Binary G-system

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Binary G-system

Binary system G(2,k) is special case of G-system, n=2. It is suitable for classification of musical structures.

Distance schema

In a binary system (base E(A) ={0, 1}) it is easy to express structure of instances.
Let level of class L be the number of digits "1" in an instance and let interval be the distance between two digits "1". Distance schema is the list of all neighboring intervals, last interval in brackets.

 k-gon                                     distance schema
     0
   0   0                                           (0)
     0
     0         0         0         1
   1   0     0   0     0   1     0   0             (4)
     0         1         0         0
     0         0         1         1
   1   0     0   1     0   1     1   0           1 (3)
     1         1         0         0
     0         1
   1   1     0   0                               2 (2)
     0         1
     0         1         1         1
   1   1     0   1     1   1     1   0         1 1 (2)
     1         1         0         1
     1
   1   1                                     1 1 1 (1).
     1
 

Powers of (a+b)

The powers of (a+b) are an example of the system G(2,k). Nesting mechanism is marked with [] brackets.

 ( a + b )^1 = a + b
 ( a + b )^2 = [a]^2+ 2ab + [b]2
 ( a + b )^3 = [a]^3+ 3a^2b + 3ab^2 + [b]3
 ( a + b )^4 = [a]^4+ 4a^3b + 4a^2b^2+ 2[ab]^2+ 4ab^3+ [b]^4
 ( a + b )^5 = [a]^5+ 5a^4b + 10a^3b^2+ 10a^2b^3+ 5ab^4+ [b]^5
 ( a + b )^6 = [a]^6+ 6a^5b + 12a^4b^2+ 3[a^2b]^2+ 18a^3b^3+ 2[ab]^3+ 12a^2b^4+ 3[ab^2]^2+ 6ab^5+[b]^6

From viewpoint of G-systems structures the expression a^2 * b^2 is not equal to the [a*b]^2 (similarly a^4 * b^2 and [a^2 * b]^2, ...).

The Pascal triangle arise from g-classes in following way:

   k = 1:                   {1} + {1}
                             x     x
                             1     1
   k = 2:               [1]1+ {2} + [1]1
                               x
                               1
   k = 3:           [1]1+ {3} + {3} + [1]1
                           x     x
                           1     1
                                [2]2
                                 +
   k = 4:           [1]1+ {4} + {4} + {4} + [1]1
                           x     x     x
                           1     1     1
   k = 5:       [1]1+ {5} + {10} + {10} +  {5} + [1]1
                       x      x      x      x
                       1      1      1      1
                          [3]3  [2]2  [3]3
                           +     +     +
   k = 6:     [1]1+ {6} + {6} + {6} + {6} + {6} +  [1]1
                     x     x     x     x     x
                     1     2     3     2     1
   k = 7:  [1]1+ {7} + {7} + {7} + {7} + {7} + {7} +  [1]1
                  x     x     x     x     x     x
                  1     3     5     5     3     1
                                  [2]2
                                    +
                       [4]4       [4]4        [4]4
                       +            +           +
   k = 8: [1]1+ {8} + {8} + {8} + {8} + {8} + {8} + {8} + [1]1
                 x     x     x     x     x     x     x
                 1     3     7     8     7     3     1

Substructures of classes

Substructure of class g in G(2,k) is such class g* for that one its instance u* satisfies the relation:

   u*=u* AND g (binary).

For example [0001] in G(2,4) is substructure of [0101], because [0001]=[0001] AND [0101].
On the contrary e.g.[0011] is not substructure of [0101].

Class g=1387 (known from music), i.e. [101011010101], has distance scheme 221221(2) and 65 substructures. In the following outline these substructures are listed - ordered by levels.

 0/ 0
 1/ 1
 2/ 3,5,9,17,33,65
 3/ 11,13,21,35,37,41,49,67,69,73,81,97,133,137,145
 4/ 43,45,53,75,85,99,105,139,141,149,163,165,169,
 177,197,209,291,293,297,329
 5/ 107,171,173,181,213,299,301,331,355,397,461,421,
 425,597,661
 6/ 363,427,429,685,693,725
 7/ 1387

Schematic algebra