Algebraic action on Diminished chords

Diminished chords are always ignored by musicians when they learn music theroy. Musicians and music theorists always study the properties and relations of Major and Minor chords or Consonant chords. In Neo-Riemannian theory of TI and PLR groups, he suggest a way of applying Dihedral group, D 12 to study the relation of Consonant chords. In this paper we will introduce some concepts of group theory to understand the algebraic relationship between the Diminished chords.


.1 Dihedral group
A dihedral group is the group of symmetries of a regular polygon, which includes rotations through an angle and reflections about some axes.In algebra, D n refers to the dihedral group. A regular polygon with n sides has 2n different symmetries,i.e. n rotational symmetries and n reflection symmetries. Usually, we take n ≥ 3 here. The associated rotations and reflections make up the dihedral group D n . If n is odd, each axis of symmetry is the lines that pass through each vertex and the midpoint of its opposite edge.If n is even, then the axes of symmetry is the lines that pass through a vertex and its opposite vertex and also the the lines that pass through the midpoint of an edge to the midpoint of the opposite edge of the polygon. In either case, there are n axes of symmetry.
Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.

Group action
Let (G, ·) be a group and let S be a non-empty set. We say that G acts on S if we are given a function Given an action of G on S we can define the following sets. Let s ∈ S.Define the orbit of s as, Note that Orb (s) is a subset of S, equal to all the images of the element s under the action of the elements of the group G. We also define the stabilizer of s as, Stab(s) = {g ∈ G : g ⋆ s = s}.
Note that Stab (s) is a subset of G. In fact, it is a subgroup. Note that every element g ∈ G defines a function S → S by s → gs.

Music theory
Music is basically the combination of sounds. Sounds are produced by vibrations that propagates in a medium, different frequency of vibration produce differnt sounds. For example,a middle C note in a piano produces a frequency of 262Hz, whereas a middle F note produces a frequency of 349.228Hz. In music theory, an Octave is the interval above a musical note which produces twice the frequency of that lower note. For example, the note A 4 produces a frequency of 440Hz,while the note A 5 produces a frequency of 880Hz and A 3 produces the frequency of 220Hz. There are 12 notes in an octave. The interval from a note to the next note is called a Semi-tone. A semi-tone higher to a note is denoted by the symbol ♯, pronounced as Sharp.A semi-tone lower note is denoted by the symbol ♭. For example, C ♯ denote the note which is immediately next to the C note and D ♭ denotes the note which is immediately before D. Note: C ♯ and D ♭ denotes the same musical note.
Note: The C in the table represents the 'Middle C' of the piano and the note A is in a standard tuning at 440Hz.

Algebraic action
Since it is useful to translate pitch class to integer mod12 , we will consider 0 to be C and construct the pitch class as in the following table 0  1  2  3  4 5  6  7  8  9 10 11 For instance, the number 0 refers to the note C, since 0=0mod12 and the number 15 refers to the note D ♯ ,since 15=3mod12.Thus, we can map every note in music to an element in Z 12 . Now we can define a Triad as a set of three elements in Z 12 . For instance we take a C-major triad {C, E, G} as {0,4,7}.
In general any Major triad is defined as {x, x + 4, x + 7}, any minor triad is defined as {x, x + 3, x + 7} and any Diminished triad is defined as {x, x + 3, x + 6}, where x is the root note of the Triad. Note that this is an ordered set.

TI group
The Transposition is defined mathematically as a function from Z 12 to Z 12 such that, where, n ∈ Z 12 and The Inversion is defined as a function from Z 12 to Z 12 such that, where, n ∈ Z 2 These transposition and inversion forms a group under the operation Composition defined as, For example:

Musical clock
The idea of converting Musical notes to elements in Z 12 help us to construct a Polygon with 12 vertices, where each vertex corresponds to a musical note. Thus we can now use Dihedral group to study the Algebraic structure of Diminished chords. The following figure shows us how the musical notes and elements in Z 12 are related with the 12-gon.

