User:Contribution/Factor Limit: Difference between revisions

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* 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=

=Prime-bounded 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.

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-~~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

* 5-7-3-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-~~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

* 5-13-2-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-~~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

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

[[harmonic limit|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.

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 * 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=
+=Prime-bounded 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.
+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-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
+* 5-7-3-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-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
+* 5-13-2-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-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
+* 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 limit|Harmonic limits]], factor limits, minimal and maximal primes factor limits, and all other kinds of just intonation subsets, are sets of rational numbers.
+[[harmonic limit|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.