Proceedings IX CIM 1991, Genova, November 13-16 - AIMI

An Illustration: "Temporal" and "Metrical" Representations of Rhythm
As an illustration of the application of the policy of parsimony I present here a comparison of two
possible representations of rhythm. No attempt has been made to incorporate a sophisticated
representation of pitch into either of these representation schemes, though the possibility of this is not
ruled out. The first, which I call the "temporal" scheme, represents a piece of music as a simple
sequence of symbols (numbers, perhaps) separated by spaces, with each group of three symbols
representing a single note by its onset time, pitch and duration respectively. Notes are represented in
order of increasing onset time. The second representation, called the "metrical" scheme, consists of a
sequence of symbols or symbol sequences, separated by spaces, representing metrical units all of equal
duration. The onset time of a unit is coincident with the end of the previous unit, or immediate if the
unit is the first of the sequence. A unit may be a single note, represented by a single symbol indicating
its pitch; or a unit may be a group of simultaneous notes, in which case it is represented in the form x,y
etc. where x and yare symbols indicating pitch; or a unit may be split into two subunits represented as
[X Y], where the sum of the durations of X and Y is the duration of one metrical unit, and the onset
time of Y is coincident with the end of X; or a unit may be both split and have one or more
simultaneous notes lasting its entire duration, represented for example as [X Y],x,y; or a unit may be
null (i.e. consist of no symbol) in which case it represents a rest of one unit's duration. Subunits have
the same syntax and semantics as full metrical units, so there can be subunits of subunits etc. The actual
duration of metrical units and subunits is given by a "metre indication" at the head of a sequence. The
format for this is similar to the format ofmetrical units so that a number in the metre indication gives the
duration of a note occupying an analogous position in the metrical unit. A triple metre, for example,
could be indicated by [2 1],3. Where a subunit's duration is not explicitly indicated, it is taken to be
half the appropriate unit's duration. Where a note extends across one or more metrical boundaries, it is
necessary to represent the note by more than one symbol, in fact one in each of the metrical units
through which it lasts. The first of these symbols will have an underscore character "_" following it to
indicate that the note continues beyond the end of the metrical unit (similar in function to a tie in music
notation). The last symbol will have an underscore before it to indicate that the note begins before the
beginning of the metrical unit, and any intervening symbols will have underscores both before and
after. (This "metrical" representation scheme has some similarities to the scheme worked out on a
wonderfully rational basis by Bernard Bel (1990).)
Properly, one should include the "size" of system necessary to evaluate instantiation functions in
the test of a representation's parsimony. In a real computer representation this is quite possible of
course, but here I have instead aimed to keep the evaluation of functions simple, hence presumably
requiring only a small evaluator. Functions are applied in the order listed. The representation string is
searched from its start for a match with the left-hand side of the function. When a match is found the
matching segment in the representation string is replaced by the right-hand side. Lower case Roman
letters can match anyone symbol, including any preceding or following underscore. Lower case Italic
letters can match any number of symbols (including none). Upper case Roman letters can match the
whole of a metrical unit (which might be null). Where there is a question mark, the predicate following
it must be true for the function to be applied. Functions for rational number arithmetic are assumed to
have been already defined with priority greater than the functions listed here. Note that spaces are
significant but ends of lines are not, and numbers are taken to be single symbols.
The "metrical" scheme is clearly more complex, and so will require larger instantiation functions
than the "temporal". In fact something like the "temporal" scheme is likely to be required as an
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