1. Introduction
1.1 Rationale
In the initial sentence of Romance of the Three Kingdoms, a classic Chinese historical novel in the fourteenth century, its author pointed that this world must unite after lengthy separation and separate after lengthy union. As we can see, interdisciplinary research blooms in recent decades. “Music until the seventeenth century was one of the four mathematical disciplines of the quadrivium beside exploited numerous algorithmic composition methods. We could have a glimpse through many books. They convey valuable aspects of algorithmic composition: Robert Rowe distinguished between “symbolic processes” and “sub-symbolic processes”
(Rowe, 2001, 1) in interactive music; Eduardo Reck Miranda divided algorithmic compositions into “microscopic level, note level, and building-block level” (Miranda, 2001, 2); Heinrich K. Taube exemplifies notes from “metalevel” (Taube, 2004); Gareth Loy provided comprehensive “mathematical foundations of music” (Loy, 2006) with applications to composition; Gerhard Nierhaus differentiated between “style imitations”
and “genuine composition” (Nierhaus, 2009, 3). Those boundaries, however, are becoming more and more ambiguous in modern music technology. In fact, Rowe has demonstrated how to “bridge the levels between sub-symbolic and symbolic systems”
(Rowe, 2009). In my opinion, to reconsider the phenomena of higher levels from the point of view which we have in lower levels can assist us in composing genuinely but with imitations by adopting the principles in old styles.
In the evolvement of western art music, the principle of the interaction between voices originates from the perspective of contrapuntal polyphonic music; the degree of consonance is measured by the vibration through physical presentation and auditory perception. Those two fundamentals form diverse music vocabularies and styles. They further develop into various complex or simplified theories. Even in the latter Equal Temperament instrument system and homophonic music style, the core thought of
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classical compositional methods is still unvaried. Nevertheless, since people invent and exploit music technology to engage in computer-aided composition, automated arrangement, and digital music information retrieval, symbolic pitch-class theories and homophonic composing techniques dominate most research. Although they have the convenience and bring respectable accomplishment, their spirit is essentially different from it in the art content which they imitate.
I have hungered for the underlying rules in melody and in the relations between because I had taken honors physics during senior high school. Since then, I realized that some physical laws exist in arts, too. After I went to the graduate school, I saw more and more scientific ways to analyze, reorganize, devise, and actualize new sound and music. On account of my academic training, I intuitively associated the skill of melody pitches control with “kinetic energy” and “potential energy” (Thornton, 2004, 78) in classical dynamics.
My primary idea was to mathematically mimic the energy of horizontal melodic motion and vertical interval tension, from classical music composers’ perspectives originated in polyphonic vocal style and measured by the vibration through physical presentation and the auditory perception. Most prevalent methods of automated music generation treat notes on staff as “neutral pitch-classes” or “tonal pitch-classes”
(Temperley, 2001, 118). In those kinds of system, all pitch interval degrees are generalized (e.g. Lewin, 1987; Morris, 1987; Krumhansl, 1990; Straus, 1990).
Nevertheless, the energy consumption to make every diatonic pitch with the identical amplitude is an uneven distribution. It can be estimated by the ratio of audio frequencies, which is easy to obtain especially in terms of “Just Diatonic Scale” (Campbell, 1987, 172). With the values, the natural law of kinetic energy and potential energy may be applied to the motion of melody pitches.
1.2 Review
As far as musical pitches are concerned, people (e.g. Morris, 1998; Schell, 2002;
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Tymoczko, 2008; Callender, 2008; Toussaint, 2010) usually refer to rules of harmony and voice leading (e.g. Kostka, 2000 and Aldwell, 2003). Moreover, Fred Lerdahl and Ray Jackendoff’s famous “Generative Theory” (Lerdahl, 1983 and 2005) creates a hierarchical tree structure to vertically divide the melody and rank the pitches in each voice part. It is very effective in homophonic music. Unfortunately, it sacrificed a true melody’s independence from chord construction and chord succession. They admit that
“in truly contrapuntal music there is an important sense in which each line should receive its own separate structural description” (Lerdahl, 1983, 116). Actually, Llewelyn Southworth Lloyd and Hugh Boyle tell us about the connection between melody and harmony. “The melodic line was everything: harmony was in the making, it was being formed by writing concurrent melody” (Lloyd, 1979, 71). Hence, notwithstanding the fact that the melody composing is the most difficult genius to acquire and to instruct, it is indispensable for composers in order to accomplish elegant pieces no matter whether they intend to compose polyphonic music or not. Melody is the horizontal connection of pitches and rhythms; harmony is the vertical one of melodies. When we discover the essence of melody composing, we had better consider only itself and disregarding harmony.
