Harmonic limit: Difference between revisions

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== Prime limits as subgroups ==

Prime limits are essentially special cases of [[just intonation ~~subgroup|~~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 {{w|free abelian group}} The [[~~rank~~]] of this group is~~ equal to ~~π (''p''),~~ the ~~{{w|Prime-counting function|~~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.

Prime limits are essentially special cases of [[just intonation subgroups]] that do not skip any primes.

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.

== Harmonic class ==

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 == Prime limits as subgroups ==
-Prime limits are essentially special cases of [[just intonation subgroup|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 {{w|free abelian group}} The [[rank]] of this group is equal to π (''p''), the {{w|Prime-counting function|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.
+Prime limits are essentially special cases of [[just intonation subgroups]] that do not skip any primes.
+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.
 == Harmonic class ==