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Score Generation in Voice-Leading and Chord Spaces

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Score Generation in Voice-Leading and Chord Spaces Score Generation in Voice-Leading and Chord Spaces Michael Gogins [email protected] Irreducible Productions Abstract searched for a means of imbuing the abstract spaces used by mathematical algorithms with intrinsic musical structure. Common principles of voice-leading can be represented using Recently, I chanced across Dmitri Tymoczko’s work on an orbifold (i.e. a quotient space) in which each point is a the geometry of musical chords (Tymoczko 2006). He identi- chord, and the smoothness of voice-leadings corresponds to fies some mathematical spaces that do have inherent musical the closeness of their chords in the space (Tymoczko 2006). structure. I think these spaces could be used by a variety of The set of such voice-leadings for the major and minor triads mathematically based compositional algorithms. includes the Tonnetz, which leads to an understanding of how The present work demonstrates such an algorithm, a Lin- to use related geometries for all trichords, higher arities of denmayer system that generates scores by moving a turtle chord, voice-leading solutions, and chord progressions. The representing a chord about in two related musical spaces: a present work demonstrates the use of such spaces for algo- voice-leading space, and a chord progression space. rithmic composition. A Lindenmayer system takes an axiom and symbol replacement rules, and recursively rewrites the axiom by applying the rules. A turtle that represents a chord, 2 Musical Spaces which moves within either a voice-leading space or a chord progression space, then interprets the symbols in the final Tymoczko and colleagues (Tymoczko 2006; Callender, production of the Lindenmayer system as commands for writ- Quinn, and Tymoczko 2006) have begun developing a geo- ing a score. This shows how score generators can work in metric approach to music theory. The starting point is to rep- mathematical spaces which have intrinsic musical structure. resent chords as points in a linear chord space, which has one dimension of pitch per voice, distinguishing inversions, oc- taves, and order of voices. This space extends up and down 1 Introduction to infinity from a reference pitch at the origin. Chord space is continous, and every equally tempered and non-equally tem- The use of algorithms to generate scores is as old as com- pered chord can be defined in it. puter music (Hiller and Isaacson 1959). Today there are many Music theorists distinguish various equivalence classes for methods of algorithmic composition (Burns 1993). They can chords. For each equivalence class (such as octave equiv- broadly be categorized as either grammatical, i.e. based on alence, permutational equivalence, inversional equivalence, some sort of generative grammer, or mathematical, i.e. based transpositional equivalence), as well as for each combination on mathematical operations or forms such as fractals. Some of equivalence classes, there is a corresponding orbifold in notable examples of grammatically-generated pieces are the chord space. The equivalence defines which points of the works of David Cope (Cope 1991; Cope 1996; Cope 2000). space are identified to form an orbifold (Callender, Quinn, Some examples of mathematically-generated pieces are Viola and Tymoczko 2006). Elegy (Dodge 1994) and Gendy 3 (Xenakis 1995). An orbifold is a quotient space whose points are equiva- I find that the mathematical algorithms appeal more to my lent under some group action (any combination of translation, sense of musical taste, yet I also find them difficult to control rotation, glide rotation, and reflection) that defines a symme- with respect to harmony and counterpoint, whether in the tra- try. In other words, the group action permutes the corners ditional sense, or in the more general sense of having large- of the orbifold, thus gluing faces of the quotient space to- scale pitch structure and being well-formed. I do hear pro- gether. For example, a strip of paper is a quotient space of ductions from grammatical algorithms that are harmonically the plane; gluing two ends together creates a ring, which is and contrapuntally quite well-formed; yet to my taste, they an orbifold. Giving the ends a half twist before joining them do not sound sufficiently original. Consequently, I have long creates a Moebius band, another orbifold. 