Majestazic system

The Majestazic system ^{[idiosyncratic term]} is an elementary-algebra description for regular temperaments. It describes them in terms of using different numbers in place of the primes, or more informally, "redefining" the primes, to satisfy an equation. For example, meantone can be described as follows: [3]⁴ = [5] × [2]⁴. Any values can be used for [3], [5], and [2] as long as they satisfy this equation. Quarter-comma meantone, for example, keeps [2] and [5] equal to 2 and 5, while setting [3] to 2×5^1/4. 12edo, when taken as a meantone temperament, sets [2] equal to 2, [3] equal to 2^19/12, and [5] equal to 2^7/3.

This allows any arbitrary temperament to be described in terms of an equation (more precisely, a system of multivariate binomial equations - i.e. one term on each side of the = sign, and only integer exponents) where the variables are the redefined primes. For example, the temperament described by [2]⁷ = [5]³ tempers out 128/125.

A temperament can be described by a system of equations to define it more precisely, for example the temperament {[3]⁴ = [5] × [2]⁴, [2] = 2, [2]¹⁹ = [3]¹²} has precise values for all three primes, because it is a system of 3 equations. Specifically, it is 12edo in the 5-limit.

The rank of a temperament, in this case, is how many degrees of freedom the system of equations has. A particular tuning can be described by defining one of the variables in the equation, which gives the answer to the other. A precise meantone tuning can be described by defining two variables: for example, defining [2] and [5] (i.e. the octave and the major third) gives you the value of [3].