The Scale-based
model
Assumptions
This model is largely based on the same assumptions as the Chord-based model, with one essential
difference: the skeleton is not formed of chord tones, but consists of
a scale fragment.
For a few examples click here.
The idea that a melody can be reduced by stepwise removing less
important notes thereby revealing the underlying framework or
skeleton has a long history in music theory. The idea has been given a
theoretical foundation by Schenker (1935), part of which was captured
in a set of rules by Lerdahl & Jackendoff (1983). Below we describe
the formalization of the
reverse process, in which a skeleton is transformed into a melody,
proposed by Baroni, Dalmonte & Jacoboni (2003) as part of
their computer model that generates melodies in the style of cantatas
by the Italian composer Legrenzi. Marsden (2001) developed a similar
model of melody construction based on a network of elaborations (see:
http://www.lancs.ac.uk/staff/marsdena/software/novagen/index.htm).
The scale-based skeleton according
to Baroni, Dalmonte & Jacoboni (2003).
Baroni et al. analyzed melodies from a period of almost ten
centuries,
from Gregorian music to the Romantic Lied and beyond, and arrived at
the following interesting conclusions:
- A melodic phrase consists of a central
body which may be preceded by an anacrusis and followed by a feminine ending. Thus, both
anacrusis and feminine ending are optional.
- The central body is a melodic fragment beginning and ending on a
down beat. The anacrusis consists of notes in weak metrical position
preceding the central body, while the feminine ending consists of notes
in weak metrical positions following the central body.
Here are two examples, the first having all three parts, the second
one only a central body and a feminine ending:
These are their main theses:
- The central body “...can be reduced, at its
deep level, to a kernel which not only progresses by conjunct step, but
which is also mono-directional.” (p. 292) In the examples above the
kernels are respectively F# E D and F E D C#
- Actual melodies can be conceived as resulting from the
application of a limited number of transformations upon the
mono-directional scale fragment. These transformations include melodic and harmonic
transformations.
Melodic
transformations:
- Repetition (R)
- Neighbor (N)
- Skip (S)
Harmonic
transformation:
- Chord transposition (T), resulting from the "substitution of
one note with another belonging to the same chord". (p. 374).
Thus, in
a sense, T takes the form of a repetition figure.
According to Baroni et al., melodic transformations are ubiquitous in
both pre-tonal and tonal
music; harmonic transformations are mainly found in tonal music.
Formal description of the
transformations
(Context: the scale of C major)
A. Melodic transformations
Repetition (R).
Parameters: NumberOfRepetitions
Formalization: nR(C) = C, n*C. E.g. 3R(C) = C C C C.
Neighbor tone (N).
Parameters: PreFigurePitch, Length
(multiple of 2), Direction
Formalization: N+(C) = C D C . N-(C) = C B C.
N can be applied recursively: E.g.
N2+(a) = a, b, c, b, a. N3-(a) = a,
g, f, e, f, g, a. Here is an example:
Comments:
- A single N transformation adds 2 notes to the skeleton.
- The recursive application of the N transformation leads to an
up-down or down-up figure. See note example m. 2. However, such a
figure is
most likely not perceived
as a combined Neighbor figure, but indeed as an up-down figure.
- A Neighbor tone figure should perhaps be seen as a second-order
ornament. I.e. it is only applied if a melody already has two
successive
identical pitches (resulting from an earlier R transformation).
This presupposes a hierarchical melody construction in which in
each
step only one transformation is applied. This notion will be pursued in
an alternative implementation described below.
Skip (S)
Parameters: PreFigurePitch,
PostFigurePitch, Length, Ascending (boolean), Filled (boolean).
Currently only Filled skips are implemented.
Requirements: Interval between Pre- and PostFigurePitch = 1 step (on
the scale).
Dependent variable: Leap. Leap is a function of: Direction of the
interval between Pre- and PostFigurePitch, Length, Ascending.
The
application of a Skip figure of length 3, ascending in bar 1,
descending
in bar 2. The red notes are the skeleton.
