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A Framework for Automated Schenkerian Analysis

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A Framework for Automated Schenkerian Analysis ISMIR 2008 – Session 3b – Computational Musicology A FRAMEWORK FOR AUTOMATED SCHENKERIAN ANALYSIS Phillip B. Kirlin and Paul E. Utgoff Department of Computer Science University of Massachusetts Amherst Amherst, MA 01003 {pkirlin,utgoff}@cs.umass.edu ABSTRACT [1] in a hierarchical manner. Schenker’s theory of music al- lows one to determine which notes in a passage of music In Schenkerian analysis, one seeks to find structural de- are more structurally significant than others. It is important pendences among the notes of a composition and organize not to confuse “structural significance” with “musical im- these dependences into a coherent hierarchy that illustrates portance;” a musically important note (e.g., crucial for artic- the function of every note. This type of analysis reveals ulating correctly in a performance) can be a very insignifi- multiple levels of structure in a composition by construct- cant tone from a structural standpoint. Judgments regarding ing a series of simplifications of a piece showing various structural importance result from finding dependences be- elaborations and prolongations. We present a framework tween notes or sets of notes: if a note X derives its musical for solving this problem, called IVI, that uses a state-space function or meaning from the presence of another note Y , search formalism. IVI includes multiple interacting compo- then X is dependent on Y and Y is deemed more structural nents, including modules for various preliminary analyses than X. (harmonic, melodic, rhythmic, and cadential), identifying The process of completing a Schenkerian analysis pro- and performing reductions, and locating pieces of the Ur- ceeds in a recursive manner. Starting from the musical score satz. We describe a number of the algorithms by which IVI of a composition, one may locate any number of structural forms, stores, and updates its hierarchy of notes, along with dependences. After no more can be found, an abstracted details of the Ursatz-finding algorithm. We illustrate IVI’s score is produced by rearranging or removing the less struc- functionality on an excerpt from a Schubert piano composi- tural notes. The new abstracted score will reveal new de- tion, and also discuss the issues of subproblem interactions pendences: when their subservient neighbors are moved or and the multiple parsings problem. eliminated, various structurally important notes in the origi- nal score will be deemed less significant to their new neigh- 1 SCHENKERIAN ANALYSIS bors. This iterative process illustrates Schenker’s conception A number of types of music analysis are concerned with “la- that tonal compositions consist of a “continuum of interre- beling” individual objects in a musical score. In harmonic lated structural levels,” [1] where each structural level of the analysis, labels are assigned to chords and notes correspond- music represents that composition at a different level of ab- ing to harmonic function; in contrapuntal voice segregation, straction. Each successively abstract level expands or pro- labels are assigned to notes indicating voice assignment. A longs various aspects of the previous one. The most abstract rhythmic analysis may assign different levels of metrical im- level of the piece is the background level. At the other end of portance to notes. These styles of analysis are similar in that the spectrum is the foreground level: an analysis at this level they often describe musical components in isolation, or only usually still contains most of the notes of the score and most in relation to their immediate neighbors on the musical sur- closely represents the musical surface. Between these levels face. is the middleground level. While Schenker’s own analyses Structural analysis, on the other hand, emphasizes dis- usually only contain these three levels, there can be many covering relationships among notes and chords in a com- levels in between the surface level music and the ultimate position, rather than studying individual tones in a vacuum. background level; rarely are the levels clearly delineated. The word “structure” here refers to “the complete fabric of Schenker theorized that at the background level, all tonal the composition as established by melody, counterpoint, and works could be represented by one of three basic outlines, harmony in combination” [1]. consisting of a three chord harmonic progression (the Bass- Schenkerian analysis is the most well-developed type of brechung or bass arpeggiation) supporting a three-, five-, structural analysis. This type of analysis examines the “in- or eight-note descending melodic line (the Urlinie or funda- terrelationships among melody, counterpoint, and harmony” mental line). Together, these form the Ursatz or fundamen- 363 ISMIR 2008 – Session 3b – Computational Musicology tal structure. The Ursatz is a basic structure that appears be useful in automating Schenkerian analysis, though the over most of the length of a single composition, though it concept of hierarchical