# Harmonic limit

A positive rational number *q* belongs to the ** p-limit**, called the

**or**

*p*harmonic**prime 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. Another way to express the

*p*-limit is that it consists of the ratios of

*p*-smooth numbers, where a

*p*-smooth number is an integer with prime factors no larger than

*p*.

## Examples of *p*-limits

With increasing limits, the tonal space becomes more dense.

- 2-limit contains only multiples of the octave (2/1), see 1edo
- 3-limit contains 3/2, the just perfect fifth
- 5-limit contains 5/4, the perfect major third
- 7-limit contains 7/4, the harmonic seventh or septimal subminor seventh
- 11-limit contains 11/8, the undecimal tritone or "Alphorn-Fa"
- 13-limit
- 17-limit
- 19-limit
- 23-limit
- 29-limit
- 31-limit
- 41-limit
- 47-limit
- 61-limit

## See also

- Odd limit
- Harmonic Class (HC)
- Consistency
- Wikipedia: Limit (music) (covers also the distinction between odd-limit and prime-limit)
- Wikipedia: Størmer's theorem