Without the theory of harmony, contrapuntal techniques antecedently show the craft to compose and to organize melodies. We have a wealth of laws, rules, customs, and suggestions in modal or tonal counterpoint textbooks (e.g. Jeppensen, 1992 and Kennan, 1987). After Bill Schottstaedt’s and Dian-Foon Wu’s expert systems (Schottstaedt, 1984 and Wu, 1988), Mary Farbood, Bernd Schoner, Kamil Adiloglu, Ferda N. Alpaslan, and Gabriel Aguilera et al made contributions to the first species counterpoint, too (Farbood, 2001; Adiloglu, 2007; Aguilera, 2010). Beside counterpoint, David Temperley’s probabilistic models and David Cope’s analyses as well as re-syntheses also pay attention to melody (Temperley, 2007 and Cope, 2001, 2005, and 2008). Even so, none of above went below the note level.
In respect of melody pitches at the note level, there are three layers: the pitch, the interval, and the contour. First, Richard Parncutt explains that musical notes play a role in categorization. “Notes belong to the world of information. The attributes of a note correspond not to the physical attributes of the tone to be played but to its perceptual attributes, expressed by means of labelled categories” (Parncutt, 1989, 23). Such categorization is capable of reducing “the amount of information carried by the pitches of a passage of music to a manageable level, removing information about the precise tuning of pitch or interval, and retaining only its semitone category” (Parncutt, 1989,
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44). Second, Kenneth J. Hsu reveals that rather than acoustic frequencies (pitches), “the incidence of the frequency intervals, or of the changes of acoustic frequency, has a fractal geometry” (Hsu, 1991, 3507). Third, not merely Arnold Schoenberg but Leon Dallin mention the importance of a balanced contour to melody composing as well (Schoenberg, 1967 and Dallin, 1974). By contrast with Parncutt, Robert D. Morris claims that “musical contour is one of the most general aspects of pitch perception, prior to the concept of pitch or pitch class” (Morris, 1993, 205). William Thompson also argues that melodic contour is more manageable. “In general, research indicates that listeners’ mental representations of novel melodies contain contour information but relatively little information about absolute pitch or exact interval size” (Thompson, 2008, 95).
On the other hand, we can look those layers through the lower level below notes.
Mark Schmuckler surveys models of melodic contour and builds an effective one by Fourier analysis, which can predict the “melodic similarity” (Schmuckler, 2010 and Hewlett, 1998). Nonetheless, its rigid unit of pitch interval (semitones) do not have the capability which Ali C. Gedik has developed, to be “represented in a continuous pitch space in contrast to discrete pitch space representation in western music with 12 pitch-classes” (Gedik, 2010). For this reason, with an eye on a more satisfactory delineation of melodic contour, we could directly count the ratio of audio frequencies instead of the symbolic pitch interval.
Finally, Victor Zuckerkandl describes the effort to cross an interval. “Stepwise motion can be considered normal motion, in the sense that it involves the least effort in the move from tone to tone; while every skip goes beyond the norm in that it expresses a greater effort by taking us to a more distant tone more rapidly than the normal succession of intervening steps would permit” (Zuckerkandl, 1971, 65). In place of the symbolic degree of musical scale, we could measure the effort in ratio of sound energy.
1.3 Purpose
A fairly large body of literature exists on the algorithmic composition of melody.
Nevertheless, we cannot apply the predominant symbolic system of pitch class to non-tempered tuning systems which consist of unequal intervals. What is even worse, there is little research has been done on the conception of energy in the motion of melody pitches. Thereby, we need the more universal comprehension with intent to, just as I addressed before, reconsider the phenomena of higher levels from the point of view
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which we have in lower levels.
In this thesis, I will propose a model for algorithmic composition of melody. More specifically, the melody only comprises monophonic pitches, which melodic contour corresponds with the natural law of kinetic energy and potential energy. Its input and output data could be simply sub-symbolic audio frequencies, but it is also able to accept and produce symbolic musical pitches through translation. After that, I will demonstrate how to utilize the model to implement a system with very limited guidance from music theory.
“Johannes Tinctoris … takes into account the crucial fact that the composer's judgment must be based not only on what he hears at a given moment but what he must keep in mind in the continuity of hearing” (Mann, 1965, viii). By this research, we will have the capability to maintain the continuity of hearing in the motion of melody pitches without statistical processes such as higher-order Markov models, which has “an often overlooked deficiency ... lies in their inability to indicate information which is provided in lower-order models” (Nierhaus, 2009, 81).
Albeit Chih-Fang Huang and I proposed a similar method last year (Lin, 2010), it is not close enough to the natural law of kinetic energy and potential energy. Thereupon I have amended the formulae so that they are simpler yet more logical. I will illustrate the effects in this thesis.
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