A Computer Model for the Schillinger System of Musical Composition

A Computer Model for the Schillinger System of Musical Composition

Matthew Rankin

A thesis submitted in partial fulfillment of the degree of

Bachelor of Science (Honours) at The Department of Computer Science Australian National University

August 2012

c Matthew Rankin

Except where otherwise indicated, this thesis is my own original work.

Matthew Rankin 28 August 2012

Acknowledgements The author wishes to sincerely thank Dr. Henry Gardner for his extremely valuable assistance, insight and encouragement; Dr. Ben Swift also for his continuous encouragement and academic mentorship; Jim Cotter for igniting what was a smouldering interest in algorithmic composition and more recently providing participants for the listening experiment; and Mia for her unyielding, belligerent optimism.

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Abstract A system for the automated composition of music utilising the procedures of Joseph Schillinger has been constructed. Schillinger was a well-known music theorist and composition teacher in New York between the first and second World Wars who developed a formalism later published as The Schillinger System of Musical Composition [Schillinger 1978]. In the past the theories contained in these volumes have generally not been treated in a sufficiently rigorous fashion to enable the automatic generation of music, partly because they contain mathematical errors, notational inconsistencies and elements of ‘pseudo-science’ [Backus 1960]. This thesis presents ways of resolving these issues and a computer system which can generate compositions using Schillinger’s formalism. By means of the analysis of data gathered from a rigorous listening survey and the results from an automatic genre classifier, the output of the system has been validated as possessing intrinsic musical merit and containing a reasonable degree of stylistic diversity within the broad categories of Jazz and Western Classical music. These results are encouraging, and warrant further development of the software into a flexible tool for composers and content creators.

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Contents Acknowledgements

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Abstract 1

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Background 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Introduction to the Schillinger System . . . . . . . . . . . . . . 1.2.1 Schillinger in Computer-aided Composition Literature 1.2.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Summary of this Thesis . . . . . . . . . . . . . . . . . . . . . . .

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Overview of Computer-aided Composition 2.1 Dominant Paradigms in Computer-aided Composition . . . . . 2.1.1 Style Imitation versus Genuine Composition . . . . . . . 2.1.2 Push-button versus Interactive . . . . . . . . . . . . . . . 2.1.3 Data-driven versus Knowledge-engineered . . . . . . . . 2.1.4 Musical Domain Knowledge versus Emergent Behaviour 2.2 Formal Computational Approaches . . . . . . . . . . . . . . . . 2.2.1 Markov Models . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Artificial Neural Networks . . . . . . . . . . . . . . . . . 2.2.3 Generative Grammars and Finite State Automata . . . . 2.2.4 Case-based Reasoning and Fuzzy Logic . . . . . . . . . . 2.2.5 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . 2.2.6 Chaos and Fractals . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Cellular Automata . . . . . . . . . . . . . . . . . . . . . . 2.2.8 Swarm Algorithms . . . . . . . . . . . . . . . . . . . . . . 2.3 The Automated Schillinger System in Context . . . . . . . . . . Implementation of the Schillinger System 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 A Brief Refresher . . . . . . . . . . . . . . . . . . . . 3.1.2 The Impromptu Environment . . . . . . . . . . . . . 3.2 Theory of Rhythm . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Rhythms from Interference Patterns . . . . . . . . . 3.2.2 Synchronisation of Multiple Patterns . . . . . . . . . 3.2.3 Extending Rhythmic Material Using Permutations . ix

