Classification

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)

  • Cadinal. There are 6 groups (dual complemented moitif: PCS and his complemented PCS) 0/12, 1/11, 2/10, 3/9, 4/8, 5/7, 6/6. Base of the musaics numbering, from 1 (empty set - 0/12) to 88 (checkerboard - 6/6).
      • representation : couple of integers in {0/12, 1/11, 2/10, 3/7, 4/8, 5/7, 6/6}
  • is-operations. Multiplicative and complemented operations than depend of n, their number is : phi(n) * 2. With n=12, is-operations set is :{M1, M5, M7, M11, CM1, CM5, CM7, CM11)
  • Transposition degree (or transposition cycle) . Cardinal of cyclic-orbit. In other words, determines number of distinct PCS in cyclic-orbit, if is less than, then it corresponds to a limited transposition PCS. All is-motif of a musaic have same transposition degree, so we can attach this property to a musaic.
      • representation : Integer in { 1, 2, 3, 4, 6, 12 }
  • Stabilizer set. Set of operations of transformation that keep musaic invariant. In twelve-tone equal temperament (n=12), there are 13 stabilizer classes.
      • representation : ddo (decorated diamond model)
  • Transformer set : A set of operations which, taken individually, when applied to a is-motif give another is-motif. This is dual complement to stabilizer class.
      • representation : ddo
  • Motivation degree. Determines, for a given musaic, the number of its distinct is-motif, from 1 to 8, exactly 1, 2, 4 or 8.
    • Motivation is defined as : #is-operations divided by #stabilizers, where #is-operations depend of n and #stabilizers depend of PCS given. For example, musaic n° 88 (checkerboard) is complementary to itself and also invariant by all rotations, so its fecundity degree is 8/8=1
      • representation : integer in { 1, 2, 4, 8 } (reminder, for n = 12)
  • MusaicFecundity : This is number of distinct PCS in orbit of a musaic (cardinal of orbit). This is also sum of cyclic-orbit cardinal of each distincti is-motif of musaic. For example, musaic n°45 (called "pointing"), fecundity is motivation degree multiplied by transposition degree = 8 * 12 = 96
      • representation : integer in set motivation degree X transposition degree, so { 1, 2, 4, 8 } X { 1, 2, 3, 4, 6, 12 } = { 1, 2, 3, 4, 6, 8, 12, 16, ..., 96}

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

  • cardOrbitMode : This is number of distinct PCS in set (orbit) of all modes of a PCS.
    • representation : integer less than or equal to cardinal of PCS
  • cardOrbitCyclic : This is number of distinct PCS in cyclic orbit of PCS.
    • representation : integer value is : (n x cardOrbitMode) / cardinal, value is less than or equal to cardinal of n.
    • formula derived from equality in these two ratios : n/cardOrbitCyclic and cardinal/cardOrbitMode

under construction...