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 : octotrope
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 (octotrope), which we will use to represent twice "transformer set" ans "stabilizers set" (see below). Let's show them on octotrope :
Octotrope
Criteria for classifying musaics
Let's take an example. Some musaics have, as a set of is-stabilizers, the only operations m1 and m7. This characteristic can be considered as a classification feature that we could represent by a singular octotrope, the stabilizing operations are highlighted, as shown in the following figure :
m1-m7
Others examples of stabilizers class :
m1-m7-cm5-cm11
m1-m7-cm5-cm11
(undressed)
m1-m11-cm1-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 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-motif. All PCS into same cyclic orbit share same intervallic structure (Ex : CMaj, C#Maj, DMaj, ... all are Maj)
representation : musaic, clock, ...
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 }
IS-Stabilizer set. Set of operations of transformation that keep musaic is-motif invariant. In twelve-tone equal temperament (n=12), there are 13 is-stabilizer classes.
representation : octotrope)
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 : octotrope complement
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 distinct 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
formula in PCS context : (this.cardinal * this.cyclicPrimeForm().orbit.cardinal) / this.n
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...