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3.3

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3.7 3.8 4

3.2.4 Rhythms from Algebraic Expansion . . . . . . . . . . . . . . . . . Theory of Pitch Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Flat and Symmetric Scales . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Tonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Nearest-Tone voice-leading . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Deriving Simple Harmonic Progressions From Symmetric Scales Variations of Music by Means of Geometrical Progression . . . . . . . . . 3.4.1 Geometric Inversion and Expansion . . . . . . . . . . . . . . . . . 3.4.2 Splicing Harmonies Using Inversion . . . . . . . . . . . . . . . . . Theory of Melody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 The Axes of Melody . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Superimposition of Rhythm and Pitch on Axes . . . . . . . . . . . 3.5.3 Types of Motion Around the Axes . . . . . . . . . . . . . . . . . . 3.5.4 Building Melodic Compositions . . . . . . . . . . . . . . . . . . . Structure of the Automated Schillinger System . . . . . . . . . . . . . . . 3.6.1 Rhythm Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Harmonic and Melodic Modules . . . . . . . . . . . . . . . . . . . 3.6.3 Parameter Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . Parts of Schillinger’s Theories Not Utilised . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Results and Evaluation 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 Common Methods of Evaluation . . . . . . . . . . . 4.3 Automated Schillinger System Output . . . . . . . . 4.4 Assessing Stylistic Diversity . . . . . . . . . . . . . . 4.4.1 Overview of Automated Genre Classification 4.4.2 Choice of Software . . . . . . . . . . . . . . . 4.4.3 Classification Experiment . . . . . . . . . . . 4.4.4 Preparation of MIDI files . . . . . . . . . . . 4.4.5 Classifier Configuration . . . . . . . . . . . . 4.4.6 Classification Results . . . . . . . . . . . . . . 4.5 Assessing Musical Merit . . . . . . . . . . . . . . . . 4.5.1 Listening Survey Design . . . . . . . . . . . . 4.5.2 Listening Experiment . . . . . . . . . . . . . 4.5.3 Quantitative Analysis and Results . . . . . . 4.5.4 Qualitative Analysis . . . . . . . . . . . . . . 4.5.4.1 Methodology . . . . . . . . . . . . . 4.5.4.2 Analysis and Results . . . . . . . . 4.5.4.3 Genre and Style . . . . . . . . . . . 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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Conclusion 95 5.1 Summary of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Avenues for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

A Samples of Output A.1 Harmony #1 . A.2 Harmony #2 . A.3 Harmony #3 . A.4 Melody #1 . . A.5 Melody #2 . . A.6 Melody #3 . .

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B Listening Survey C Function List C.1 Rhythmic Resultants — Book I: Ch. 2, 4, 5, 6, 12 . . . . . C.2 Rhythmic Variations — Book I: Ch. 9, 10, 11 . . . . . . . C.3 Rhythmic Grouping and Synchronisation — Book I: Ch. C.4 Rhythmic Generators . . . . . . . . . . . . . . . . . . . . C.5 Scale Generation — Book II: Ch. 2, 5, 7, 8 . . . . . . . . C.6 Scale Conversions — Book II: Ch. 5, 9 . . . . . . . . . . C.7 Harmony from Pitch Scales — Book II: Ch. 5, 9 . . . . . C.8 Geometric Variations — Book III: Ch. 1, 2 . . . . . . . . C.9 Melodic Functions — Book IV: Ch. 3, 4, 5, 6, 7 . . . . . . Bibliography

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Contents

Chapter 1

Background

1.1

Introduction

Almost since the inception of the discipline of computing, people have been using computers to compose and generate music. This is perhaps unsurprising given the importance of algorithmic principles in much compositional thinking throughout musical history. The use of computers for music has mostly been driven by the desires of composers to generate interesting and unique new material. Recognising the distinction between the composition of musical scores and other forms of music and sound generation, [Anders and Miranda 2011] have proposed the use of the term ‘computer-aided composition’ to refer to one area of what is more broadly known as ‘computer music’, a discipline which also encompasses the arts of sound synthesis and signal processing [Roads 1996]. This thesis is concerned with computer-aided composition: in particular, the computer-realisation of the musical formalism of Joseph Schillinger [Schillinger 1978]. Some authors prefer the term ‘algorithmic composition’ to refer to computer-aided composition [Nierhaus 2009]. In this thesis the two terms will be used interchangeably. Joseph Schillinger was a Ukrainian-born composer, teacher and music theorist who was active in New York from the 1920s until his death in 1943. Schillinger’s lasting influence as a theorist and teacher exerted itself through famous students such as George Gershwin, Benny Goodman and Glenn Miller; and several distinguished television and radio composers [Quist 2002]. The distillation of his life’s work is contained in three large volumes. Two of these constitute The Schillinger System of Musical Composition [Schillinger 1978]. The third volume, The Mathematical Basis of the Arts [Schillinger 1976] was intended to be broader in scope and generalise much of his prior work in music to visual art and design. The Schillinger System attempted to differentiate itself from other accepted musical treatises by pursuing a more ‘scientific’ approach to composition. It consequently eschewed restrictive systems of rules created from the empirical analysis of Classical styles, as well as the notion of composition by ‘intuition’. Instead it promoted a range of quasi-mathematical methods for the construction of musical material. The system was intended to be of practical use by working composers — George Gershwin famously wrote the opera Porgy and Bess while studying under Schillinger [Duke 1947]. 1

