Prime interval
A prime interval or prime harmonic is a harmonic which as a ratio of frequencies is a prime number; that is, a number such as 2, 3, 5, 7, 11, … which is divisible only by itself and 1.
Any interval of just intonation (JI) can be expressed in terms of a product of prime intervals, allowing us to decompose a complex JI interval into simpler parts. A prime interval itself cannot be expressed by other prime intervals, so no prime intervals are redundant for reconstructing the entirety of JI. For those reasons and for the fact that prime intervals occur in harmonic series, they form a very important basis (literally and mathematically) for JI.
For example, the octave is a prime interval whereas the intervals 5/3 or even 1/1 are not. In traditional ratio notation, the prime intervals are 2/1, 3/1, 5/1, 7/1, 11/1 etc.
The monzo notation of each prime interval consists of all-zeros except for a single unity entry: (2: [1⟩, 3: [0 1⟩, 5: [0 0 1⟩, 7: [0 0 0 1⟩, 11: [0 0 0 0 1⟩, …)
The opposite of a prime interval is a highly composite interval.
Individual pages
See Category: Prime harmonics.