Harmonic limit: Difference between revisions

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== 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'''~~, but this is discouraged because people e.g. often use "7-limit" to denote the proper 7-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 n is the highest prime number found in its factorization. For example:

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 {{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 regular harmonic limits, proper harmonic limits are mutually exclusive categories~~, so that JI is a disjoint union of proper prime limits~~.

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 ==

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 == 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''', but this is discouraged because people e.g. often use "7-limit" to denote the proper 7-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 n is the highest prime number found in its factorization. For example:
+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 {{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 regular harmonic limits, proper harmonic limits are mutually exclusive categories, so that JI is a disjoint union of proper prime limits.
+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 ==