Schematic algebra - Application

Musical structures

Groupings of tones

Tone groupings have some specific features. Chords – e.g. major triad {c,e,g} – can be usually derived from each tone. There is 12 distinct major triads in 12-tone system (12 traspositions of each triad). But augmented triad {c,e,g#} does no have so many variants. Already fifth transposition {e,g#,c} is (disregarding of tone order) equivalent to original triad. Triad {c,e,g#} has only 4 transpositions.

In 12-tone system there are 2¹², i.e. 4096 distinct chords (including silence and 12-tone cluster).

These chords can be partitioned into 352 classes (4096 /12 is c. 341.33).

This partitioning was made in details by Czech professor of music and musical theorist Karel Janeček in book Modern Harmony.

Each triad can be substituted by binary number. We will write these binary numbers in decimal system and then observe distribution of these numbers into classes.

Numbers of instances

To understand tone structures it is useful to draw all L-sounds (i.e. chords having L tones) existing in systems with k tones (for k=1,2,3,4,5,....)

The mathematical structures corresponding to triads are so called instances. By writing of particular transpositions into rows we get characteristic schemas.

Generally we will speak about G-systems of degree n and order k, G(n,k).

Distance schemas

Distances between tones, i.e. between digit 1 in schemas, are so called (musical) intervals. Distance schema is list of all adjacent intervals.

We put the last intervals into brackets – according to Janeček convention (see Janeček’s orientation schema in the book Modern Harmony).

Level L (of instance, of class) is number of tones, i.e. number of digit 1 (in instance, in class).

Modalities and modes

Modalities are structures in which certain parts of musical phrases are build (e.g. one short song cover usually just one modality).

Mathematical structures

G-systems & groups

We have seen G-systems from standpoint of theory of numbers. Let us now try to use theory of groups.

Groups and G-systems have some properties in common:

• nesting of lower systems into higher reminds subgroups of groups,
• there is an analogy of Lagrange’s theorem of groups in G-systems.

Group of instances in class

Instances (given by numbers u) are rotations of basic instance of class g. Composing of rotations is associative (AB)C=A(BC) and commutative AB=BA.

Rotation of elements in G-systems of order k makes commutative group of order k. (E.g. each class of system G(2,4) corresponds to rotations of square with marked apexes).

Group of instances in G-system

Numbers {1,n,n²,n³..,}, where n is base of G-system, makes cyclical group with generator n. There are numbers {1,n,n²,..,n^k−1} mod r in the first row of G-system. The next rows of self classes are g-multiples of the first row (according to module r), where g is number of the class.

G(4,2)
   0
   1 2 4 8
   3 6 12 9
   5 10
   7 14 13 11
  15

 12, 4, 2, 1
 12, 6, 2, 1
 12, 6, 3, 1

      12
    4    6
  2    2    3
       1

Partitioning of classes

G-relation separates instances into classes. Some operations (addition and multiplication according to module r) can operate also with classes.

Let us assume, that given class can be substituted by any of its instance. We will distinguish the following types of relation of classes:

   0
   1  2  4  8 16
   3  6 12 24 17
   5 10 20  9 18
 ──────────────────
   7 14 28 25 19
  11 22 13 26 21
  15 30 29 27 23
  31

Factor groups

Let G be multiplicative group and R congruence relation on this group. Then structure G/R made by break-up of G according to congruence R is also group.

This group is called factor group of group G according to congruence R.

Inverse classes

Inverse classes

Let us observe operations (addition, multiplication) applied to classes. We will look for unique inverse class to a given class.

Inverse classes to all classes exists only in systems with prime module (rεP) where all classes (excepting 0) are relative prime according to module r.

In G(2,5) are all g-numbers (excepting 0) relative prime to r=31 and makes group of order 6:

When g and r=n^k−1 are not relative prime, then problem with inverse elements can be expected.

 
In G(2,4) there does not eists inverse elements to 3 and 5. There is 3∙5 (mod 15) = 0, (3,15) ≠ 0 and (5,15) ≠ 0.

Multiplicative group is made only by classes 1 and 7.

