User:Contribution/Factor Limit: Difference between revisions
Contribution (talk | contribs) Created page with "====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 a..." |
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===Examples=== | ===Examples=== | ||
* 1-factor-limit contains only the harmonic and subharmonic series. | |||
* 2-factor-limit contains also 2<sup>2</sup>, 2<sup>-2</sup>, 2*3, 2<sup>-1</sup>*3<sup>-1</sup>, 2<sup>-1</sup>*3, 2*3<sup>-1</sup>, 3<sup>2</sup>, 3<sup>-2</sup>, etc... | |||
* 3-factor-limit contains also 2<sup>3</sup>, 2<sup>-3</sup>, 2<sup>2</sup>*3, 2<sup>-2</sup>*3<sup>-1</sup>, 2<sup>-2</sup>*3, 2<sup>2</sup>*3<sup>-1</sup>, 2*3<sup>2</sup>, 2<sup>-1</sup>*3<sup>-2</sup>, 2<sup>-1</sup>*3<sup>2</sup>, 2*3<sup>-2</sup>, 3<sup>3</sup>, 3<sup>-3</sup>, etc... | |||
3-factor-limit contains also 2<sup>3</sup>, 2<sup>-3</sup>, 2<sup>2</sup>*3, 2<sup>-2</sup>*3<sup>-1</sup>, 2<sup>-2</sup>*3, 2<sup>2</sup>*3<sup>-1</sup>, 2*3<sup>2</sup>, 2<sup>-1</sup>*3<sup>-2</sup>, 2<sup>-1</sup>*3<sup>2</sup>, 2*3<sup>-2</sup>, 3<sup>3</sup>, 3<sup>-3</sup>, etc... | |||
=Minimal and maximal primes factor limit= | =Minimal and maximal primes factor limit= | ||
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===Examples=== | ===Examples=== | ||
* 5-7-2-mmpfactor-limit contains only 5*7, 5*7 | |||
Revision as of 07:25, 12 June 2020
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
- 1-factor-limit contains only the harmonic and subharmonic series.
- 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 minimal prime number into q factorization is minp, the maximal prime number into q factorization into primes is 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 5*7, 5*7