A Foundations-of-Computation Approach to Formalizing Musical Analysis Halley Young University of Pennsylvania [email protected] ABSTRACT the relevance of such a computation. I shall then invoke type theory to describe the type-theoretic foundations for In the last century, two trends have increased the scope deriving such properties from the “basic” components of of musical analysis: music theorists have provided mathe- sound (pitch, duration, loudness, and timbre). I will dis- matical insight into specific musical scenarios, while mu- cuss comparisons both between different property types, as sicologist have examined the nature of musical analysis as well as between different property values and between dif- a cultural, cognitive, and scholarly endeavor [1][2][3] ferent musical pieces as mathematical objects from which [4][5]. This paper intends to bring these two strands of properties are derived. I will then focus on a particular research together by providing a constructive mathemati- type of property whose definition is recursive, and will in- cal foundation for the process of musical analysis. By es- troduce two such properties - tonal function and meter in tablishing a mathematical description of the generation of common practice music. an analysis of a piece of music, useful mathematical tools for performing operations frequently used in analysis, and possible precise definitions for loaded terms such as “mu- 2. THE sical similarity” and “musical form”, I will extend the an- HORIZONTAL-VERTICAL-ONTOLOGICAL alyst and the meta-analyst’s ability to create abstractions COMPLEX from musical surfaces, the core of every process of analy- The juxtaposition of “horizontal” and “vertical” views of sis. music and the realization that both need to be considered simultaneously is well understood in music pedagogy (for 1. INTRODUCTION instance, the title to New York University’s introduction to music course is “Harmony and Counterpoint”). It is In this age of globalism and prestige given to scientific equally well understood that such simultaneous compre- thought, it is unsurprising that both the study of musical hension is cognitively not trivial. Since at least the 1760’s, universals and mathematical abstractions of music are re- musicologists have complained about the relative impor- ceiving much attention. Authors such as Steven Brown tance assigned by composers to one dimension to the ne- and Joseph Jordania [6], Leonard Meyer [7], Jay Rahn [8], glect of the other (see Rousseau on harmony and melody) and most recently Samuel Mehr [9] have taken various ap- [10], and it is widely agreed that Baroque music was “more proaches including the methodologies of cognitive science, contrapuntal” than the “harmony-based” music that fol- semiotics, corpus and field studies, and anthropological lowed. However, I argue that in every type of music where theory in order to make statements about ”musical univer- more than one thing is occurring in a temporal moment, sals,” or facts that seem to describe the way many peoples there are properties of the music that emerge out of changes perform, compose, and listen to music. At the same time, that occur as the piece progresses, and properties that emerge specific musical practices, from African rhythms to the out of the simultaneity of parts being conceived as a unified Classical diatonic scale, have been scrutinized in order to whole. understand the mathematical abstractions that can explain However, the horizontal and vertical dimensions of mu- how and why these practices manifest [1][2]. However, sic only describe half of the picture. Consider a single, few scholars have studied the mathematical universals of monophonic melody. Its perception is not only as a gestalt, music analysis, or formalizations which could apply to the but also as a complex of different properties that it exhibits analysis of any musical phenomenon. Theory of compu- - one hears the song ”Hot Cross Buns”, but also the nar- tation and foundations of computation provide a paradigm row range, the diatonicity, the downwards motion, the rep- for making statements about music analysis in general with etition, and so forth. Each of these properties is a part mathematical rigor. of my experience of the song. Similarly, consider a sin- In this paper, I shall introduce several concepts borrowed gle chord. Musicologists will probe that chord for its root from computer science to generalize aspects of music anal- tone, for its overall consonance, for its interval vector, and ysis. I will first discuss the basic features of a relevant for its interval set class, because all of these are supposed computation on a piece of music, which I term a “prop- to provide additional information that’s useful for analysis. erty”, and will provide a constructive basis for determining Notice that from one perspective, information doesn’t in- crease when you compute the interval