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

• 5-7-3-mmpfactor-limit contains only 1/1, 5/1, 1/5, 5*5/1, 1/5*5, 5*5*5/1, 1/5*5*5, 7/1, 1/7, 5*7/1, 1/5*7, 7/5, 5/7, 7*7/1, 1/7*7, 5*5*7/1, 1/5*5*7, 7/5*5, 5*5/7, 5*7*7/1, 1/5*7*7, 7*7/5, 5/7*7, 7*7*7/1, 1/7*7*7

• 5-13-2-mmpfactor-limit contains only 1/1, 5/1, 1/5, 5*5/1, 1/5*5, 7/1, 1/7, 5*7/1, 1/5*7, 7/5, 5/7, 7*7/1, 1/7*7, 11/1, 1/11, 5*11/1, 1/5*11, 11/5, 5/11, 7*11/1, 1/7*11, 11/7, 7/11, 11*11/1, 1/11*11, 13/1, 1/13, 5*13/1, 1/5*13, 13/5, 5/13, 7*13/1, 1/7*13, 13/7, 7/13, 11*13/1, 1/11*13, 13/11, 11/13, 13*13/1, 1/13*13

• 5-31-1-mmpfactor-limit contains only 1/1, 5/1, 1/5, 7/1, 1/7, 11/1, 1/11, 13/1, 1/13, 17/1, 1/17, 19/1, 1/19, 23/1, 1/23, 29/1, 1/29, 31/1, 1/31

Operations on sets

Harmonic limits, factor limits, minimal and maximal primes factor limits, and all other kinds of just intonation subsets, are sets of rational numbers.

Set theory features binary operations on sets: union, intersection, set difference, symmetric difference, cartesian product, power set.