  Musical frequencies in Just Intonation are comprised of rational numbers. The
structure of rational numbers is determined by prime factorisations. Just
Intonation frequencies can be split into two components. The larger component
uses only integer powers of the first two primes, 2 and 3. The smaller
component decomposes into a series of microtonal adjustments, one for each
prime number 5 and above present in the original frequency. The larger 3-limit
component can be notated using scientific pitch notation modified to use
Pythagorean tuning. The microtonal adjustments can be notated using rational
commas which are built up from prime commas. This gives a notation system for
the whole of free-JI, called Rational Comma Notation. RCN is compact since all
microtonal adjustments can be represented by a single notational unit based on
a rational number. RCN has different versions depending on the choice of
algorithm to assign a prime comma to each prime number. Two existing algorithms
SAG and KG2 are found in the literature. A novel algorithm DR is developed
based on discussion of mathematical and musical criteria for algorithm design.
Results for DR are presented for primes below 1400. Some observations are made
about these results and their applications, including shorthand notation and
pitch class lattices. Results for DR are compared with those for SAG and KG2.
Translation is possible between any two free-JI notations and any two versions
of RCN since they all represent the same underlying set of rational numbers.
