Harmonic limit: Difference between revisions

Line 16:

== 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.

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.

~~[[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~~.

A ratio belongs to harmonic class n (HCn) 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 = 3<sup>2</sup>).

~~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~~]].

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.

* [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean 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.

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

@@ Line 16: / Line 16: @@
 == 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.
+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.
-[[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.
+A ratio belongs to harmonic class n (HCn) 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 = 3<sup>2</sup>).
-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]].
+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.
+* [[2.3-equivalent class and Pythagorean-commatic interval naming system|Pythagorean 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.
 == Individual pages of ''p''-limit JI ==