Generally there exists inverse classes only for classes with numbers g relative prime to module r, i.e. for such g, that (g,r)=1. E.g. in G(2,6) the following group is made by numbers relative prime to r=2⁶−1 = 63:

D(7,*)
   1 7
 ──────
 1 1 7
 7 7 1

D(15,*)
    1 3 5 7
  ───────────
  1 1 3 5 7
  3 3 3 0 3
  5 5 0 5 5
  7 7 3 5 1

D(31,*)
    1 3 5 7 11 15
 ────────────────────
  1 1 3 5 7 11 15
  3 3 5 15 11 1 7
  5 5 15 7 1 3 11
  7 7 11 1 5 15 3
 11 11 1 3 15 7 5
 15 15 7 11 3 5 1

 D(63,*)
    1 5 11 13 23 31
  ────────────────────
  1 1 5 11 13 23 31
  5 5 11 31 1 13 23
 11 11 31 23 5 1 13
 13 13 1 5 23 31 11
 23 23 13 1 31 11 5
 31 31 23 13 11 5 1

Groups of higher orders

Only numbers 1 and 5 in T(6,∙) are inversible, i.e. there exists 1/x to given x. Let us mark In the group of inversible elements in Zn.

• If pε P then Ip is cyclical group.
• In is cyclical group, if n=2,4,p^t or 2∙p^t (for pεP, p>2, t≥1).

    Group            R-systems  
    1 2 4 5 7 8      R(2,6,9)         R(5,6,9)
    2 4 8 1 5 7       0                0
    4 8 7 2 1 5       1,2,4,8,7,5      1,5,7,8,4,2
    5 1 2 7 8 4       9                9  
    7 5 1 8 4 2    
    8 7 5 4 2 1

    Group        R-system  
    1 3 5 7      R(5,2,8)    R(7,2,8)   
    3 1 7 5       0           0
    5 7 1 3       1,5         1,7
    7 5 3 1       3,7         3,5
                  8           8

Physical structures

Interactions of nucleons

Schemes of binary system G(2,k) show to be suitable also for describtion of interactions of the nucleons in nuclei of atoms. Consider a set of pairs of two types of elements P, N, ie. the analogy of atomic helium nuclei (2 protons + 2 neutrons).

There are four possible ties between the elements:

    1.p1 - n1
    2.p1 - n2
    4.p2 - n1
    8.p2 - n2

All combinations of existence and nonexistence of linkages are instances of G(2,4).

We also get an analogous scheme when considering generic unidirectional links:

    1.p ->p     
    2.p ->n     
    4.n ->p     
    8.n ->n

(D.Bruncko has used a similar schemes of interactions in Čs.čas.fyz 41, 1991, when studying elastic and inelastic interactions of nucleons).

Distribution functions

Consider a set of k-particles, each of which may be in any of n possible states. Assuming that each arrangement occurs with the same probability, we get n^k groups identified by variations with repetition.

For k=2, n=3 total n^k=3²=9 skupin:

00, 01, 10, 02, 20, 11, 12, 21, 22 Maxwell-Boltzmannovo distribution.

Assumption of distribution with equal probability showed to be incorrect in microcosm. Particles are indistinguishable, they creates   groups stated by combinations with repetition.

For k=2, n=3:  =   = 6 groups:

    00
    01 or 10
    02 or 20
    11          
    12 or 21
    22

Under conditions that two particles can not be in the same state (ie. the Pauli exclusion principle), the number of groups in the file (according to combinations without repetition) narrows to three, i.e.  =  =3:

    01 or 10
    02 or 20  
    12 or 21

In a group of particles that make up the G-instance system, we expect that the states will mutually indistinguishable when they are of the same class.

Such a criterion is not used in physics, it is not respected by any of those statistics. To the structure of G(3,2) Bose-Einstein statistics is the closest, but it is generally based differently.

Binary schemes            Instance
─────────────────────────────────────
1/ 0000                   0
2/ 0001 0010 0100 1000    1  2  4  8
3/ 0011 0110 1100 1001    3  6 12  9
4/ 0101 1010              5 10
5/ 0111 1110 1101 1011    7 14 13 11
6/ 1111                  15

In G(2,4) is a total of 16 instances.

According to Maxwell-Boltzmann statistics all instances are equally probable.

Bose-Einstein's statistics make 5 groups, e.g. 0011 will be in the same group as 0101.

Sign systems

Sign system S(k) = S(±,k) is similar to binary systému G(2,k). We are looking for possible arrangements of oriented arrows (eg. spins of electrons in atoms).

Some classes, which are distinguished in binary system are in the system with directions considered equivalent. Negation (change of − to + and vice versa) does not alter the characteristics of the class.

