About 5-limit Just Intonation

The Just Intonation scales are composed of intervals defined by ratios of small integers. When "small" is taken to mean "5", the resulting system is composed of intervals n/m where n and m can factored by the prime numbers 2, 3, and 5 (and no others). This is called 5-limit Just Intonation and the intervals are often listed by ordering them within a single octave. Here's one list:

Selected Intervals in 5-limit Just Intonation

1/1	the unison
16/15	the just minor second
9/8	the just major second
6/5	the just minor third
5/4	the just major third
4/3	the just fourth
3/2	the just fith
8/5	the just minor sixth
5/3	the just major sixth
16/9	the just minor seventh
15/8	the just major seventh
2/1	the octave

Observe that there are many numbers in this table larger than five. These can all be factored into just the three primes. For example, 16=2*2*2*2, 15=5*3, etc.

A complete list of all the possible intervals (even within a single octave) is impossible because there are an infinite number. The general principle is that any ratio 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 the operation of raising a number to a power (thus 2^3 means 2 multiplied by itself three times, 2*2*2, or 8). For example, the interval 27/20 can be represented as 2^-2 3^3 5^-1. As usual, negative powers correspond to factors in the demoninator. Thus 20 is factored as 2^2 5^1 but it appears in the denominator so the powers are negative.