## Objects of classification

We have seen in 88MusaicsExplained that the 88 musaics alone represent all possible combinations of the 4096 PCS of the 12-PC set.

The first reduction is represented by the 352 equivalence classes transposition, which translates geometrically in a translation of same is-motif (matrix pattern).

352 is the number of interval structures (or classes of polygons) that can be inscribed in the chromatic circle.

For example, the interval structure of the minor 3-note chord (3,4,5) is identified by a triangle structure type (and a is-motif in matrix representation), one among the 352:

(3, 4, 5) intervallic structure and its PCS in cyclic-orbit (12 transposed PCS)

We have also seen that a single is-motif can, by the bias of geometrical operations, hide as many as seven other than it!

Geometry representation of is-motif  (in the center of the figure above) is a structure with 8 possible "views" (8 is-motifs), that we call musaic

First, we will look at the peculiarities of the geometrical transformation operations we have used, namely the rotation of a half-turn around a point and the left and right diagonals, as well as complementarity.

## Singularity of transformation operations

### 2-cycle

All operations (rotations and complement) are of the 2-cycle type.

Applying the same operation twice to a musaic brings it back to its initial state: Rotation of half a turn with respect to the same point, the same diagonal, including the complementarity operation. Illustration below.

(this feature is highlighted by the diagonal of the table below)

Sample of 2-cycle operation

### Remakable identies between operations

There is a close link between the rotation operations of half a turn around a point and diagonals : one of these three being the compound of the other two, as shown in the table below and the animated example.

It is a well-known structure of mathematics: Klein four-group

Here is an illustration of these characteristics.

Musaic n°45

« pointing  »

Klein group in action

Example of group actions on a musaic

2-cycle operations and remakable identites

By adding the neutral operation (identity), we can summarize these relationships by this geometric representation, In which colors of the different is-motifs have been reported

4 operations and their relationships

However, the complementary operation (also 2-cycle) is missing here, this operation extends the four others and give, in all, 8 operations (4 + 4 complemented).  Below, the table in the same logic as the previous one.

Table of musaic transformations

8 operations (4 + 4 complemented) and their relationships

8 operations in a single geometric figure : decorated diamond operations model

4 operations (disc) + 4 complemented same opérations (ring)

As we saw in the 88MusaicsExplained, transformation operations can be expressed algebraically : identity = m1, rotation by left diagonal = m5, by right diagonal = m7 and around a point = m11.

By focusing on transformation operations, we obtain an efficient classification tool, which is why we have iconographed the concept (decorated diamond operations model - DDOM), which we will use to represent twice  "transformer set"  ans "stabilizers set" (see below).  Let's show them on DDOM :

DDOM (Decorated Diamond Operations Model)

## Criteria for classifying musaics

Let's take an example. Some musaics have, as a set of stabilizers, the only operations m1 and m7. This characteristic can be considered as a classification feature that we could represent by a singular ddom undressed ;-), the stabilizing operations are removed from the ddom, as shown in the following figure :

ddom without stabilizers { m1, m7 }

Others examples :

ddom without stabilizers { m1, m5, cm7, cm11 }

For this one, chastely, we keep the empty summits dressed up - they characterize the relationship "complementary to oneself".

ddom : { m1, cm11, m11, cm11 }

A musaic can be characterized by its stabilizers (operations that do not affect  is-motif) or its transformers (operations that produces another is-motif): transformers and stabilizers are complementary. In particular, with n = 12,  there are only 13 classes of ddom (transmorfers/stabilizers) :

So, 88 musaics can be partitioned in 13 stabilizer classes :

Some musaic characteristics (n = 12)

-- PCS scope------------------

under construction...