Computational harmonic analysis with rhythmical weights

Abstract

Analysis of harmony is the first step in the analysis of common practice (Baroque, Classical and Romantic) period of Western music because in these genres musical structure is aligned with tonal motion and organicity is created primarily by harmony. The analysis includes finding regions of tonality/key, labeling chords and cadences, assigning functions to chords, finding prolonged harmonic functions and consequently forming a tree-like hierarchy. The result of this effort is the discovery of the harmonic motion and how musical entities function within this motion. This is how a music theorist analyses harmony based polyphonic music, i.e., music of the common practice period. A chord's function in its tonal context has an emotional projection that is perceived by the listeners: a harmonic tension that rises or falls, held in a suspension or resolved. A rise in harmonic tension raises an expectation for resolution and its resolution is an emotional relief. The fine balance between increase and decrease in harmonic tension through time is perceived by the emotionally sensitive listener. This ebb and flow in tension is a critical determinant of the aesthetics of harmonic language. In this thesis, we describe an algorithm for harmonic analysis of polyphonic music and demonstrate its implementation on the RUBATO Composer music composition and analysis environment. Our harmonic analysis model completes Riemann's unfinished program by assigning a function to any chord –not only triads and sevenths– based on the pitch content of the chord and the harmonic tension created between consecutive chords. As a background of music modeling, we overview mathematical approaches to music analysis at the symbolic level (i.e., note level and above) and then we examine how computational power can be used for modeling and analysis of music and musical processes. Then, we review similarities as well as differences between music and language and also musical structure analysis methodologies borrowed from linguistics research such as grammars and parsers. In Chapter 3, we give an overview of RUBATO Composer music composition and analysis environment including its historical line of development. We summarize its mathematical pillars and the software architecture. We also describe a number of rubettes that we designed and programmed on RUBATO for computational analysis purposes: a rubette that enables mixing of weights, and another rubette to translate a MIDI file into a MIDI denotator, and another one to be able to trim MIDI files. We also explain a rubette that translates harmonic analysis output to Lilypond, a music typesetting format. In Chapter 4, we give our motivation for computational harmonic analysis by reviewing related concepts such as tonality, harmony and tonal tension as well as a review of computational models for analysis of tonal tension and harmony. In Chapter 5, we describe our mathematical and computational models and their software implementation for analysis of harmony. During this thesis, we added some components to the Computational Harmonic Analysis Network –a suite of rubettes to analyze harmony. The additions are implementation of Viterbi algorithm for optimum path computation and direct-thirds method for Riemann Matrix computation.s The core of the thesis is however is being able to analyze harmony using also metric/rhythmic information of musical events. Our harmonic analysis model had previously assumed that chords have the same metric importance. However as musicians we know that meter in music imposes a hierarchy in perception of musical events in time, i.e., not every instant, and not every beat is equally important during perception. Thus, we extended our model to include metric importance of musical events. In Chapter 6, we give a review of recent research on perception of time and periodicity based on recent neuroscience research. Then, we focus on temporality in perception of music. We consider meter and rhythm as the skeleton system that span time, whereas melody and harmony are the flesh over the bones. This ontological order of where meter and rhythm is primordial is relevant for a vast majority of genres in music including music of the common practice period. Then, we overview metrical analysis algorithm based on Mazzola's metric analytics which reveals local (inner) meters in music and its implementation as the new MetroRubette. Finally, in Chapter 7, we describe a computational model for harmonic analysis of music where, next to pitch content and temporal position of neighboring chords, metric position of chords is also considered. Music is given to the analysis algorithm at a symbolic level as a MIDI file. We explain the new algorithm in detail and also give sample analyses with the algorithm's implementation as software on the RUBATO Composer. We compare harmonic analyses with and without metrical proximity, examine their differences and discuss results.