Background

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Schillinger’s work has frequently been mentioned in passing by researchers working in the field of computer-aided composition, but rarely addressed in any detail. There are several examples of similar individual algorithms that have been incorporated into computer-aided composition systems, but most of these systems focus on specific computational paradigms which are unrelated to the rest of Schillinger’s work. To the best of the author’s knowledge, only one other system dedicated specifically to the automation of Schillinger’s procedures exists in the form of publicly available software, and no such system has been referred to in the academic literature. This thesis will therefore provide the first formal presentation and evaluation of an ‘automated Schillinger System’. From here onwards, this term will be used to refer to the computer implementation being presented, while the term ‘Schillinger System’ will be used as a short form of The Schillinger System of Musical Composition.

1.2

Introduction to the Schillinger System

The two volumes of the Schillinger System [Schillinger 1978] consist of twelve books presented as individual ‘theories’. Each of these theories is an exposition of Schillinger’s musical philosophy combined with his technical discussions pertaining to general principles and explicit procedures. They include numerous examples of the procedures being carried out by hand, and lengthy annotations by the editors who published the work after Schillinger’s death. The collection of theories is listed below. The work in its entirety is a formidable 1640 pages. Consequently, the scope of this thesis has only allowed for the first four theories to be considered in detail. I Theory of Rhythm II Theory of Pitch-scales III Variations of Music by Means of Geometrical Projection IV Theory of Melody V Special Theory of Harmony VI Correlation of Harmony and Melody VII Theory of Counterpoint VIII Instrumental Forms IX General Theory of Harmony X Evolution of Pitch Families XI Theory of Composition XII Theory of Orchestration

§1.2 Introduction to the Schillinger System

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An existing software program known as StrataSynch by David Mc Clanahan1 is the only other known automated system to make explicit use of Schillinger’s theories. It implements the generation of four-part diatonic harmony using books V and VIII, and a single chapter from book I. The system described in this thesis extends beyond the scope of that system to a more versatile form of harmony generation utilising books I, II and III; and to the generation of single-voice melodic compositions utilising books I–IV.

1.2.1

Schillinger in Computer-aided Composition Literature

In an extended commentary on computer music from 1956–1986, Ames acknowledged the algorithmic nature of Schillinger’s work without pointing the reader to any known computer implementation, and noted that it had become ‘all but forgotten’ [Ames 1987]. Schillinger’s work was discussed in greater detail by Degazio, who again pointed out how much of it was presumably amenable to computer implementation, and highlighted how particular properties of the Theory of Rhythm would enable self-similar musical structures to be generated, thus relating it to the exploration of fractals in computer music [Degazio 1988]. The ability of the system to generate fractal structures was also identified by Miranda [Miranda 2001]. Miranda further noted the interesting rhythmic possibilities of using algebraic expansions and symmetrical patterns of interference, both of which are also explored in the Theory of Rhythm. More recently Nierhaus gave a cursory mention of Schillinger in the epilogue of a survey of algorithmic composition, implicitly acknowledging that it is possible to be adapted but also failing to cite any example of an implementation [Nierhaus 2009]. Although the discussion of a specific implementation is lacking, algorithms similar to those in Schillinger’s Theory of Melody were used with apparent success in early work by Myhill (cited in [Ames 1987]) and later by Miranda as part of a musical data structure used by agents in a swarm algorithm [Miranda 2003]. Furthermore, there are numerous examples of algorithms which use permutation in a similar manner to Schillinger’s Theory of Rhythm, and plenty of examples of systems which use inversion and retrograde techniques in a manner similar to Schillinger’s ‘geometrical projections’. There is no suggestion being made that these particular techniques originate from Schillinger’s system alone; indeed their use can be found throughout the history of Western musical composition [Nierhaus 2009].

