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 [[Free abelian group|finitely generated 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.

== Harmonic class ==

'''Harmonic class''' ('''HC''') classifies [[JI]] [[ratio]]s based on the highest [[prime interval|prime]] they contain in either the numerator or denominator. HC tells us that the ratio must contain the prime of whatever class it falls into and will contain no higher prime.

[[Harmonic limit]] refers to the highest prime allowed in the ratios and includes all simpler ratios of lower limit, but HC only contains those which contain that prime. For example, while 5/4 falls into the 7-limit, it is not considered a HC7 interval because the highest prime it contains is 5 not 7. Therefore, HC7 must contain a 7 and no higher prime. 9/7 however would be considered HC7 because 9 is not prime but rather a multiple of 3. Therefore, HC9 does not exist.

It has been criticized by some schools that the sound of JI is not well characterized by this classification system. Specifically, it is believed that each harmonic class lacks a consistent sound quality. Rather, [[primodality]] classifies intervals by their common denominator, and meanwhile, the [[2.3-equivalent class and Pythagorean-commatic interval naming system|2.3-equivalent class]] may be used as an enhancement suitable for traditional JI and/or [[regular temperament theory]].

== Individual pages of ''p''-limit JI ==

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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 [[Free abelian group|finitely generated 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.
+== Harmonic class ==
+'''Harmonic class''' ('''HC''') classifies [[JI]] [[ratio]]s based on the highest [[prime interval|prime]] they contain in either the numerator or denominator. HC tells us that the ratio must contain the prime of whatever class it falls into and will contain no higher prime.
+[[Harmonic limit]] refers to the highest prime allowed in the ratios and includes all simpler ratios of lower limit, but HC only contains those which contain that prime. For example, while 5/4 falls into the 7-limit, it is not considered a HC7 interval because the highest prime it contains is 5 not 7. Therefore, HC7 must contain a 7 and no higher prime. 9/7 however would be considered HC7 because 9 is not prime but rather a multiple of 3. Therefore, HC9 does not exist.
+It has been criticized by some schools that the sound of JI is not well characterized by this classification system. Specifically, it is believed that each harmonic class lacks a consistent sound quality. Rather, [[primodality]] classifies intervals by their common denominator, and meanwhile, the [[2.3-equivalent class and Pythagorean-commatic interval naming system|2.3-equivalent class]] may be used as an enhancement suitable for traditional JI and/or [[regular temperament theory]].
 == Individual pages of ''p''-limit JI ==