Equave limit

The equave limit^{[idiosyncratic term]} generalizes the concept of odd limit, extending for any equave what the odd-limit represents specifically for the equave 2/1.

Definition

The q-equave-n-limit is defined as the set of all positive rationals [math]\displaystyle{ \displaystyle {q^z}\cdot\frac{u}{v} }[/math], where:

• [math]\displaystyle{ q \in \mathbb{Q}^{+} }[/math]​ (i.e., [math]\displaystyle{ q }[/math] is a positive rational number), called the equave,
• [math]\displaystyle{ z \in \mathbb{Z} }[/math]​ (i.e., [math]\displaystyle{ z }[/math] is an integer, positive or negative),
• [math]\displaystyle{ u,v \in \mathbb{Z}^{+} }[/math]​​ (i.e., [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are positive integers) such that [math]\displaystyle{ u \leqslant n }[/math] and [math]\displaystyle{ v \leqslant n }[/math],
• [math]\displaystyle{ n \in \mathbb{Z}^{+} }[/math]​​ (i.e., [math]\displaystyle{ n }[/math] is a positive integer), called the limit.

The parameter [math]\displaystyle{ n }[/math] places an upper bound on the values of the integers [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math], meaning that both [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are less than or equal to [math]\displaystyle{ n }[/math]. Thus, the q-equave-n-limit consists of ratios generated by multiplying a power of [math]\displaystyle{ q }[/math] by ratios [math]\displaystyle{ \displaystyle \frac{u}{v} }[/math]​, where the numerator and denominator are constrained by the limit [math]\displaystyle{ n }[/math].

Additional constraints can be applied to the ratios [math]\displaystyle{ \displaystyle \frac{u}{v} }[/math]​​ by restricting them to a specific just intonation subgroup.