1.2.2

Motivation

If many of the procedures expounded by Schillinger are not unique (this is not to suggest that none of them are), then the value of his treatise is that it collates them together, with each one presented in the context of the others and potentially useful interrelationships drawn. One of the motivations for adapting the Schillinger system is therefore the fact that it incorporates many algorithmic techniques which are demonstrably useful in computer-aided composition on their own, but have not been ex1

www.capsces.com/stratasync

Background

4

tensively tested together in the absence of other prevailing computational paradigms. Another motivation is the fact that other oft-cited treatises on music theory in algorithmic composition contain rules which are derived from existing music, such as Piston’s Harmony [Piston 1987]. Conversely, Schillinger’s work purports to have taken a more universal approach that does not draw its rules from the analysis of any particular musical corpus. For this reason it is ostensibly likely to be able to produce compositions which do not fall into the category of ‘style imitation’, which Nierhaus identified as being overwhelmingly dominant in the field [Nierhaus 2009]. Instead, it should allow for a measure of stylistic diversity. As will be discussed in chapters 3 and 4 of this thesis, these notions are contentious and worthy of investigation.

1.2.3

Criticism

The very premise of Schillinger’s work is controversial by virtue of the fact that it effectively condemns previous theories and methodologies as inadequate [Backus 1960]. As a result it has attracted rigorous scrutiny by various authors. A 1946 review by Barbour [Barbour 1946] examined each of the ‘achievements’ of the Schillinger System listed in a preface by the editors, and concluded that none of them were substantiated. Barbour also listed a number of errors and inconsistencies which highlighted the work’s fundamental lack of a sound scientific or mathematical basis. Schillinger’s work was derided extensively by Backus [Backus 1960]. Dubbing it both ‘pseudo-science’ and ‘pseudo-mathematics’, he surveyed the first four volumes in some detail, pointing out that many descriptions of procedures are unnecessarily verbose and laced with undefined jargon; that the musical significance of them is based on numerology rather than any appropriately cited research; that much of the symbolic notation serves to obfuscate rather than clarify the expression of sometimes trivial mathematical ideas; and finally that several mathematical definitions are simply incorrect. Backus thus raised many important issues concerning the formal interpretation of Schillinger’s techniques which are tackled in chapter 3 of this thesis. Neither Backus nor Barbour commented on whether Schillinger’s procedures were of any use by contemporary composers for generating musical material. In light of their resounding criticism, it is significant that other authors have considered many of the theories to be demonstrably useful in practice, or cited testimony from successful composers suggesting as much [Degazio 1988]. The composer Jeremy Arden published a PhD thesis documenting the study and utilisation of the Schillinger System from a compositional perspective [Arden 1996], concluding that the Theory of Rhythm and Theory of Pitch Scales offered many useful techniques. Although he swiftly dismissed the Theory of Melody as ‘too cumbersome’ to be of practical use, similar principles to those contained in that theory have been found useful in other contexts as mentioned above in section 1.2.1. There is therefore no absolute consensus which would wholly discourage computer implementations of the Schillinger System.

§1.3 Summary of this Thesis

1.3

5

Summary of this Thesis

In this thesis, the automated Schillinger System designed by the author will be presented and evaluated. To begin with, chapter 2 will survey both the dominant paradigms and the specific computational approaches in the field of computer-aided composition. This theoretical basis will serve to position the automated Schillinger System within the academic literature. The details of the software implementation of the four initial books of the Schillinger System listed in section 1.2 will be presented in chapter 3. Alongside the requisite technical discussion, chapter 3 will provide a comprehensive outline of the bulk of the procedures contained in these books. Perhaps more importantly, it will also identify the inherent difficulties in translating a formalism designed for composers into a model able to be represented computationally, including the resolution of Schillinger’s notational and practical inconsistencies and the necessity for a raft of new procedures to sensibly link the theories together. The evaluation of musical output is a perennial problem in this inter-disciplinary field, and few authors tend to venture beyond subjective conclusions drawing on their own musical backgrounds. However, one method of more rigorous evaluation consists of the enlisting of a ‘team of experts’ to supply qualitative data for analysis. Such an approach has been used to study the output of the system presented here. Additionally, the burgeoning field of automatic genre classification has been engaged as a means of quantitatively assessing the statistical characteristics of the output. Together these forms of analysis aim to establish both the intrinsic musical merit and stylistic diversity of the automated Schillinger System. These experiments and their results will be presented in chapter 4. The recently released four-part harmony system by Mc Clanahan and the active pursuit of new forms of representation for Schillinger’s ideas, embodied by the online Schillinger CHI Project2 , suggest a resurgence of interest in automating parts of the Schillinger System. The software presented in this thesis aims to contribute to this momentum, and is amenable to development beyond its current state as a ‘push-button’ music generator into a modular interface that could be used by composers and multimedia content creators. Many potential avenues for future research are explored in chapter 5.

