User:Contribution/Factor Limit

Purpose

A common way to determinate a subset of Just Intonation intervals is to demarcate an harmonic limit.

Another possible way would be to delimit a maximal amount of primes allowed in the factorization of the rational numbers.

Factor limit

Definition

A positive rational number q belongs to the f-factor-limit, called the factor limit, for a given positive integer f if and only if the sum of the exponent absolutes of its factorization into primes is less than or equal to f.

Examples

• 0-factor-limit contains only 1

• 1-factor-limit contains also the prime harmonic series (2, 3, 5, 7, 11, 13, 17, etc...) and the prime subharmonic series (2^-1, 3^-1, 5^-1, 7^-1, 11^-1, 13^-1, 17^-1, etc...), called prime intervals.

• 2-factor-limit contains also 2², 2^-2, 2*3, 2^-1*3^-1, 2^-1*3, 2*3^-1, 3², 3^-2, etc...

• 3-factor-limit contains also 2³, 2^-3, 2²*3, 2^-2*3^-1, 2^-2*3, 2²*3^-1, 2*3², 2^-1*3^-2, 2^-1*3², 2*3^-2, 3³, 3^-3, etc...

Minimal and maximal primes factor limit

Definition

A positive rational number q belongs to the minp-maxp-f-mmpfactor-limit, called the minimal and maximal primes factor limit, for a given prime number minp, a given prime number maxp with maxp>=minp and a given positive integer f if and only if the mininal prime of q factorization into primes is more than or equal to minp, the maximal prime number into q factorization into primes is less than or equal to maxp, and the sum of the exponent absolutes of q factorization into primes is less than or equal to f.

Examples

• 5-7-2-mmpfactor-limit contains only 1, 5, 5², 7, 5*7, 5^-1*7^-1, 5^-1*7, 5*7^-1, 7²