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 S 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=( 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 ( 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 S
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 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) (a)
(b)
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
S To illustrate
permutational proportional repetition we will examine two
rhythmic entities (e.g., two measures) S |