Note
If Length = 2 and the Direction of the interval between Pre- and
PostFigurePitch is the opposite of the Direction of the Skip, the
resulting Figure is similar to a Neighbor figure:
If the Leap of a Skip
Figure is one step, the resulting Figure is a Neighbor note Figure
B. Harmonic transformation
Transposition (T)
Parameters: PreFigurePitch,
PostFigurePitch, Length, Harmony, Direction, StepSize (within triadic
scale)
In this transformation one or more notes are added that are
transpositions of the PreFigure note within the current harmony. Here
is an example:
Example
of the application
of the Transposition operation to the skeleton G A B C
Example
The following example shows how the first phrase of the beginning of
the Allegretto from Mozart's Fantasia in d minor could be constructed
by applying transformations to an ascending scale fragment (other very
interesting examples can be found in Marsden (2005)):
Mozart KV 397 Allegretto
Starting from the following ascending scale fragment:
A B C# D E (1)
Apply Transpositions + 2, + 1 to the 1st and 2nd and 5th skeleton
notes, this yields:
A F# D B G E C# D A F# E (2)
Next fill the gaps between the applied transpositions and skeleton
notes 2 and 3:
A F# D C B G E D C# D A F# E (3)
Metrical placement
As shown, these operations can be applied to a skeleton (on a
harmonic basis) without consideration of the timing, or metrical
placement. The sequence (3) could be positioned on a meter such that
the skeleton coincides with the downbeats of the 2/4 meter. However,
the composer choose to place notes 2 and 3 of the skeleton on the
upbeats, creating appoggiatura’s on bars 2 and 3.
The next figure shows the stepwise development of the melody.
Play
Two remarks are in order: (1) The current implementation does not
include a transformation that creates an appoggiatura (Marsden's set
does). (2) The fact that we are able to describe
how the first phrase could be
generated by elaborating a scale
fragment, does not mean that MGII would ever actually produce it,
because there are an almost infinite number of orders of applying
elaborations to an increasing number of notes. And only one very
specific strategy will actually generate the phrase. Moreover, it
should be noted, that the actually melody constructed by Mozart is
co-determined by considering the perceptual effects of the
elaborations, an aspect so far not covered by Melody Generator II.
Implementation/Operation
The technique of melody construction explained above has been
implemented as follows. Starting from a rhythm, a skeleton is
positioned on the downbeats, a harmonic progression matching the
skeleton is set, and the remaining notes
are assigned pitches by applying one or more of the
transformations.
At present, the user can select three ways to realize the ornaments:
1. Ornament all measures in the same way, with one of
the four transformations described above R, N, S, or T. See note
examples above.
2. Randomly let the program decide which
transformation is used for each measure. In the example below the
ornaments in the 3 measures are respectively N, T, S
3. Realize an ornamentation
using a mix of the 4 transformations: the notes between two
skeleton notes are subdivided in a few groups (size depending on the
total
number of ornamental notes) and to each group an ornamental figure is
assigned randomly (or weighted). A flow chart of this method can be
inspected here. This mode of
ornamentation becomes mainly
interesting
for
melodies having a high rhythmical density as in the two following
examples.
Issues
Meter
and Ornamentation
In the implementation, the transformations are applied without
taking
into account the metrical weight of the notes. However, in tonal music
relatively important tones tend to be placed on metrically strong
locations
Perceptual relevance of the
skeleton
There is an important and interesting issue regarding
the perceptual relevance of the skeleton notion. It seems
questionable whether the skeleton remains
perceivable (even unconsciously) when the density is increased
and the skeleton becomes relatively hidden.
References
- Baroni, M., Dalmonte, R., & Jacoboni, C. (2003). A computer-aided inquiry in music
communication. The rules of music. Lewiston: The Edwin Mellen
Press.
- Baroni, M., & Jacoboni, C. (1978). Proposal for a grammar of
melody. The Bach chorales. Montréal: Les Presses de
l’Université de Montréal.
- Lerdahl, F., & Jackendoff, R. (1983). A generative theory of
tonal music. Cambridge, MA: M.I.T. Press.
- Marsden, A. (2001). Representing melodic patterns as networks of
elaborations. Computers and the
Humanities, 35, 37-54.
- Marsden, A. (2005). Generative structural representation of tonal
music. Journal of New Music Research, 34, 409-428.
- Schenker, H. (1979, originally published 1935) Free composition. New York:
Longman.
- Schenker, H. (1980, originally published 1954). Harmony. Chicago:
Chicago University Press.
(For a complete list of references click here)
ScaleBasedModel.html (last update 18.5.2009)
Created in SeaMonkey
© D.J. Povel, 2008
