Harmonic limit

English Wikipedia has an article on:

Limit (music)

In just intonation, the p-limit or p-prime-limit consists of the ratios of p-smooth numbers, where a p-smooth number is an integer with prime factors no larger than p.

A positive rational number q belongs to the p-limit for a given prime number p if and only if it can be factored into primes (with positive or negative integer exponents) of size less than or equal to p. For any prime number p, the set of all rational numbers in the p-limit defines a finitely generated free abelian group. The rank of this group is equal to π (p), the number of prime numbers less than or equal to p. Hence, for example, the rank of the 7-limit is 4, as it is generated by 2, 3, 5 and 7.

Individual pages of p-limit JI

2-limit	3-limit	5-limit	7-limit	11-limit	13-limit
17-limit	19-limit	23-limit	29-limit	31-limit	37-limit
41-limit	43-limit	47-limit	53-limit	59-limit	61-limit
67-limit	71-limit	73-limit	79-limit	83-limit	89-limit

See also

• Odd limit
• Harmonic class
• Wikipedia: Størmer's theorem

External links

• Tonalsoft Encyclopedia | Limit