Intro To Mappings
When one val just won't do...
A val maps JI onto one chain of generators, or relates that generator chain back to JI. However, many temperaments incorporate more than one of chain of generators. The familiar meantone temperament is an example, as it requires two: the fifth and the octave. However, vals only relate a single generator chain to JI. If we want to evolve out of the realm of isolated EDOs and consider higher-dimensional temperaments in their full glory, we're going to have to raise the bar on what vals can do for us. Luckily, the mathematics to do so is simple enough.
A temperament's "rank" denotes how many independent chains of generators exist within the temperament. This is a mathematical term that's borrowed from the field of group theory. It can also be viewed as the "dimensionality" of the temperament.
- An equal temperament is rank 1, as it exists in its entirety as a stack of one single generator.
- Temperaments which consist of two generators, or more commonly a "period" and a generator, are rank 2. Meantone is a good example, as its separate chain of fifths and chain of octaves constitute two independent generator chains
- Temperaments which consist of three generators, or more commonly a period and two generators, are rank 3. 5-limit JI, while not being a "temperament" in the traditional sense, would nonetheless be considered rank 3, as its three generators are 2/1, 3/1, and 5/1 (or 2/1, 3/2, and 5/4 if you'd like).
- 7-limit JI would be rank 4, etc.
A single val in isolation only maps JI onto temperaments that are rank 1. For us to deal with temperaments of rank > 1, we simply need to use more than one val. In general, the number of vals that it requires to fully map a temperament is equal to the temperament's rank.
At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:
<a b c] - period
<d e f] - generator
The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.
When mapping a prime in JI onto a rank 2 temperament, one must think about how many steps of each type of generator it takes to reach the final tempered prime interval. For an example, we'll look at meantone temperament, and we'll start by mapping 2/1. We'll assume that the period is 2/1, and the generator is 3/2. 2/1 maps to one period and zero generators, and hence we arrive at
<1 _ _]
<0 _ _]
3/1 is slightly more complicated - it requires one step along the 2/1 period chain, plus one step along the 3/2 generator chain, to get to 3/1. This is represented by the following mapping:
<1 1 _]
<0 1 _]
5/1 is simpler - we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:
<1 1 0]
<0 1 4]
This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.
Change of Basis
In the above example, we wrote out the meantone mapping matrix from the perspective of the two generators 2/1 and 3/2. What if we instead wanted to treat the generators as being 2/1 and 4/3? Or, what if we wanted to write it out from the perspective that the generators are 2/1 and 3/1? All of these will lead to different val lists, but will still represent the same temperament.
In the language of mathematics, you've simply changed the basis for your temperament, and the resulting temperamental spaces will be isomorphic to one another. This is just a fancy way of stating that they're the same temperament.
If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:
<1 2 4]
<0 -1 -4]
This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods -minus- a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's DOWN, plus four octaves - it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.
If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):
<1 0 -4]
<0 1 4]