Harmonic limit: Difference between revisions
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{{Wikipedia|Limit (music)}} | {{Wikipedia|Limit (music)}} | ||
In [[just intonation]], the '''''p''-limit''' (or '''''p''-prime-limit''') | In [[just intonation]], the '''''p''-limit''' (or '''''p''-prime-limit''') is the set of [[frequency ratio]]s 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 [[prime factorization|factored]] completely into prime numbers no larger than ''p'' (with positive or negative integer exponents). In mathematics, such numbers are known as {{w| | A frequency ratio belongs to the ''p''-limit if and only if both its numerator and denominator can be [[prime factorization|factored]] completely into prime numbers no larger than ''p'' (with positive or negative integer exponents). In mathematics, such numbers are known as {{w|smooth number|''p''-smooth numbers}}. | ||
An interval | 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. | ||
Conversely, containing the prime ''p'' | |||
All prime limits contain infinitely many intervals. Even if we [[octave reduction|restrict]] our consideration to intervals within a single octave, all prime limits except the [[2-limit]] still contain infinitely many distinct ratios. | All prime limits contain infinitely many intervals. Even if we [[octave reduction|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 as subgroups == | ||
Prime limits are essentially | 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. | ||
For any prime number ''p'', the p-limit creates a well-defined mathematical structure. 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. | 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. | ||
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While harmonic limit encompasses all ratios up to a given prime, '''harmonic class''' ('''HC''') 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. | While harmonic limit encompasses all ratios up to a given prime, '''harmonic class''' ('''HC''') 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. | ||
A ratio belongs to harmonic class n ( | A ratio belongs to harmonic class ''n'' (HC-''n'') if and only if n is the highest prime number found in its factorization. For example: | ||
* [[7/4]] is HC7 because 7 is the highest prime in its factorization. | * [[7/4]] is HC7 because 7 is the highest prime in its factorization. | ||
* [[5/4]] is HC5, not HC7, even though it's within the 7-limit. | * [[5/4]] is HC5, not HC7, even though it's within the 7-limit. | ||
* [[9/7]] is HC7 because the highest prime is 7 (since 9 = 3<sup>2</sup>). | * [[9/7]] is HC7 because the highest prime is 7 (since {{nowrap| 9 {{=}} 3<sup>2</sup> }}). | ||
This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime. Unlike harmonic limits, harmonic classes are mutually exclusive categories. | This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime. Unlike harmonic limits, harmonic classes are mutually exclusive categories. | ||
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Various alternative classification systems exist for characterizing intervals, such as: | Various alternative classification systems exist for characterizing intervals, such as: | ||
* [[Odd-limit]] classifies intervals based on the complexity of the ratio itself. | * [[Odd-limit]] classifies intervals based on the complexity of the ratio itself. | ||
* [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean interval naming]] characterizes just intervals by their deviation from basic Pythagorean intervals. | * [[2.3-equivalent class and Pythagorean-commatic interval naming system|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. | * [[Overtone scales]] and [[primodality]] classify scales and chords based on their relative position in the harmonic series. | ||
Revision as of 11:42, 24 April 2025
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.
Harmonic class
While harmonic limit encompasses all ratios up to a given prime, harmonic class (HC) 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.
A ratio belongs to harmonic class n (HC-n) if and only if n is the highest prime number found in its factorization. For example:
- 7/4 is HC7 because 7 is the highest prime in its factorization.
- 5/4 is HC5, not HC7, even though it's within the 7-limit.
- 9/7 is HC7 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 harmonic limits, harmonic classes 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.