Modeling Tonality: Applications to Music Cognition - eScholarship

Modeling Tonality: Applications to Music Cognition Elaine Chew (eniale@alum.mit.edu) University of Southern California Integrated Media Systems Center, and Department of Industrial and Systems Engineering Los Angeles, CA 90089-1450 USA Abstract Processing musical information is a task many of us per- form effortlessly, and often, unconsciously. In order to gain a better understanding of this basic human cognitive ability, we propose a mathematical model for tonality, the underlying principles for tonal music. The model simul- taneously incorporates pitch, interval, chord and key rela- tions. It generates spatial counterparts for these musical entities by aggregating musical information. The model also serves as a framework on which to design algorithms that can mimic the human ability to organize musical in- put. One such skill is the ability to determine the key of a musical passage. This is equivalent to being able to pick out the most stable pitch in the passage, also known as “doh” in solfege. We propose a computational algorithm that mimics this human ability, and compare its perfor- mance to previous models. The algorithm is shown to predict the correct key with high accuracy. The proposed computational model serves as a research and pedagog- ical tool for putting forth and testing hypotheses about human perception and cognition in music. By designing efficient algorithms that mimic human cognitive abilities, we gain a better understanding of what it is that the human mind can do. Introduction Music cognition is a complex task requiring the integra- tion of information at many different levels. Neverthe- less, processing musical information is an act with which we are all familiar. The mind is so adept at organiz- ing and extracting meaningful patterns when listening to music that we are often not even aware of what it is that we do when comprehending music. Some of this uncon- scious activity includes determining the tonal center 1 , the rhythm, and the phrase structure of the piece. I illustrate our unconscious ability to process music by a short anecdote from my own experiences. In my first semester as a pianolab 2 instructor at MIT, I encountered a few students who had no prior musical background. I asked one such student, after he carefully traced out the melodic line for Yankee Doodle, “What is the key 3 of 1 The tonal center, also called the tonic of the key, is the pitch that attains greatest stability in a musical passage. 2 A keyboard skills class for students enrolled in Music Fun- damentals and Composition courses. 3 Excerpted from the Oxford Dictionary of Music: A key implies adherence, in any passage, to the note-material of one of the major or minor scales. When the pitches in a scale are this piece?” He responded with a reasonable question: “What do you mean by key?” I began singing the piece and stopped mid-stream. I then asked the student if he could sing me the note on which the piece should end. Without hesitation, he sang the correct pitch 4 , thereby successfully picking out the first degree, and most sta- ble pitch, in the key. The success of this method raised more questions than it answered. What is it we know that causes us to hear one pitch as being more stable than oth- ers? How does the mind assess the function of this stable pitch over time as the music evolves? Before we can study music cognition, we first need a representation for musical structure. In this paper, we propose a mathematical model for tonality, the underly- ing principles of tonal music. According to Bamberger (2000), “tonality and its internal logic frame the coher- ence among pitch relations in the music with which [we] are most familiar.” The model uses spatial proximity to represent perceived distances between musical entities. The model simultaneously incorporates representations for pitch, interval, chord and key relations. Using this model, we design a computational algo- rithm to mimic human decisions in determining keys. The process of key-finding precedes the evaluation of melodic and harmonic structure, and is a fundamental problem in music cognition. We relate this new represen- tation to previous models by Longuet-Higgins & Steed- man (1971) and Krumhansl & Schmuckler (1986). The computational algorithm is shown to identify keys at a high level of accuracy, and its performance is compared to that of the two previous models. The Representation Western tonal music is governed by a system of rules called tonality. The first part of the paper proposes a ge- ometric representation, the Spiral Array model, that cap- tures this system of relations among tonal elements. The Spiral Array model offers a parsimonious description of the inter-relations among tonal elements, and suggests ordered, the first degree of the scale gives the scale its name. This is also the most stable pitch, known as the tonic. 4 A pitch is a sound of some frequency. High frequency sounds produce a high pitch, and low frequency sounds pro- duce a low pitch. This is distinct from a note, which is a symbol that represents two properties, pitch and duration.