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http://schillinger.destinymanifestation.com/

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Background

Chapter 2

Overview of Computer-aided Composition

This chapter will give a broad overview of the field of computer-aided composition, in order to place the automated Schillinger System in context, and to position this thesis as an addition to the computer music literature. As remarked upon by Supper [Supper 2001], the distinctions between compositional ideas, realisation in the musical score, and auditory perception are clearly bounded in a computing context. As this thesis is focusing on computer-aided composition rather than attempting to encompass the entire field of computer music, this overview does not include algorithms which take music generation beyond the level of symbolic representation into digital audio. Instead, it is presumed that the symbolic data generated by composition algorithms can be further mapped to musical notation, MIDI data1 or audio data depending on the application. [Supper 2001] made a further taxonomic observation which is relevant to this chapter. He distinguished between: 1. the modelling of musically-oriented algorithmic procedures to produce encodings of established music theories; 2. procedures individual to a ‘composer-programmer’ where the code produces a unique class of pieces based upon the composer’s individual expertise; and 3. experiments with algorithms from extra-musical fields such as dynamic systems or machine learning. In fact, there are many instances where individual implementations bear relevance to two or three of Supper’s categories, and his is only one of a number of possible taxonomies for describing computer-aided composition — section 2.1 lists a variety of other significant distinctions within the algorithmic composition literature. However, it is safe to observe that much recent academic research in computer-aided composition is based primarily on the application of pre-existing extra-musical algorithms to music, thus falling into Supper’s third category. Section 2.2 describes this literature. 1

MIDI stands for Musical Instrument Digital Interface. It is the dominant protocol for handling symbolic musical information in computer systems and hardware synthesizers.

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Overview of Computer-aided Composition

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Figure 2.1 provides a visualisation of the array of computational approaches used in the field, as discussed in section 2.2. These are connected by dashed lines which represent their algorithmic or mathematical similarity, and roughly partitioned in terms of their use within the various paradigms discussed in section 2.1. Non-musical Data Streams

Not data-driven

Musical domain knowledge

Automated Schillinger System

IGAs

Genetic Algorithms

Sometimes data-driven

Musical "Expert Systems"

Constraint Programming

Chaos

Genetic Programming

Generative Grammars

L-systems

FSA

Cellular Automata

Fuzzy Logic

ATNs Swarm Algorithms Markov Chains

Data-driven

Fractals

Case-based Reasoning

Artificial Neural Nets

Figure 2.1: Approaches to Computer-aided Composition

§2.1 Dominant Paradigms in Computer-aided Composition

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As this chapter will be limited to the discussion of systems designed with the ultimate goal of composing music, other research areas such as computer auralisation, computational creativity and automated musicological analysis, despite being closely related to the success of particular algorithmic composition approaches, will not be explored per se. Discussions of computer style recognition, expressive musical performance and output evaluation are relevant to the experiments presented in chapter 4 and will be included there in the appropriate places.

2.1

Dominant Paradigms in Computer-aided Composition

Before commencing a description of the common algorithm families used in this field, it will be useful to outline several overarching (and often competing) paradigms. These are partly representative of differing philosophical approaches to automatic music generation, and partly to do with historical shifts in emphasis on computational approaches, which are in turn the result of past developments in artificial intelligence and the modelling of natural phenomena.

