User:Contribution/Factor Limit: Difference between revisions

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Another possible way would be to delimit a maximal amount of primes allowed in the factorization of the rational numbers.

=~~Factor~~ limit=

=Maximal 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.

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 interval|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===

* 1-factor-limit contains ~~only~~ the ~~harmonic and subharmonic series~~.

* 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=

* 2-factor-limit contains also 22</~~sup>~~, ~~2-2~~, ~~2*3~~, ~~2-1*3-1, 2-1*3, 2*3-1, 32, 3-2~~, ~~etc..~~.

[[User:Contribution/Minimal_Prime_Limit|Minimal harmonic limits]], [[harmonic limit|maximal harmonic limits]], minimal factor limits, maximal factor limits, and all other kinds of Just Intonation subsets are sets of strictly positive rational numbers.

* 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..~~.

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

=~~Minimal~~ and ~~maximal primes~~ factor limit=

=Prime-bounded and factor-bounded 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~~.

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===

* 5-7-2-~~mmpfactor~~-limit contains only 5*7, 5*7

* (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 [[User:Contribution/(31-Limit_Factor_Gradient)_x_(26_Tritaves_Chain)_x_(Infinite_Octaves)|this table]].

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 Another possible way would be to delimit a maximal amount of primes allowed in the factorization of the rational numbers.
-=Factor limit=
+=Maximal 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.
+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<sup>-1</sup>, 3<sup>-1</sup>, 5<sup>-1</sup>, 7<sup>-1</sup>, 11<sup>-1</sup>, 13<sup>-1</sup>, 17<sup>-1</sup>, etc...), called [[prime interval|prime intervals]].
+* 2-max-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-max-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 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===
-* 1-factor-limit contains only the harmonic and subharmonic series.
+* 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 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...
+=Operations on sets=
-* 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...
+[[User:Contribution/Minimal_Prime_Limit|Minimal harmonic limits]], [[harmonic limit|maximal harmonic limits]], minimal factor limits, maximal factor limits, and all other kinds of Just Intonation subsets are sets of strictly positive rational numbers.
-* 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...
+Set theory features binary operations on sets: union, intersection, set difference, symmetric difference, Cartesian product, power set.
-=Minimal and maximal primes factor limit=
+=Prime-bounded and factor-bounded 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.
+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===
-* 5-7-2-mmpfactor-limit contains only 5*7, 5*7
+* (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, 3<sup>2</sup>, 3<sup>-2</sup>, 3<sup>3</sup>, 3<sup>-3</sup>, 3<sup>4</sup>, 3<sup>-4</sup>, 3<sup>5</sup>, 3<sup>-5</sup>, 3<sup>6</sup>, 3<sup>-6</sup>, 3<sup>7</sup>, 3<sup>-7</sup>, 3<sup>8</sup>, 3<sup>-8</sup>, 3<sup>9</sup>, 3<sup>-9</sup>, 3<sup>10</sup>, 3<sup>-10</sup>, 3<sup>11</sup>, 3<sup>-11</sup>, 3<sup>12</sup>, 3<sup>-12</sup>, 3<sup>13</sup>, 3<sup>-13</sup>, 3<sup>14</sup>, 3<sup>-14</sup>, 3<sup>15</sup>, 3<sup>-15</sup>, 3<sup>16</sup>, 3<sup>-16</sup>, 3<sup>17</sup>, 3<sup>-17</sup>, 3<sup>18</sup>, 3<sup>-18</sup>, 3<sup>19</sup>, 3<sup>-19</sup>, 3<sup>20</sup>, 3<sup>-20</sup>, 3<sup>21</sup>, 3<sup>-21</sup>, 3<sup>22</sup>, 3<sup>-22</sup>, 3<sup>23</sup>, 3<sup>-23</sup>, 3<sup>24</sup>, 3<sup>-24</sup>, 3<sup>25</sup>, 3<sup>-25</sup>, 3<sup>26</sup>, 3<sup>-26</sup>.
+* (2;2;0;+∞)-limit contains only 1/1, 2/1, 1/2, 2<sup>2</sup>, 2<sup>-2</sup>, 2<sup>3</sup>, 2<sup>-3</sup>, 2<sup>4</sup>, 2<sup>-4</sup>, 2<sup>5</sup>, 2<sup>-5</sup>, 2<sup>6</sup>, 2<sup>-6</sup>, 2<sup>7</sup>, 2<sup>-7</sup>, 2<sup>8</sup>, 2<sup>-8</sup>, 2<sup>9</sup>, 2<sup>-9</sup>, 2<sup>10</sup>, 2<sup>-10</sup>, 2<sup>11</sup>, 2<sup>-11</sup>, 2<sup>12</sup>, 2<sup>-12</sup>, 2<sup>13</sup>, 2<sup>-13</sup>, 2<sup>14</sup>, 2<sup>-14</sup>, 2<sup>15</sup>, 2<sup>-15</sup>, 2<sup>16</sup>, 2<sup>-16</sup>, 2<sup>17</sup>, 2<sup>-17</sup>, 2<sup>18</sup>, 2<sup>-18</sup>, 2<sup>19</sup>, 2<sup>-19</sup>, 2<sup>20</sup>, 2<sup>-20</sup>, 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 [[User:Contribution/(31-Limit_Factor_Gradient)_x_(26_Tritaves_Chain)_x_(Infinite_Octaves)|this table]].