 Instance S0(k):
    k=1:   -+
    k=2:   00
           -+ +-
           0- 0+ -0 +0
           -- ++ 

E.g. Instances 00011 and 00111 in S(5) belong to two distinct classes.

In the system S(5): −−−++ and −−+++, similarly as −−−−− and +++++ are of the same class.

Instances S(k): 

    k=1: -+
    k=2: --  ++  -+ +-
    k=3: --- +++ --+ -+- +-- ++- +-+ -++

    k=4: 
    −−−− ++++
    −−−+ −−+− −+−− +−−− +++− ++−+ +−++ −+++
    −−++ −++− +−−+ ++−−
    −+−+ +−+− 

In the case, when we admit next to + and –, also neutral possibility 0 (as e.g. with charge of particles), we get more schemes:

Genetic structures

Name "G-systems" originated as an abbreviation of the "Genetic Systems". Structures that occur in these systems exhibit some signs of heredity. Higher systems take lower systems into itself - simultaneously in the two planes: by nesting (Chapter 8) and by segmentation (Chapter 9). This analogy encourages to compare G-systems with existing knowledge about genetic structures.

Inheritance

Consider four possible variants of connection of two elements of genetic conditions A,a:

    a   A         0   1        a   A
a  aa  aA     0  00  01     a  0   1 
A  Aa  AA     1  10  11     A  2   3

Connections aa, AA are called homozygous, connections aA, Aa heterozygous. Every connection is a configuration of a single gene. Configurations are possible to interbreede, a new connection contains just one element of the two configurations of ancestors. Configuration of all genes produce so called genotype.

Platí:

By crossbreeding of homozygotes (bastardation) aa x AA only connection AA or Aa can arise. ie. heteozygots ("non-purebred" individuals, mongrels, bastards). All new individuals have the same configuration, are united (uniform).

·Crossbreeding of heterozygotes aA x aA may give rise to any connection aa,aA,Aa,AA.

Dominant and recessive elements

Element A is dominant if it controls the existing connection and determine its display; element a which is 'weaker' than A, is called recessive. When crossing heterozygotes connection of elements {A, a} forms two groups:

. aA, Aa, AA - have features determined by element A

· aa - has features determined by element a

Properties determined by element a are in the case of dominant element A reflected in average in 1/4 of all offspring.

Dominance and intermedierity

If no of elements predominates, we are talking about so called intermedierity (incomplete dominance).

 Dominance:             Intermedierity:   
 Configuration Display   Configuration Display
    00    0              00     0
   ─────────             ──────────
    01                   01     1
    10    1              10
    11                   ──────────
                         11     2

Manifestations of dominance of genes resemble mathematical structures called ideals:

Imagine a set of elements G, which is a subset of the dominant elements D, where it is possible to interbreed any two elemens (each element with each other). If from interbreeding of dεD and gεG arise descendant belonging to D, then the set D is called the ideal. Similar " dominant " behavior has zero due to multiplication.

Crossing of the pairs of genes

Offspring for which the dependence on 2 (resp. 3, ...) elements is observed is called double (resp. triple, ...) mongrel - dihybrid (trihybrid ...).
Consider now two genes with possible elements a, A, b, B.
We will write the first gene on the first two places, and second gene on the two others places. We will replace lowercase letters by zeros, and uppercase by ones, for example: aaBb = 0010 AaBB = 1011. Possible connection variants (with writing of binary numbers in decimal on the right):

Experiments have shown that if the number of offspring with configurations from the set {1,2,4,8,7,14,13,11} is t, then the number of remaining {0,5,10,15},{3,6,12,9} is proportional to t² or 1.

Two cases are distinguished: 1/CIS (pairing) and 2/ TRANS (repulsion):

Dominant pairs of genes

When A, B are dominant configurations form four groups:

Distribution of the number of descendants in groups (ab:aB:Ab:AB) corresponds to the experimental results. E.g. for CIS (t=7) 49:15:15:177, for TRANS (t=127) 1:16383:16383:32769 [Hončariv].

Breakdown of G-system

Let us mark now connections aa=bb=0, aA=bB=1, Aa=Bb=2 and AA=BB=3. E.g. aaBb = 02, AaBB = 23, .... We get these combinations for mongrels:

k=2 (dihybrid), G(4,2) k=3 (trihybrid), G(4,3)