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:

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

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.

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. 

Criteria for classifying musaics

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...