Permutational equality and permutational proportionality of rhythmic entities

Let us suppose a finite set (e.g., measure) S is given. The permutation of the elements of set S (note durations) is any sequence that can be formed from its elements. If we wish to form all the permutations of the elements of set S we can do so by using the so-called lexicographic order of elements - the principle of compiling a dictionary or a telephone directory, for example. If S={a,b,c}, we can get six permutations from it by using this method: (abc), (bac), (bca), (cab) and (cba). Since the obtained permutations together make up an ordered set, we may denote them as S₁, S₂, ..., S₆. In the case that all elements of set S are different, the resulting permutations are called permutations without repetition. If set S has n different elements, from it we derive P_n=1×2×... ×n=n! permutations without repetition. If all the elements of set S are not different, the result obtained are permutations with repetition. If set S which contains n elements consists of n₁ elements of the first kind, n₂ elements of the second kind,..., and n_k elements of k-th kind (n=n₁+n₂+...+n_k), the result is [`P]<sub>n</sub>=[`P]_{n=n1+n2+...+nk}=[n!/n1!n2!...nk!] permutations with repetition. For example, from set S={a,a,b} we get [`P]<sub>3</sub>=[`P]₃₌₂₊₁=[3!/2!1!1!]=3 permutations with repetition: (aab), (aba), and (baa).

In order to determine rhythmic entities of the same duration (e.g., measures) which are equal in terms of permutations, we will apply the following coding. To each note duration (including rests) a corresponding numerical value (fraction) will be assigned: [1/1] ( ), [1/2] ( ), [1/4] ( ), [1/8] ( ),... In the case of dotted notes the corresponding fractions will be [3/2] ( ), [3/4] ( ), [3/8] ( ), [3/16] ( ),...

To every rhythmic entity S which is given by a sequence of note durations expressed in fractions, e.g., to S=(¹/₄, ¹/₁, ¹/₈, ¹/₈), may be assigned a number, T(S) which is equal to the sum of the denominators of the given fractions. In our example, this would be T(S)+4+4+8+8=24.

An interesting possibility for the study of rhythmic structure and its invariants is the determining of rhythmic entities (e.g., measures) which are equal in terms of permutation. Two rhythmic entities of the same duration are permutationally equal (or they have the same rhythmic content) if one of them can be obtained by permutation of the note durations of the other one. For example, rhythmic entities (¹/₄, ¹/₄, ¹/₈, ¹/₈) and (¹/₄, ¹/₈, ¹/₄, ¹/₈) are permutationally equal. The use of permutationally equal rhythmic entities of the same duration will be called permutational repetition.

If we apply this to all rhythmic entities of equal duration, e.g., to every measure, we observe a very interesting property: two rhythmic entities S₁ and S₂ with the same duration are permutationally equal only if T(S₁)=T(S₂). In other words, the same numerical value T(S) will correspond to all permutationally equal entities of the same duration (e.g., measures). Another interesting question would be: which permutation of the original note duration sequence are we looking at? This means that to every number T(S) can be assigned a lower index k - the ordinal number of the corresponding permutation. For example, the set of note durations S=(¹/₄, ¹/₄, ¹/₈, ¹/₈) where T(S)=24, gives six permutations with repetition: (¹/₄, ¹/₄, ¹/₈, ¹/₈), (¹/₄, ¹/₈,¹/₄, ¹/₈), (¹/₄, ¹/₈, ¹/₈,¹/₄), (¹/₈, ¹/₄, ¹/₄, ¹/₈), (¹/₈, ¹/₄, ¹/₈, ¹/₄), and (¹/₈, ¹/₈, ¹/₄, ¹/₄). These permutations can now be coded as 24₁, 24₂, 24₃, 24₄, 24₅, and 24₆, respectively. Note also in 24₁, 24₂, 24₅, and 24₆ the existence of a center of antisymmetry - the transformation which translates [1/4] into [1/8] and vice versa (Fig. 2.3).

Figure 2.3 Sequences 24₁ - 24₆ and the center of antisymmetry *.

To every value of the number T(S) corresponds a uniquely determined sequence with the index k. Depending on whether the permutations are with or without repetition, to k are assigned the values P_n, or [`P]_n.

The previously analyzed geometric symmetries: translational repetition and retrograde repetition are merely special cases of the permutational equality of rhythmic entities. Translational repetition appears when the entities have the same codes T(S)_k, whereas retrograde repetition appears when the permutations are equally distanced from the beginning and the end of the permutational sequence: 24₁ and 24₆, 24₂ and 24₅ (Fig. 2.4).

(a)

(b)

Figure 2.4. Translational repetition (a) and retrograde repetition (b) as forms of permutational repetition.

For every value of the number T(S) within a musical piece some or all possible permutations of note durations may be used. Furthermore, the order in which they are used is also relevant. Thus we speak about the regular (e.g., cyclic) or irregular permutational repetition of rhythmic entities of the same duration.

Another aspect of permutational regularity may be proportional permutational repetition. We say that two rhythmic entities S₁ and S₂ are permutationally proportional if one of them consists of proportionally augmented or diminished note durations that also make up the other rhythmic entity. If the coefficient of proportionality k (most often k=2ⁿ, where n is an integer), then the durations of the rhythmic entities S₁ and S₂ have the same proportion ratio. Furthermore, if k=2, we define this as permutational augmentation, and if k=¹/₂, then it is defined as permutation diminishment. As to the order of permutations in the case of permutational proportionality, we may apply a process analogous to the one used in the case of permutational equality. With this in mind, it is important to stress that proportional repetition and its corresponding geometric symmetries (dilatation, dilatational reflection) represent only a specific case of permutational proportional repetition.

To illustrate permutational proportional repetition we will examine two rhythmic entities (e.g., two measures) S₁=(¹/₂, ¹/₄) and S₂=(¹/₂,¹/₈, ¹/₈), where T(S₁)=6 and T(S₂)=18. If all note durations of these entities are diminished by two (k=¹/₂) we get a new set of note durations S=(¹/₄,¹/₈, ¹/₄,[1/16],[1/16]), where T(S)=48. In this case T(S)=[(T(S₁)+T(S₂))/k]=2(6+18)=48. By defining the rhythmic entities whose numerical value T(S) satisfy such relationships, we may find examples of proportional permutational repetition.