Equave limit: Difference between revisions
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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. | 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== | == Definition == | ||
The '''q-equave-n-limit''' is defined as the set of all positive rationals | The '''''q''-equave-''n''-limit''' is defined as the set of all positive rationals <math>\displaystyle | ||
{q^z}\cdot\frac{u}{v} | {q^z}\cdot\frac{u}{v} | ||
</math | </math>, where: | ||
* <math> | * <math>q \in \mathbb{Q}^{+}</math> (i.e., <math>q</math> is a positive rational number), called the '''equave''', | ||
* <math> | * <math>z \in \mathbb{Z}</math> (i.e., <math>z</math> is an integer, positive or negative), | ||
* <math>u, | * <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> | * <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 ''' | 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} | \frac{u}{v} | ||
</math>, where the numerator and denominator are constrained by the limit <math>n</math>. | </math>, where the numerator and denominator are constrained by the limit <math>n</math>. | ||
Latest revision as of 15:39, 9 September 2024
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.