User:Contribution/Factor Limit: Difference between revisions

(17 intermediate revisions by the same user not shown)

Line 11:

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-limit if and only if the sum of the exponent absolutes of its factorization into primes is right-bounded 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===

Line 28:

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-limit if and only if the sum of the exponent absolutes of its factorization into primes is left-bounded 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}

* 0-min-factor-limit contains ℚ+\{0}

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

Line 41:

=Operations on sets=

[[harmonic limit|~~Maximal~~ harmonic limits]], minimal ~~harmonic~~ limits, maximal ~~factor limits, minimal~~ factor limits, and all other kinds of Just Intonation subsets are sets of strictly positive rational numbers.

[[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.

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

Line 49:

==Definition==

A positive rational number q belongs to the ~~[minp~~;~~maxp~~;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

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===

* ~~[0;~~5;7;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;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.

* ~~[0;~~5;13;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;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.

* ~~[0;~~5;31;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

* (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]].

@@ Line 11: / Line 11: @@
 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-limit if and only if the sum of the exponent absolutes of its factorization into primes is right-bounded 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===
@@ Line 28: / Line 28: @@
 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-limit if and only if the sum of the exponent absolutes of its factorization into primes is left-bounded 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}
+* 0-min-factor-limit contains ℚ+\{0}
 * 1-min-factor-limit contains the above excluding 1
@@ Line 41: / Line 41: @@
 =Operations on sets=
-[[harmonic limit|Maximal harmonic limits]], minimal harmonic limits, maximal factor limits, minimal factor limits, and all other kinds of Just Intonation subsets are sets of strictly positive rational numbers.
+[[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.
 Set theory features binary operations on sets: union, intersection, set difference, symmetric difference, Cartesian product, power set.
@@ Line 49: / Line 49: @@
 ==Definition==
-A positive rational number q belongs to the [minp;maxp;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
+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===
-* [0;5;7;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;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.
-* [0;5;13;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;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.
-* [0;5;31;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
+* (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]].