vector of a chord - Copyright: c 2020 Halley Young et al. This is an open-access article dis- you’re merely describing a property that is always deriv- tributed under the terms of the Creative Commons Attribution License 3.0 able from the information you have, that of the chord’s Unported, which permits unrestricted use, distribution, and reproduction pitch classes. Nonetheless, calculating the interval vector in any medium, provided the original author and source are credited. tells the composer something important about the chord, as it allows comparison to other chords in order to esti- 5. COMMON TYPE CONSTRUCTORS WHICH mate its potential effect. The potential for calculation is not PRODUCE MANY PROPERTIES the same as the actualization of calculation, and our under- standing of music is fundamentally shaped by the calcula- In computer science and category theory, there are sev- tions we (consciously or unconsciously) produce on hori- eral standard type constructors. These are constructs which zontal, vertical, and horizontal-and-vertical aspects of the take types as arguments, and return a more complex type music. We will refer to every calculable aspect of music as which depends on the input types. Many properties fre- a “property”. This label includes the four properties (pitch, quently referenced in music analysis can be thought of as duration, dynamics, timbre) which are often cited in edu- the result of passing a primitive type (time, duration, pitch, cational literature as the “fundamental elements of music”, dynamics) to a type constructor. as well as more complicated properties like harmonic func- tion or a representation of musical contour such as CSEGs. 5.1 n-Product Type A product type, as defined by type theory, is a type that re- 3. MUSICAL OBJECTS, CONSTRUCTIVISM, AND sults from a combination of two or more other types [11]. EXPRESSIVITY For instance, the assignment of specific intervals to spe- cific instruments, as can be heard in Elliot Carter’s string Constructive mathematics, which became the foundation quartet [12], can be thought of as a property which consists for much programming language theory in the 1920’s, is of a pair of an interval and an instrument. Another product different than classical mathematics in its insistence that if type might be the association of a certain scale degree with one wants to show that an object exists, one has to con- a certain type of ornamentation, as happens in Hindustani struct the object in a well-defined logic, rather than prov- music [13]. If we limit ourselves to triads harmonically, ing that it exists indirectly. In terms of music, we may we can represent chords as products of a pitch-set class extend the metaphor of constructive and classical logics to and one of the interval-sets f[037]; [047]g. What we typ- distinguish between “classical” and “constructive” defini- ically hear as an instantiated note can be thought of as an tions of properties. A property could be defined as any n-product of a pitch, a duration, a timbre, and a dynamic function from a piece of music to some domain, including (the four primitive properties). one which is non-computable (or non-expressible in any interpretable language). However, as an analytical tool, 5.2 n-Vector Type the musical properties which can be described in an ap- propriately expressive language are much more useful, as An n-Vector Type consists of a fixed number of values this definition can then be applied to any arbitrary piece of the same type. Examples include all dyads, triads, or of music to draw similarities between the two. In con- tetrads; a traditional score (which is a fixed number of trast, claiming that “there exists a property f such that parts), and a representation of a 12-tone scale as a set of f(mozart sonata in C) = 2 is rather meaningless for 12 0/1 bits. the analyst. Furthermore, in computer science, it is under- stood that using the least complicated language possible 5.3 List Type for expressing a given property is optimal. One of the goals A list consists of a variable number of values of the same of this paper is to figure out how to define common musi- type. Examples include a motif, which can be any num- cal properties in constructive ways and using as simple a ber of notes long; a chord, which can be any number of language as possible. notes long; or a chord progression, which can contain any number of chords. 4. WHAT MAKES A PROPERTY USEFUL/RELEVANT? 5.4 Transformation Type We have informally claimed that certain properties are rel- Transformation types are homeomorphic functions from evant to some styles but not others. It is difficult to quantify one value to another of the same type. Examples include relevance absolutely, but we shall do so relative to a classi- the function which maps one rhythmic pattern to the same fication task T and a classification model type M with fixed rhythmic pattern in augmentation.