593 Of course, the orbifold most familiar to musicians is pitch under octave equivalence, or pitch-class set space, R/12Z, where the octave is defined as 12 semitones. For another example, equivalence under range R for n voices defines the orbifold (R/RZ)n. I call this ranged chord space. It has a top and bottom pitch for each voice, like a score. Voices that move past the top of the score re-appear at the bottom. Figure 1 shows ranged chord space for 3 voices. Augmented triads are white, major triads red, and minor tri- ads blue, but this is hard to see in the printed grayscale. Be- cause its opposing faces are identified, the cube in the figure can be considered a 3-dimensional torus. Figure 2: Tonnetz for trichords and minor triads (blue, light gray as printed) surrounding the central column of 4 augmented triads (white) that defines the orthogonal “axis” of symmetry of the orbifold. One end of the prism can also be visualized as being rotated 120◦ and glued to the other end to form a torus. n As a generalization of the Tonnetz, (R/12Z) /Sn illu- minates basic symmetries and constraints of Western music. The major and minor triads are so flexible with respect both to harmonic progression and voice-leading because they not only surround the orbifold’s axis of symmetry, but also lie near each other. Similarly, for tetrachords, commonly used seventh chords surround the orthogonal column of diminished seventh chords. I find understanding the Tonnetz to be ex- Figure 1: Ranged chord space for 3 voices, 2 octaves tremely helpful for understanding other musical geometries. For an in-depth derivation of these spaces and more on their relationships to other parts of music theory, see (Ty- moczko 2006; Callender, Quinn, and Tymoczko 2006). 2.1 The Tonnetz My approach here is inspired by Tymoczko’s work, al- though I derive other spaces in order to represent the specific (Tymoczko 2006) has identified the orbifold that divides operations that I require for score generation. pitch-class set space for n voices by the symmetry group for n n voices: (R/12Z) /Sn. This could be called the completed 2.2 Representing Scores and Musical Opera- Tonnetz. Its fundamental domains are n −1 dimensional sim- tions plexes. The points of the orbifold are identified by a group action that consists of a rotation (for odd dimensions) or a ro- In the present work, musical scores are represented as tation plus a reflection (for even dimensions). Figure 2 shows functions of time onto chord space. Arpeggios, passing tones, this Tonnetz for trichords. The lines connecting chords indi- counterpoint, and so on are considered to be composed of cate movements of one semitone, so that joined chords are more or less fleeting or elaborated chords. closest neighbors and the lines indicate all minimal voice- I do not directly use the completed Tonnetz as a space leadings. for compositional algorithms, although it deserves further in- The classical Tonnetz of Oettingen, Euler, and Riemann is vestigation for that purpose. My reason is that in the Tonnetz, simply the six columns of major (red, dark gray as printed) whereas a primitive operation such as a translation vector that 594 does not cross a face of the orbifold has of course always the 3. Compare the voice-leading from each chord in the lat- same orientation, if the same translation does cross a face it tice to the source chord, first by smoothness, then by can change orientation as it is reflected from a symmetry hy- parsimony (it is easy to add the option of excluding perplane. I find it simpler to use spaces in which there are parallel fifths). My current implementation uses a brute no mirrors, so that (for example) adding a vector to a chord force search to compare voice-leading sizes, but as long always moves it along the same orientation, whether or not as this need not be done in real time, it is fast enough. the chord is near a face of the quotient space. I have found two spaces to be the most useful for rep- 4. Return the target chord with the closest voice-leading. resenting the primitive operations executed by Lindenmayer Of course, with its chord-by-chord solutions, this algorithm systems: ranged chord space, which I use to find voice-leading does not perform backtracking to obtain voice-leadings that solutions (so it could also be called voice-leading space), and are optimal across a span of chords as do such rule-based normal chord space (defined below), which I use to imple- counterpoint generators as (Schottstaedt 1984) or some of its ment chord progressions (so it could also be called chord pro- descendants. But it would be easy to add memory to find gression space). voice-leadings that are optimal across a span of chords. Other properties of music may also, for convenience in implementing algorithms, be represented as additional, higher 2.2.2 Chord Progression in Normal Chord Space dimensions of chord space. For example, three dimensions (0, 1, 2) suffice to represent the pitches of a trichord. Three Just as ranged chord space is a space in which applying a more dimensions (3, 4, 5) may be used to represent the re- given voice-leading translation to any chord always produces spective loudnesss of the voices. Dimensions (6, 7, 8) may the same pitch-class set, it is valuable to have a space in which be used to represent the durations of each voice in the chord.
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