Harmonic limit: Difference between revisions
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A positive rational number ''q'' belongs to the '''''p''-limit''', called the '''''p'' harmonic''' or '''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 [[Wikipedia: Free_abelian_group|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 [[Wikipedia: Smooth number|''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''', called the '''''p'' harmonic''' or '''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 [[Wikipedia: Free_abelian_group|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 [[Wikipedia: Smooth number|''p''-smooth numbers]], where a ''p''-smooth number is an integer with prime factors no larger than ''p''. | ||
== Examples of p-limits == | == Examples of ''p''-limits == | ||
With increasing limits, the tonal space becomes more dense. | With increasing limits, the tonal space becomes more dense. | ||
Revision as of 15:46, 11 June 2021
A positive rational number q belongs to the p-limit, called the p harmonic or 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
- 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