2.1.1

Style Imitation versus Genuine Composition

The reproduction of specific musical styles (‘style imitation’) constitutes the majority of algorithmic composition literature. Its dominance was testified to by Nierhaus in the epilogue of his comprehensive survey of algorithmic composition [Nierhaus 2009]. The styles in question are either those of particular individual composers, or those exemplified by the music of a particular culture or historical period. Style imitation is not limited to any particular group of computer algorithms, but is frequently the paradigm used by most of the the approaches in figure 2.1 that encode musical domain knowledge. The reason for the dominance of style imitation is somewhat evident when one considers the large quantity of work dedicated specifically to four-voice chorale harmonisation [Pachet and Roy 2001]. This form of composition is perhaps the most thoroughly studied in the musicological literature due to the enormous quantity of ‘exemplar’ works courtesy of European Baroque and Classical composers. Consequently, a well-established set of rules of varying levels of strictness has been empirically derived from this corpus over the course of several centuries, and this theoretical framework lends itself to being expressed as an optimisation problem in the context of ‘correct’ four-part harmony writing. Since optimisation problems sit comfortably within the realm of computer science, this style of composition is the most readily approachable by computer scientists. It has been pointed out by Allan that chorale harmonisation is “the closest thing we have to a precisely defined problem” [Allan 2002]. Any music generated within formal, recognisable stylistic boundaries is able to be evaluated either objectively or with a degree of authority by human listeners. Conversely, the concept of ‘genuine composition’ [Nierhaus 2009] is problematic in computer music for the reason that genuinely new and different results are virtually impossible to validate using quantitative methods, and very much at the mercy

10

Overview of Computer-aided Composition

of individual musical taste when it comes to human scrutiny. Nevertheless, while academic work in this area is traditionally less common it is still pursued in earnest, especially by researchers utilising chaos theory or algorithms with emergent behaviours.

2.1.2

Push-button versus Interactive

An algorithmic composition system which delivers a self-contained musical fragment, complete composition or an endless stream of musical material with real-time playback requiring no human intervention after the setting of initial parameters may be referred to as a ‘push-button’ or ‘black-box’ system. Examples of well-documented push-button systems range from Hiller and Isaacson’s early experiments forming the Illiac Suite [Hiller and Isaacson 1959] to Cope’s Experiments in Musical Intelligence [Cope 2005]. Most four-part harmonisation systems also fall into this category. Systems which generate music using continual human feedback are perhaps more frequently cited as being successful. This paradigm has been referred to in terms of a human-computer ‘feedback loop’ [Harley 1995] and features in a variety of composition algorithms which are designed to either incorporate real-time human behaviour into their generative process or perform a gradual optimisation tailored to a user’s musical preference. Examples include interactive genetic algorithms using ‘human fitness functions’ [Biles and Eign 1995]; systems which allow a user to generate raw material and then modify a set of parameters to develop it further [Zicarelli 1987]; systems which allow the user to influence the generation of material from a more abstracted perspective [Beyls 1990]; systems which learn iteratively by ‘listening’ to a user’s live performance [Thom 2000]; and systems which map a user’s physical movement [Gartland-Jones 2002] or brain-wave activity [Miranda 2001] to a subset of the algorithm’s parameter space in real-time. Many authors have argued that these areas of research hold greater promise than push-button systems, based on the notion that the acts of composition (and improvisation) are fundamentally human activities dependent on human interaction. There also exists a body of software which functions as a kind of ‘blank slate’ for composers. These programs are usually modular in the sense that individual pre-existing algorithms can be interfaced arbitrarily, and there is often the scope for ‘composer-programmers’ to extend their functionality. Examples range from the early MUSICOMP by Robert Baker [Hiller and Baker 1964] to the more advanced Max by David Zicarelli [Zicarelli 2002]. Such environments are interactive by their very definition, however once the template for a composition is completed by the composer, in many cases they arguably function as push-button systems. More recently, the advent of ‘live coding’ has been made possible by environments like Impromptu [Sorensen and Gardner 2010]. These environments are specifically designed to facilitate the coding of musical procedures during performance or improvisation.

