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...

Prime-bounded factor limit

Definition

A positive rational number q belongs to the minp-maxp-f-prime-bounded-factor-limit, called the prime-bounded 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-prime-bounded-factor-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-prime-bounded-factor-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-prime-bounded-factor-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, prime-bounded 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.