Harmonic limit

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In just intonation, the p-limit or p-prime-limit consists of ratios of integers whose prime factors are 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. In math, such a number is known as a p-smooth number. An interval does not need to contain p as a factor to be considered within the p-limit. For instance, 3/2 is considered part of the 13-limit, since the primes 2 and 3 are smaller than 13. Also, an interval with a p in it is not necessarily within the p-limit. 23/13 is not within the 13-limit, since 23 is a prime number higher than 13.

Prime limits as subgroups

Prime limits are essentially special cases of subgroups that include all primes up to the limit rather than skipping any. 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. In many cases, it is often more useful to speak of subgroups of the prime-limit, rather than the full limit, and this becomes increasingly true for higher limits as the number of useful temperaments with a good approximation of full limits dwindles, and for that purpose, the term "p-horizon" can be used to refer to an entire umbrella of subgroups encompassed by the p-limit.

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

External links