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Musical frequencies in Just Intonation are comprised of rational numbers. Thestructure of rational numbers is determined by prime factorisations. JustIntonation frequencies can be split into two components. The larger componentuses only integer powers of the first two primes, 2 and 3. The smallercomponent decomposes into a series of microtonal adjustments, one for eachprime number 5 and above present in the original frequency. The larger 3-limitcomponent can be notated using scientific pitch notation modified to usePythagorean tuning. The microtonal adjustments can be notated using rationalcommas which are built up from prime commas. This gives a notation system forthe whole of free-JI, called Rational Comma Notation. RCN is compact since allmicrotonal adjustments can be represented by a single notational unit based ona rational number. RCN has different versions depending on the choice ofalgorithm to assign a prime comma to each prime number. Two existing algorithmsSAG and KG2 are found in the literature. A novel algorithm DR is developedbased on discussion of mathematical and musical criteria for algorithm design.Results for DR are presented for primes below 1400. Some observations are madeabout these results and their applications, including shorthand notation andpitch 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 versionsof RCN since they all represent the same underlying set of rational numbers.

An algorithm to assign musical prime commas to every prime number and construct a universal and compact free Just Intonation musical notation