User:Zeta Function/Half-cubic limit: Difference between revisions

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The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors, where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.

The '''half-cubic limit'''{{idio}} for a [[prime limit]] is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors (including multiplicity), where all intervals are octave-reduced. In this way, it is analogous to the reduced 2-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.

~~The monzos of such intervals must take the form where one prime factor can have up to a maximum of 2, the other prime factor can have up to a maximum of -2, and all other non-2 primes are 0s~~.

Here is the set of all intervals in the 3-half-cubic limit.

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This limit was specifically designed to be useful for composition and scale-building purposes, as it attempts to chart a middle path between the restrictiveness of the odd limit and the excessive complexity of intervals that is acceptable with prime-limits.

== Generalization ==

In general, the (p, q, a)-half-cubic limit is the set of octave-reduced intervals in the p-prime limit and a-odd limit whose numerators and denominators do not contain more than q prime factors (including multiplicity). Thus, the previously defined limit would correspond to a (p, 2, ∞)-half cubic limit. This generalization is designed to give more flexibility during composing than the first framework. A good example of such a limit might be the (13, 2, 65)-half cubic limit, which is suited to 13-limit composing.

[[Category:Limit]]

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-The half-cubic limit for a prime limit is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors, where all intervals are octave-reduced. In this way, it is analogous to the reduced 1-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.
+The '''half-cubic limit'''{{idio}} for a [[prime limit]] is defined to be the set of all intervals within a certain prime limit whose numerators and denominators both cannot contain more than two non-2 prime factors (including multiplicity), where all intervals are octave-reduced. In this way, it is analogous to the reduced 2-cubic limit but is significantly more restricted, which explains the reason for its naming as the half-cubic limit.
-The monzos of such intervals must take the form where one prime factor can have up to a maximum of 2, the other prime factor can have up to a maximum of -2, and all other non-2 primes are 0s.
 Here is the set of all intervals in the 3-half-cubic limit.
@@ Line 133: / Line 131: @@
 |2
 |}
+This limit was specifically designed to be useful for composition and scale-building purposes, as it attempts to chart a middle path between the restrictiveness of the odd limit and the excessive complexity of intervals that is acceptable with prime-limits.
+== Generalization ==
+In general, the (p, q, a)-half-cubic limit is the set of octave-reduced intervals in the p-prime limit and a-odd limit whose numerators and denominators do not contain more than q prime factors (including multiplicity). Thus, the previously defined limit would correspond to a (p, 2, ∞)-half cubic limit. This generalization is designed to give more flexibility during composing than the first framework. A good example of such a limit might be the (13, 2, 65)-half cubic limit, which is suited to 13-limit composing.
 [[Category:Limit]]