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.

Maximal factor limit

Definition

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

In other words, a positive rational number q belongs to the fmax-max-factor-limit if and only if the sum of the exponent absolutes of its factorization into primes is right-bounded to fmax.

Examples

• 0-max-factor-limit contains only 1

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

• 3-max-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 factor limit

Definition

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

In other words, a positive rational number q belongs to the fmin-min-factor-limit if and only if the sum of the exponent absolutes of its factorization into primes is left-bounded to fmin.

Examples

• 0-min-factor-limit contains ℚ+\{0}

• 1-min-factor-limit contains the above excluding 1

• 2-min-factor-limit contains the above excluding prime intervals

• 3-min-factor-limit contains the above excluding 2², 2^-2, 2*3, 2^-1*3^-1, 2^-1*3, 2*3^-1, 3², 3^-2, etc...

Operations on sets

Minimal harmonic limits, maximal harmonic limits, minimal factor limits, maximal factor limits, and all other kinds of Just Intonation subsets are sets of strictly positive rational numbers.

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

Prime-bounded and factor-bounded limit

Definition

A positive rational number q belongs to the (pmin;pmax;fmin;fmax)-limit, called the prime-bounded and factor-bounded limit, if q ∈ pmin-min-prime-limit ∩ pmax-max-prime-limit ∩ fmin-min-factor-limit ∩ fmax-max-factor-limit.

Examples

• (5;7;0;3)-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;0;2)-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;0;1)-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.

• (3;3;0;26)-limit contains only 1/1, 3/1, 1/3, 3², 3^-2, 3³, 3^-3, 3⁴, 3^-4, 3⁵, 3^-5, 3⁶, 3^-6, 3⁷, 3^-7, 3⁸, 3^-8, 3⁹, 3^-9, 3¹⁰, 3^-10, 3¹¹, 3^-11, 3¹², 3^-12, 3¹³, 3^-13, 3¹⁴, 3^-14, 3¹⁵, 3^-15, 3¹⁶, 3^-16, 3¹⁷, 3^-17, 3¹⁸, 3^-18, 3¹⁹, 3^-19, 3²⁰, 3^-20, 3²¹, 3^-21, 3²², 3^-22, 3²³, 3^-23, 3²⁴, 3^-24, 3²⁵, 3^-25, 3²⁶, 3^-26.

• (2;2;0;+∞)-limit contains only 1/1, 2/1, 1/2, 2², 2^-2, 2³, 2^-3, 2⁴, 2^-4, 2⁵, 2^-5, 2⁶, 2^-6, 2⁷, 2^-7, 2⁸, 2^-8, 2⁹, 2^-9, 2¹⁰, 2^-10, 2¹¹, 2^-11, 2¹², 2^-12, 2¹³, 2^-13, 2¹⁴, 2^-14, 2¹⁵, 2^-15, 2¹⁶, 2^-16, 2¹⁷, 2^-17, 2¹⁸, 2^-18, 2¹⁹, 2^-19, 2²⁰, 2^-20, etc...

• ((5;7;0;3)-limit ∪ (5;13;0;2)-limit ∪ (5;31;0;1)-limit) × (3;3;0;26)-limit × (2;2;0;+∞)-limit is represented by this table.