About Higher Limit Just Intonation

Any interval in 5-limit Just Intonation can be represented as 2^i 3^j 5^k where i, j, k are any integers and where ^ represents power. For example, the interval 81/80 can be represented as 2^-4 3^4 5^-1. Similarly, any interval in 7-limit Just Intonation can be represented as 2^i 3^j 5^k 7^n where i, j, k, and n are any integers. In the most general situation there are m primes p1, p2, p3, ..., pm and m integers i1, i2, i3, ..., im. Then any interval in m-limit Just Intonation can be written p1^i1 p2^i2 p3^i3 ... pm^im This representation of Just Intonations is used in the TransFormSynth to define tuning continua such as the syntonic, the magic, the hanson, etc. This allows the same interval in two different tuning systems to be discussed in a unified and unambiguous way. It is also what allows the vertical sliders to move between a variety of familiar (and not-so-familiar) tuning systems such as the various equal temperaments and the various comma-tunings.