Harmonic limit: Difference between revisions

Line 15:

== 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.

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 ''{{w|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.

@@ Line 15: / Line 15: @@
 == 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.
+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 ''{{w|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.