A Graph-Theoretical Approach for Pattern Matching in Post-Tonal Music Analysis

Table 1: The integer notation of pitch classes. Although plenty of research works of post-tonal music analysis have been done by musicologists, most of them focus on analyzing and comparing a limited number of music compositions. Comparative study across numerous compositions is difficult because the identification of pitch-class sets occurred in compositions—an important aspect of pitchclass set theory—is highly time-consuming and error-prone when performed manually. Large-scale comparative study requires an effective post-tonal music analysis system. There are software utilities to assist post-tonal music analysis [21, 16, 18], but none of them considers searching for pitch-class sets. [7, 9, 6] investigated how to choose groups of musical notes for analysis. Peter Castine published his Ph.D thesis on computer-aided analysis on post-tonal music [4]. However, he focused on how graphical user interface could help in pitch-class set theory analysis. A Pascal routine named Setsearch is presented in the book Pascal Programming for Music Research [3]. It locates pitch-class sets by using a sequential searching algorithm with a sliding frame. However, it is difficult to determine the frame size as the author mentioned. Improper value of the frame size may cause false dismissals, i.e. some matched patterns are missed. The fundamental problem of pattern matching in post-tonal music analysis is on the definition of matching condition(s). In Figure 1, given the pitch-class set query {7, 8, 11}, the first matched pattern is the first three notes of the melody but not including the minim 2
Abstract A graph-theoretical approach is proposed to facilitate pattern matching in posttonal music analysis with pitch-class set theory, in which music perception is taken into account by incorporating stream segregation—a model of the perceptual organization of music. A piece of music is modelled as a graph, with each musical note presented as a vertex and the relationship between a pair of musical notes as an edge. The relationship is determined by stream segregation. According the proposed matching conditions, searching for a musical pattern is equivalent to searching for a special subgraph called maximal matched CRP subgraph. The comparisons are made between the patterns identified by the graph-theoretical approach and those by the musicologists.
1
Introduction
One of the most significant areas in music research, analysis of post-tonal music—art music that began in Europe in the early 20th century, significant composers of which include Bela Bartok, Alban Berg, Arnold Schoenberg, Igor Stravinsky, Anton Webern— consists of studying how this music works by finding the building blocks as unifying cells in such a musical work. Searching for these building blocks or musical patterns can be modelled as a pattern matching problem: given a musical pattern query and a music database, the aim is to search for all matched patterns in the database. In this paper, the pattern matching problem in post-tonal music analysis is based on pitch-class set theory —the widest currency seen as the core system for post-tonal music analysis [1]. The idea of pitch-class set theory is illustrated in the following example in [14]. The opening of Arnold Schoenberg’s Klavierst¨ ucke, Op. 11, No. 1 is shown in Figure 1. Although the first three notes of the melody and the sustained notes in bars 4-5 look quite dissimilar, these two collections of notes are, in the perspective of pitch-class set theory, equivalent building blocks because they both contain a B, a G , and a G . Twentieth-century composers have greater freedom in exploring ways to unify a music composition and pitch-class set theory aims to uncover building blocks hidden in the music. A building block is called a pitch-class set, denoting a group of pitch classes. A 1
Mäßige q

Piano

p








Figure 1: The opening of Arnold Schoenberg’s Klavierst¨ ucke, Op. 11, No. 1.
Integer Pitch-class name content 0 B , C, D 1 C,D 2 C∗, D, E 3 D,E 4 D∗, E, F 5 E , F, G
Integer Pitch-class name content 6 F,G 7 F∗, G, A 8 G,A 9 G∗, A, B 10 A,B 11 A∗, B, C
A Graph-Theoretical Approach for Pattern Matching in Post-Tonal Music Analysis
Wai Man Szeto and Man Hon Wong Department of Computer Science and Engineering The Chinese University of Hong Kong Shatin, N.T., Hong Kong {wmszeto, mhwong}@.hk