A brief introduction to Regular Temperament Theory
by Dave Keenan
We like to have different flavors of consonance, which correspond to different simple ratios. We want to find tunings — sets of notes — that will give us enough consonances in enough flavors, without being too complex, and without having errors that are too large. The complexity of a tuning is partly about the number of notes per octave, and partly about the number of different step sizes.
Regular Temperament Theory is a powerful tool to aid us in finding such tunings. It observes that the problem of approximating many simple ratios can be reduced to one of approximating a few small prime numbers. It then generates all of its tunings by stacking (both up and down) a small number of intervals which are called the generators. In one extreme, the generators are the small prime numbers themselves, giving just intonation (JI) as a lattice, having zero errors but high complexity. In another extreme, there is a single small generator whose iterations must approximate all the desired primes, giving an equal temperament (ET), having low complexity but high errors.
These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "meantone" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).
When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.
The defining thing about a regular temperament is the the count of each generator required to approximate each prime number. This is called the temperament's mapping, and can be represented as a matrix.
We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and many resulting temperaments named and catalogued.
- where "approximating" means something like "having errors less than 30 cents".
- where "simple" means something like "involving a pair of integers whose product is less than 1000".
- or our homegrown variety of multilinear algebra, which includes two copies of exterior algebra and uses an extended bra-ket notation, for which, unfortunately, tools are not readily available.
- Strictly speaking, it is the wedge product of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament. Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators, and therefore a canonical form for the mapping matrix. For example, the first generator, called the period, can be given as a unit fraction of the lowest prime, usually the octave (2). The second generator can be given as a unit fraction of the simplest ratio that allows it to be smaller than the period, and so on.