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 {q^z}\cdot\frac{u}{v} [/math], where:

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

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

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