The action
We define the set of diminished triads as S={(x,x+3,x+6) , x∈ Z 12 }. Let the dihedral group (D 12 , ⋆) act on the set of diminished triads.The action is defined as a map from where, i ∈ Z 12 , j ∈ Z 2 , r is the rotation and f is the reflection. Then, 1. ∃ e ∈ D 12 such that (e, s) = (r 0 f 0 , s) = r 0 f 0 · s = s, ∀ s ∈ S 2. g 1 · (g 2 · s) = g 1 · (g 2 (s)) = g 1 (g 2 (s)) = g 1 ⋆ g 2 (s), ∀ g 1 , g 2 ∈ D 12 , s ∈ S 3. r 12 = f 2 = e and f rf = r −1 Theorem 3.1. The TI group (T I, •) is isomorphic to a Dihedral group (D 12 , ⋆) Proof. T I =< T n , I m >, where T 12 = I 2 = e and I 1 T 1 I 1 = T −1 D 12 =< r n , f n >, where r 12 = f 2 = e and f rf = r −1 Let φ : T I → D 12 such that φ(T n ) = r n and φ(I n ) = f n Since φ maps the generator of TI group to generator of D 12 and they have same orders, the mapping φ is an isomorphism.

Orbits and Stablilizers
The orbit of s ∈ S is {g · s|g ∈ T I}. Since the action of T n can map any triad to another triad, the orbit of any s ∈ S is S.
The stabilizer of s is {g ∈ T I|g · s = s}. Clearly, T 0 stabilizes every element of S since T 0 (x, x + 3, x + 6)=(x, x + 3, x + 6) and there is an other stabilizer for every s ∈ S which is a refection about the axis passing through the vertices (x + 3)mod12 and (x + 9)mod12.

Geometric structure
To study the geometric structure, we divide the set of Diminished triads into three families, where any two Triads in a family share two notes. For example: C diminished triad and F ♯ diminished triad belong to the same family. The following table shows the Families of diminished triads. Family By connecting the common notes in the triads we can clearly see that each family of diminished triads is Tetrahedral ie.Triangular Pyramid shaped. The following figures show the Geometric structure of Diminished triads. In the following figures each face corresponds to a unique Diminished triad.

Relation and Jump
Now we define a function from Z 12 to Z 12 as This new function is called as Relation since it takes a diminished chord to another diminished chord in its family.
In other words, R 1 takes C-diminished triad to D ♯ -diminished triad and R 2 takes D-diminished triad to G ♯ -diminished triad Next we define another function from Z 12 to Z 12 as This new function is called as Jump since it takes a diminished chord to a diminished chord in another family. Proof. The set JR has the following elements, JR={R 0 , Here any element is in the form R n J m , where n∈ Z 4 and m∈ Z 3 . The function J 0 is composing J zero times and the function J 3 is same as the function R 0 Let R n J m , R p J q be any two elements in the set JR. Now R n J m • R p J q = R n+p J m+q Since n, p ∈ Z 4 , n + p ∈ Z 4 .So,R n+p ∈ JR. Also J m+q ∈ {J 1 , J 2 , R 0 } since m, p ∈ {1, 2, 3} Thus the set JR is closed under composition.
Let R n J m ,R p J q ,R a J b be any three elements from the set JR.
So associativity holds.
The identity element e = R 0 Claim: Thus identity exists in the set JR.
Let R n J m be an arbitrary element in the set JR. Then the element R 4−n J 3−m is the inverse of R n J m , since R n J m • R 4−n J 3−m is R 0 .Thus inverse exists for every element in the set JR under the operation composition.
Let R n J m and R p J q be any two elements in the set JR. Then, Thus the set JR forms an abelian group under the operation composition.

The orders of elements in JR group
The orders of each element in JR group is, Since JR group is generated by a single element it is a cyclic group.

Subgroups of JR group
From the fundamental theorem of cyclic groups,for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each divisor. Order

The action of JR on the set of Diminished chords
The set of diminished triads,S={(x, x + 3, x + 6),x∈ Z 12 }. The action is defined as a map from JR × S → S (R n J m , s) → (R n J m (s)) = (R n J m (x), R n J m (x + 3), R n J m (x + 6)) where,n ∈ Z 4 , m ∈ Z 3 , s ∈ S such that; (i)R 0 (s) = s, f orall s ∈ S (ii)(R n J m • R p J q )(s)=R n J m (R p J q (s)), for all R n J m , R p J q ∈ JR and s ∈ S

Orbits
For any s ∈ S, Orb(s) = {R n J m (s) : R n J m ∈ JR} Orb(s)=S, for all s ∈ S Thus, the JR group acts transitively on the set of diminished triads.

Stabilizer
Stab(s) = {R n J m ∈ JR : R n J m (s) = s} Clearly, R 0 is the stabilizer of this action.