Wedge-product calculations are useful in tuning theory in two ways:
Both multimonzos and wedgies show common traits shared by different tunings belonging to the same family, and thus are a great aid in tuning taxonomy.
[how to calculate a wedge-product:]
Lets first take the simplest case worth considering. The wedge product of two 3-limit (2D) vectors.
[a1 a2> ^ [b1 b2>
The procedure is to first list every product of a coefficient from A with a coefficient from B, i.e. their ordinary scalar products. So with 2 coefficients in each there will be 2x2 = 4 products to consider,
a1*b1, a1*b2, a2*b1, a2*b2.
As you calculate each product, combine the indices of the two coefficients to make a compound index for it. It is important to keep the indices in their original order at this stage. So we have
product index a1*b1 11 a1*b2 12 a2*b1 21 a2*b2 22
There are certain rules about what to to with each product now, depending on its compound index. There are 3 possibilities:
product index a1*b2 12 -a2*b1 12
Now find any products that have the same index and add them together. So we have only
-- product -- index a1*b2 - a2*b1 12.
Now list all these sums in alphabetical order of their indices, inside as many brackets as the sum of the number of brackets in the two arguments, and pointing in the same direction. The wedge product is only defined for values having their brackets pointing the same way.
So our answer is
Now lets try something more messy. A 7-limit (4D) vector wedged with a 7-limit bivector. This might represent combining a third comma with two that have already been combined, as an intermediate result on the way to finding the ET mapping where these all vanish.
[a1 a2 a3 a4> ^ [[b12 b13 b14 b23 b24 b34>>
We first make the list of products of all pairs, with their compound indices.
product index a1*b12 112 a1*b13 113 a1*b14 114 a1*b23 123 a1*b24 124 a1*b34 134 a2*b12 212 a2*b13 213 a2*b14 214 a2*b23 223 a2*b24 224 a2*b34 234 a3*b12 312 a3*b13 313 a3*b14 314 a3*b23 323 a3*b24 324 a3*b34 334 a4*b12 412 a4*b13 413 a4*b14 414 a4*b23 423 a4*b24 424 a4*b34 434
Now we get rid of all those with two digits the same. Of course once you've got the idea, you wouldn't even bother writing them down in the first place. This leaves.
product index left-and-larger count a1*b23 123 a1*b24 124 a1*b34 134 a2*b13 213 1 a2*b14 214 1 a2*b34 234 a3*b12 312 2 a3*b14 314 1 a3*b24 324 1 a4*b12 412 2 a4*b13 413 2 a4*b23 423 2
And we do the left-and-larger counts on the indices that aren't already in alphabetical order (shown above), and negate the product if this is odd. And we end up with:
product index a1*b23 123 a1*b24 124 a1*b34 134 -a2*b13 123 -a2*b14 124 a2*b34 234 a3*b12 123 -a3*b14 134 -a3*b24 234 a4*b12 124 a4*b13 134 a4*b23 234
Now we sum the products having the same index.
------- product -------- index a1*b23 + a3*b12 - a2*b13 123 a1*b24 - a2*b14 + a4*b12 124 a1*b34 - a3*b14 + a4*b13 134 a2*b34 - a3*b24 + a4*b23 234
Now we list them in alphabetical (also numerical) order of index inside the correct number of brackets.
[[[a1*b23+a3*b12-a2*b13 a1*b24-a2*b14+a4*b12 a1*b34-a3*b14+a4*b13 a2*b34-a3*b24+a4*b23>>>