Ring number

Every edo has a ring number, which equals the GCD of the edostepspans of the octave and the fifth, or more generally the equave and the generator.

Examples

12-edo's best approximation of 3/2 is 7\12. Since 7 and 12 are co-prime, 12-edo has only one circle of fifths. Its ring number is 1, and 12-edo is said to be single-ring. An edo with a non-co-prime fifth is multi-ring, or "ringy". For example, 15-edo's best approximation of 3/2 is 9\15. The ring number is the GCD of 9 and 15, which is 3. Thus 15-edo is a triple-ring edo. Using an alternative approximation of 3/2 affects the "ringiness": 18-edo is not multi-ring, but 18b-edo is.

Properties

If N is a prime number, N-edo is a single-ring edo. Note that if N is not prime, N-edo may still be single-ring.

Circle-of-fifths notation only works for single-ring edos.

On the scale tree, multi-ring edos appear only on the spines of the kites, shown here as dotted lines:

Generalizations

The ring number can be defined using any two intervals, not just the octave or the fifth. The two intervals are treated as equave and generator. For example, 15-edo can be thought of as generated by 2\15 (Porcupine/Triyo), which makes it single-ring. And 13ed3 can be thought of as generated by 6\13 (a tempered 5/3), again single-ring.

Analogous to ring number, rank-2 temperaments have a chain number. For example, any rank-2 temperament with a pergen of (P8, P5) has a chain number of 1, and is single-chain. All other pergens are multi-chain. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m*n/|f|, where M is the multigen and f is M's fifthspan. For example (P8/2, M2/4) is quadruple-chain.

Like the ring number, the chain number can be generalized to other intervals besides the octave and the fifth.