reductions is not discussed. Temper- often manifests itself at other levels in the structural hier- ley later re-developed some of these modules and a number archy as well. While the Ursatz is an important facet of of new ones with a Bayesian statistics slant [16]; here he Schenkerian analysis, the idea of structural levels is more does present a high-level discussion on how to test or verify encompassing, as the Ursatz can be seen as another musical Schenker’s ideas from a Bayesian viewpoint. device that may appear at multiple levels of abstraction [2]. Schenkerian analysis has largely resisted formalization because it was not presented in any sort of formal man- 2 PREVIOUS WORK AND STATE OF THE ART ner by Schenker himself. Indeed, many textbooks illustrate concepts largely through examples and long paragraphs of The most widely known formalization of musical structure prose [1, 3] rather than by giving a recipe for constructing in a hierarchical fashion similar to Schenker’s is that of Ler- an analysis step-by-step. There have only been three large- dahl and Jackendoff [8]. The authors attempt to describe scale initiatives in developing a computational procedure the structure of music from a linguistic perspective by pro- for Schenkerian analysis; other relevant work (e.g., Meehan viding preference-rule systems that govern various aspects [12]) was undertaken on a smaller scale and was restricted of musical structure. They present a set of rules for group- to models, not algorithms. The first large project, the work ing of notes in a voice, and another for deducing metrical of Smoliar [14], resulted in a tool used to assist a human structure in term of strong beats and weak beats. Their most in performing an analysis, largely by confirming the valid- intriguing contributions here, however, are the preference- ity of reductions; the system did not perform any analysis rule systems for two recursive reductive systems, one based on its own. The second, by Kassler [7], described a model on “time-span analysis” that is governed by metrical and that took the analysis from the middleground level to the grouping structure, and another based on “prolongational background, but did not work directly with the surface-level analysis” that is controlled by the ideas of rising and falling musical notes (what one sees in the actual score). The third musical tension in a piece. These reductions can be illus- project in Schenkerian analysis is more current: an under- trated by trees, where the leaves are the surface-level notes taking by Marsden [9, 10, 11], which seeks to derive an of the piece and higher-level branches represent the more analysis directly from a MIDI-style music representation. structural tones, or by an equivalent musical depiction as a This project, however, is still in its infancy and is focused sequence of staves showing successive levels of the resultant on calculating all possible reductions using a dynamic pro- hierarchy of pitches. gramming algorithm and scoring each analysis. For exam- Though Lerdahl and Jackendoff frame their discussion ple, it makes no mention of locating the Ursatz (a critical in terms of discovering a grammar of music, they acknowl- step), does not take advantage of other notational informa- edge that their system is incomplete and probably cannot tion that would be given by a higher-level representation of a be directly turned into a formal algorithm. This is mainly score such as MusicXML, and the published research makes due to the lack of weightings for the preference rules and no mention of harnessing previous research in voice segre- that the authors’ representations are based primarily on ver- gation, key finding, or harmonic analysis to bootstrap the tical segmentation of the input music as chords, and do not process. give enough weight to voice-leading considerations. Never- theless, a number of people have undertaken formalization 3 THE IVI FRAMEWORK of Lerdahl and Jackendoff’s system, though the results usu- ally require tweaking of parameters. For example, Hirata We now explain the functionality of the IVI framework, a and Aoyagi [6] present a framework for the representation system to perform automated Schenkerian analysis. The IVI of musical knowledge in the spirit of Lerdahl and Jackend- name (pronounced like ivy, the plant) was chosen for the I- off, while Hamanaka et al. [4, 5] describe an implementa- V-I (tonic-dominant-tonic) harmonic structure that Schenker tion that can automatically make reductions according to a theorized was at the background level of all tonal composi- set of preference rules, but the weights on the rules must be tions. The framework consists of numerous components, adjusted by hand for each piece. which we detail in the following section. Temperley [15] presents models for meter, phrase struc- We have chosen MusicXML as the input format for IVI. ture, voice segregation, pitch spelling, harmonic structure, While the MIDI file format is more widely used for music and key structure using a preference rule system very simi- file storage than MusicXML, the richer score format of Mu- lar to that of Lerdahl and Jackendoff.
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