Mapping to lattice

Many temperaments documented on the wiki are accompanied with a mapping to lattice. These use the same extended bra-ket notation as temperament mappings but provide quite different information: they give the coordinates of the primes in a tempered lattice. The lattice for which they give coordinates does not necessarily correspond with the mapping otherwise provided.

Example

Consider supermagic. Its mapping to lattice is given as [⟨0 0 -1 3] ⟨0 1 1 0]}. We can read this in four vertical slices; the coordinate (0,0) is for prime 2, (0,1) is for prime 3, (-1,1) is for prime 5, and (3,0) is for prime 7.

Note that supermagic is a rank-3 temperament, meaning it has three generators. Or said another way, it has one period and two (other) generators. This corresponds with the fact that it has three generator maps — or rows — in its mapping matrix, which is given as [⟨1 0 0 5] ⟨0 1 0 3] ⟨0 0 1 -3]}. The reason the mapping to lattice has only two rows while the mapping has three rows is because octave equivalence has been assumed in the lattice. So the first generator, the special one, the period, which in this case is equal to the octave, is not visualized on this lattice; we use one axis for the first generator, and the other axis of the second generator.

The octave equivalence also explains why prime 2 is found at (0,0).

It should not be surprising that prime 3 is found at a coordinate that is only one step away from (0,0): because reading the column for prime 3 from the mapping, we get a generator-count vector of [0 1 0}, meaning that one of the generators is an approximation of prime 3.

However, we may be a bit confused by the next coordinate. Why is the coordinate of prime 5 one step each in both directions, despite the fact that the mapping gives a vector of [0 0 1} to prime 5, similar to the one it gives for prime 3? Well, this is simply evidence that the mapping to lattice does not correspond to the mapping given for this temperament. It means that on this particular lattice, we get to prime 5 by first moving one step of 3/1, and then one step of whatever else it takes to get from there to 5/1 (which is 5/3). In other words, this mapping to lattice appears to correspond to a different mapping (for the same temperament): [⟨1 0 0 5] ⟨0 1 1 0] ⟨0 0 -1 3]}. The difference here is that the third row of the mapping has been added to the second row, and then the third row has been negated. So now if we read off prime 5's vector, we get [0 1 -1}, meaning that the approximation of prime 5 is arrived at by moving up one of the first generator, and down one of the second generator. So the second generator is actually not a 5/1, but a 6/5. This is because moving down by that is multiplying by 5/6, and if we're starting at 3/2 (not 3/1, because of octave reduction), we need to go by 5/6 to get to 5/4 (again, not 5/1, because of octave reduction). Actually it may be the case that the mapping this lattice is for is [⟨1 1 1 5] ⟨0 1 1 0] ⟨0 0 -1 3]}, so that the second generator is understood to be an approximation of 3/2 instead of 3/1; we can't really know, because of the octave equivalence, but it seems likely that if it was important to move down from 3 to 5 within an octave under octave equivalence, that the author of this mapping would want all of the generators to be octave reduced.

Finally we have the coordinate for prime 7. Another simple one. It's just a move by 3 of the second generator. It is worth noting, though, that the generators of the mapping have their order swapped in the mapping to lattice. The first generator map (row) of the mapping corresponds with the second coordinate of the mapping to lattice, while the second generator map of the mapping corresponds with the first coordinate of the mapping to lattice. This is perhaps because the the mapping to lattice coordinates are sorted ascending by generator size, while the mapping is often sorted descending by generator size (because it's common for generators to be reduced within the size of previous generators; see https://en.xen.wiki/w/Normal_lists#Equave-reduced_generator_form and https://en.xen.wiki/w/Normal_lists#Minimal_generator_form).

So here is a small sample of this particular tempered lattice for this temperament, just enough to show what the mapping to lattice is indicating:

		g₁
		-1	0	1	2	3
g₂	0		2/1			7/1
	1	5/1	3/1