 -52 7 86||, wizard.

The interior product is also useful in finding the temperament mapping given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the mapping. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [<0 0 -1 -2 3|, <0 1 0 2 -1|, <0 2 -2 0 4|, <0 -3 1 -4 0|, <-1 0 0 5 -12|, <-2 0 -5 0 -9|, <3 0 12 9 0|, <2 5 0 0 19|, <-1 -12 0 -19 0|, <4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|], the normal val list for 11-limit marvel. In practice this method nearly always suffices.