Equave limit: Difference between revisions

The '''equave limit'''{{idiosyncratic}} 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