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A Summary of the FJS

The Functional Just System (FJS) is a notation system used to extend traditional staff notation to be able to notate all of Just Intonation (JI). Like Helmholtz-Ellis notation or Johnston notation, the FJS does this by extending staff notation with a list of special accidentals. The FJS uses a deterministic process to convert any JI ratio uniquely into an interval class (M2, m3, A4, etc.) plus additional specific FJS accidentals.

The goals of the FJS are:

The core of the FJS is the FJS master algorithm, which takes any prime number greater than 3 as its input, and makes use of a constant known as the radius of tolerance (whose standard value is 65/63, the mediant of 33/32 and 32/31). It looks for the simplest approximation of that prime number closer than the radius of tolerance within the 3-limit (simplest means lowest absolute value of the fifth shift, i.e. the exponent of three). The difference between the target and the approximation is then recorded as the so-called formal comma of that prime, always in the original direction, i.e. with the prime number factor in the numerator.

Just Intonation is notated in the following way:

In the FJS, every prime is associated with two values: the fifth shift, and the formal comma. Both are useful to describe the FJS fully, but only one of them must be known for a given prime for both to be uniquely determined without having to use the FJS master algorithm or the radius of tolerance (cf. the formal description).

Below is the FJS master algorithm.

  1. Input the desired prime interval and call it p.
  2. Let k = 0.
  3. Consider the interval of k Pythagorean fifths and call it P.
  4. Is the difference between p and P less than the radius of tolerance?
  5. If so: k is the fifth shift. Output. End.
  6. If not: move to the next k in the sequence: (0, 1, −1, 2, −2, 3, −3, …) and repeat from step 3.

In step 4, the meaning of “difference” is the absolute value of the cent size of the difference, chosen in octaves of p and P to minimize this absolute difference. For details, see the crash course.