Harmonic limit

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Limit (music)

In just intonation, the p-limit (or p-prime-limit) is the set of frequency ratios that can be expressed using only prime numbers less than or equal to p.

A frequency ratio belongs to the p-limit if and only if both its numerator and denominator can be factored completely into prime numbers no larger than p (with positive or negative integer exponents). In mathematics, such numbers are known as p-smooth numbers.

An interval does not need to contain the prime p itself to be within the p-limit. For example, 3/2 belongs to the 13-limit because both 2 and 3 are smaller than 13. Conversely, containing the prime p does not guarantee membership in the p-limit. For instance, 23/13 is not within the 13-limit because 23 is a prime number larger than 13.

All prime limits contain infinitely many intervals. Even if we restrict our consideration to intervals within a single octave, all prime limits except the 2-limit still contain infinitely many distinct ratios.

Prime limits as subgroups

Prime limits are essentially just intonation subgroups that do not skip any primes. For any prime number p, the p-limit creates a well-defined mathematical structure, called free abelian group. This structure has a dimension (or rank) equal to the number of prime numbers less than or equal to p. For example, the 7-limit works with intervals built from the primes 2, 3, 5, and 7, so it has 4 dimensions.

Often, composers and theorists find it more practical to work with smaller subsets of a prime limit rather than using all possible intervals within that limit. This becomes increasingly important for higher limits, as the number of practical tuning systems that can reasonably approximate the full set of intervals diminishes. For discussing these subsets that exist within a larger prime limit, some theorists use the term "p-horizon" to refer to the collection of all possible subsets within a p-limit.

Proper harmonic limit

While harmonic limit encompasses all ratios up to a given prime, proper harmonic limit classifies JI ratios based only based on the highest prime they contain in either the numerator or denominator. Equivalently, it is all of the intervals of a prime limit that are not found in a lower prime limit. It has been called harmonic class or HC.

A ratio belongs to the proper p-prime limit if and only if p is the highest prime number found in its factorization. For example:

• 7/4 is proper 7-limit because 7 is the highest prime in its factorization.
• 5/4 is proper 5-limit, not proper 7-limit, even though it's within the 7-limit.
• 9/7 is proper 7-limit because the highest prime is 7 (since 9 = 3²).

This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime. Unlike regular harmonic limits, proper harmonic limits are mutually exclusive categories.

Alternative classification systems

Prime limits contain infinitely many intervals, so they cannot truly function as limits on complexity. Rather, they serve as compositional constraints that help manage choices when working with just intonation or regular temperaments.

Various alternative classification systems exist for characterizing intervals, such as:

• Odd-limit classifies intervals based on the complexity of the ratio itself.
• Pythagorean-commatic interval naming characterizes just intervals by their deviation from basic Pythagorean intervals.
• Overtone scales and primodality classify scales and chords based on their relative position in the harmonic series.

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
• Cubic and octahedral limits
• Prime minimum
• Harmonic class
• Wikipedia: Størmer's theorem

External links

• Tonalsoft Encyclopedia | Limit