Schematic algebra - Application

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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).

G-system G(2,4)

In case k=4, the set of all  24=16 instances breaks into  6 classes,  corresponding G-system G(2,4) has the following structure:

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.

Distance schemas in G(2,4)

Example of instances and their distance schemas in G(2,4):

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).

Example - natural modality

In G-system G(2,12) modality with number of instance 1387, binary schema 010101101011 and distance schema 1 2 2 1 2 2( 2) corresponds to white keys of piano.

This modality has k=12 transpositions, i.e. 12 possible musical performances (scales or tonalities C major, C# major, D major, B major).

Each modality (in any of 12 performances) has in addition several modes, i.e. tone orders (regarding selection of main tone). Number of modes is equal to number of tones in modality, i.e. level of instance, in our case L=7. Modes in principle does not differ from scales (scales keep mode names,…):
1387 7 12 010101101011 1 2 2 1 2 2( 2)

Mathematical structures

Polynoms of congruence classes

Polynoms of third degree in field of congruence classes mod 2 makes system G(2,3):

Derivation reduces degree of polynoms and also number of distinct polynoms. Derivation of polynoms from G(2,3) pass (regardless of coeficients) into polynoms from G(2,2):

According to mod 2 there are only 2 results x² and x²+1:

and these are from G(2,1):

Extension of finite field

Adjunction of element q of degree h to field T makes a field, where all elements have form
a0 + a1q + a2q² +...+ ah−1qh−1


element q given by equation q²=5 extends field Z with all numbers of the form: a0 + a1q = a0 + a1√5;

element s³=5 with numbers: a0 + a1s + a2s² = a0 + a1√5 + a2(√5)².

If field of coeficients a0,a1,a2,.. is finite, it is possible to use G-systems to depict this extension.

For example, let as have field Z3 with elements {0,1,2} extended by element q, where q²=c [Blažek, Koman].

So we get system G(3,2) from the system G(3,1):

Similarly field Z2 with elements {0,1} extended by element s, s³=c, makes schema G(2,3):

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:

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,..,nk−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.

Lagrange’s theorem
   1 2 4 8
   3 6 12 9
   5 10
   7 14 13 11

Let us consider system G(n,k) as group and nested system G(n,d), d|k, as its subgroup. Then module r of G(n,k) can be seen as order of the group and nesting quotient as index of subgroups.

System G(2,4) has two “subgroups”: G(2,2) a G(2,1), where G(2,1)<G(2,2)<G(2,4). Modules of systems are r(4)=15, r(2)=3 and r(1)=1.

It corresponds to Lagrange’s theorem r(1)|r(2)|r(4), i.e. 1|3|15.

There are instances {0,5,10,15} belonging to G(2,2), i.e. after cancelling out by quotient of nesting (5) we have {0,1,2,3}.

Similarly G(2,1) has instances {0,15}, i.e. after cancelling out {0,1}.

Nesting and subgroups

There is distinction of nested systems and subgroups. Order of G-system is the number k, order of group is the number r=nk−1.

All classes of systems with prime order are self classes. But not all self classes must be relative prime to module r=nk−1. It holds only in case of prime module r, i.e. when r is Mersenne’s prime. In opposite case there is more self classes then relative prime classes.

In case G(2,11) is k prime. But number r=211−1 = 2047 = 23∙89 is not prime. In G(2,11) there exists φ(2047) = 22∙88= 1936 instances, i.e. 1936/11=176 classes relative prime to 2047. And there is 2046/11 = 186 self classes in G(2,11).

Nesting of G-systems and groups continues to structures (systems resp.subgroups) of prime order.

G-systems with prime order and composite module r can have also other then trivial subgroups!?!

Compositional sequences

Systems nested into G(n,k), having orders d1|d2|...|k reminds so called compositional sequences. E.g. for k=12 we get the following 3 sequences:

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

These sequences corresponds to three rows between numbers 1 and 12 in the following diagram:

    4    6
  2    2    3

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:

Relation of classes
Pairs of classes
Break-up to pairs of classes.

In case, when whole G-system breaks-up into pairs of classes, it holds:


   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

In system G(2,5) there is 8 contrast classes. Classes make 4 pairs (−0=31; −1=+15; −3=+7; −5=+11)

Therefore 2k|(r−1) i.e. 10|(31−1).

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=nk−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:

Inverse classes - examples
   1 7
 1 1 7
 7 7 1
    1 3 5 7
  1 1 3 5 7
  3 3 3 0 3
  5 5 0 5 5
  7 7 3 5 1
    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
    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.

Cyclic group

For n=9=3² we get after removing nested rows and columns cyclical group:

    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
Non-cyclic group
For n=8=2³  group is not cyclic  (it breaks up into cyclical subgroups of second order):
    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

Maxwell, James Clerk
Maxwell, James Clerk [], 1831-1879, (English) physicist, author of the general theory of electromagnetism. He studied the molecular physics.

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 nk groups identified by variations with repetition.

For k=2, n=3 total nk=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 f_binom_nk1_k   groups stated by combinations with repetition.

For k=2, n=3: f_binom_321_2  =  f_binom_4_2  = 6 groups:

Bose-Einstein distribution
    01 or 10
    02 or 20
    12 or 21

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. f_binom_n_k  = f_binom_3_2  =3:

Fermi-Dirac's distribution
    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: --- +++ --+ -+- +-- ++- +-+ -++
    −−−− ++++
    −−−+ −−+− −+−− +−−− +++− ++−+ +−++ −+++
    −−++ −++− +−−+ ++−−
    −+−+ +−+− 

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.

Mendel, Johann Gregor
Mendel, Johann Gregor [mendl], 1822-1884, Moravian scientist of German nationality. He -after several years of systematic experiments with plants- expressed observed relationships mathematically. Three Mendel's laws (On uniformity of crossbreeds, On splitting of characters and On free combination of aptitudes) form the basis of modern genetics.


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.


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):

Morgan Thomas Hunt
Morgan Thomas Hunt [], 1866-1945, American zoologist and geneticist. He supplemented Mendel's laws by further laws, explaining the function of chromosomes in heredity: 1/ genes in chromosomes are linearly arranged, 2/ The number of linkage groups is equal to the number of pairs of chromosomes.

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)