Rhythm
generation in Melody Generator II
A rhythm in Melody Generator II is determined by two parameters: meter
and number of syncopations.
Meter, Metrical weights, Metrical
stability
Meter is the temporal framework in which a rhythm is defined. Meters
differ in the number of beats per measure as well as the subdivision of
the beats. In MGII the beats of a n/4 meter (4/4, 3/4, 2/4) are
subdivided into 4 ‘sub-beats’, in case of the 6/8 beat, each beat is
subdivided into 6 sub-beats.
The beats in a meter differ in their weights as shown in Figure 1 which
display the meters 3/4 and 6/8 as a tree. These metrical weights
correspond to the degree of perceptual markedness of the different
positions of a meter (positions of high or low weight are often called
strong and weak beats).
Thus, a meter defines the moments in time at which a tone can be
positioned, as well as the metrical weight of each of these positions.
Figure 1. Tree-representation of a
3/4 and a 6/8 meter with the weights of the branches (metrical
positions).
Since we know that the most important characteristic of a rhythm is the
degree to which it evokes a meter in a listener, I had to be able to
generate rhythms that differed in that sense. This is the opposite of
the parsing of rhythms that I had performed in my research in order to
determine which beat is induced by a rhythm. In order to manipulate
this feature in the rhythms generated in Melody Generator, I introduced
the concept metrical stability as explained below.
Metrical
stability refers to the degree of association between a rhythm
and a meter. A rhythm is stable in the context of some meter if it
strongly induces that meter; it is unstable if it does not induce that
meter.
Povel & Essens (1985) have shown that the main factor in the
induction of meter is the distribution of accents in a rhythm: the more
this distribution conforms to the pattern of metrical weights of some
meter, the stronger that meter will be activated (and the higher the
metrical stability of the rhythm will be). The degree of accentuation
of a note in a rhythm is determined by temporal features, and by its
intensity, pitch, spectral composition and duration. Here, I only
mention the temporal features which are by far the most important
determiners:
- Inter-onset-interval (note value in musical terms): the longer
the relative inter-onset-interval (IOI) between the current note and
the next note, the higher the accentuation of the note.
- Grouping: the following tones are relatively accented due to
their position in a sequence of tones
- Isolated tones (relative isolation to be defined)
- The second tone of a group of 2 notes (group to be defined)
- The first and last tones of groups consisting of 3 or more tones
Now we can define the metrical stability of a rhythm in relation to
some meter: The metrical stability of a rhythm expresses the extent to
which the accentuation pattern of the rhythm matches the weight pattern
of a meter. This correspondence can be quantified by means of Pearson’s
coefficient of correlation.
Towards generating rhythms varying
in metrical stability
In the context of Melody Generator we want to vary the metrical
stability of rhythms generated within a given meter: from fitting very
well to being more or less syncopated.
From the accentuation rules mentioned above, it follows that metrical
stability increases if:
- longer notes occur at strong metrical positions (positions with a
high weight);
- groups of two notes are positioned with the last note of the
group on a strong metrical position;
- groups of more than two notes are positioned such that both the
first and the last note of the group coincide with relatively strong
metrical positions.
Taking into account the above considerations, we are now in a position
to generate rhythms of differing stability within a given meter. We
know that if we want to make a stable rhythm its accent pattern should
match the weight pattern of the meter. Thus, stable rhythms are those
in which the strongest metrical positions have a note, are followed by
one or more positions without a note and are preceded by one or more
notes.
Using these criteria we can indeed generate rhythms of high metrical
stability. This is accomplished as shown in the following figure:
Figure 2. Construction of a stable
rhythm within a 4/4 meter
As shown in Figure 2, the metrical stability of a rhythm is decreased
by adding one or two syncopations at randomly selected positions.
Longuet-Higgins & Lee (1984) define a syncopation as follows: “If R
is a rest or a tied note {in a musical score}, and N is the next
sounded note before R, and the weight of N is no greater than the
weight of R, then the pair [N, R] is said to constitute a syncopation.”
p. 430. Using this definition, syncopations are added in the following
way: starting at a random position within a measure (but not the
first), the first following slot is found with a weight equal to the
lowest weight (-4 for 4/4; -3 for the other meters) + 1, remove that
note and add a note to the previous slot. This results in the addition
of a note after a beat. (Both method and description can be improved, I
think)
Thus, when this step is accomplished, the locations at which notes will
be produced are fixed. There is however one other parameter which is
taken into account in this step: the setting of the local constraints
All bars same rhythm and Some bars same rhythm.
References
The generation of rhythm in Melody Generator II is mainly based on the
following articles:
Lerdahl, F., & Jackendoff, R.
(1983). Chapter 4 in: A generative
theory of tonal music. Cambridge, MA: M.I.T. Press.
Longuet-Higgins, H. C., & Lee, C. S. (1982). The perception of
musical rhythms. Perception,
11, 115 – 128.
Longuet-Higgins, H. C., & Lee, C. S. (1984). The rhythmic
interpretation of monophonic music. Music
Perception, 1, 424 – 441.
Povel, D. J. (1981). Internal representation of simple temporal
patterns. Journal of Experimental
Psychology, Human Perception and Performance, 7, 3 - 18.
Povel, D. J. (1984). A theoretical framework for rhythm perception. Psychological Research, 45, 315-337.
Povel, D. J., & Essens, P. J. (1985). Perception of temporal
patterns. Music Perception, 2,
411 - 441.
(For a complete list of references click here)
RhythmGenerationInMGII.html (last update 6.5.2008)
Created in SeaMonkey
© D.J. Povel, 2008
