## Binary G-system

#### 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

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