§2.1 Dominant Paradigms in Computer-aided Composition

2.1.3

11

Data-driven versus Knowledge-engineered

In computer-aided composition a ‘data-driven’ solution relies on a database of existing musical works on which to perform pattern extraction, statistical machine learning or case-based reasoning to derive musical knowledge. By contrast, a ‘knowledgeengineered’ system requires the coding of musical knowledge in the form of procedures or the manual population of a knowledge base. In figure 2.1, these alternative paradigms have been used to categorise various computational approaches on the left of the diagram. An expert system combines a knowledge base of facts or predicates, ‘if-then-else’ rules and heuristics, with some kind of inference engine to perform logical problem solving in a particular problem domain [Coats 1988; Connell and Powell 1990]. Such a system requires the acquisition of knowledge either automatically or through a human ‘domain expert’ [Mingers 1986]. The front end may be interactive (the user inputs queries or data) or non-interactive (fully automated). There is generally also the prerequisite that an expert system is capable of both objectively judging its output using the same knowledge base, and tracing the decision path that led to the output for the user to analyse [Coats 1988]. The inherent flaws of expert systems are well-known. One problem is that as a system’s parameter space becomes more ambitious, the knowledge base of rules tends to expand exponentially. In algorithmic composition this has lead to optimisation problems in four-part harmonisation which become computationally intractable above a ˇ [Ebcioglu ˇ certain polyphonic density or beyond a certain length, as found by Ebcioglu 1988]. Beyls also cited the ‘complexity barrier’ inherent in musical expert systems, and further noted the lack of graceful degradation in situations with incomplete or absent knowledge [Beyls 1991]. Phon-Amnuaisuk mentioned the common problem of arbitrating between contradictory voice-leading rules [Phon-Amnuaisuk 2004]. One of Mingers’ main criticisms of expert systems in general was that a rule base must always be incomplete when built from only a sample of all possible data [Mingers 1986]. In knowledge-engineered musical expert systems, the most significant obstacle is the time-consuming encoding of a sufficient quantity of expert knowledge to allow the system to compose anything non-trivial. For style imitation, a further problem is that many rules inherent to a particular style may not be obvious even to experts, or may not be possible to adequately express in the required format. Sabater et al. articulated an underlying issue of rule-based style imitation: “the rules dont make the music, it is the music which makes the rules” [Sabater et al. 1998]. For these reasons, the data-driven approach has become favoured by many researchers. Some of these authors have advocated for alternative ‘connectionist’ approaches to uncover the implicit knowledge of a musical corpus rather than attempt to find explicit rules — their solutions typically perform supervised learning of the corpus using artificial neural networks.

12

2.1.4

Overview of Computer-aided Composition

Musical Domain Knowledge versus Emergent Behaviour

In figure 2.1 the two paradigms of musical domain knowledge and emergent behaviour have been split vertically. The application of musical domain knowledge in computer-aided composition generally leads to a set of either implicit or explicit musical rules being enforced, something practically unavoidable except in cases where completely random behaviour is sought for aesthetic reasons. The approach is often, but not always, aligned with style imitation. Such examples found in the literature are usually broadly referred as ‘musical expert systems’, but not all such approaches necessarily fall into this category if the accepted meaning of the term ‘expert system’ in computer science literature is enforced [Mingers 1986]. Miranda has suggested that rule-based composition systems lack ‘expression’ due to their inability to break rules, citing a famous quote by Frederico Richter: “In music, rules are made to be broken. Good composers are those who manage to break them well” [Miranda 2001]. This perceived fundamental flaw with the knowledge-based approach has provided inspiration for many researchers to look instead to paradigms which focus on dynamic or emergent behaviour, such as chaos, cellular automata and agent interaction in virtual swarms. Evolutionary algorithms have also been explored extensively, because although they are usually designed to operate in a musical knowledge domain, they do so in a fundamentally stochastic manner rather than by applying generative rules [Biles 2007]. The dichotomy between knowledge-based music and ‘emergent’ music was identified by Blackwell and Bentley, who separated the algorithmic composition field into ’A-type’ and ’I-type’ systems [Blackwell and Bentley 2002]. These labels respectively refer to systems that rely on encoded musical knowledge, and those that map the data streams from swarms, dynamic systems, chaotic attractors, natural phenomena or human activity to musical output. Beyls posited an equivalent delineation of ‘symbolic’ versus ‘sub-symbolic’ algorithms [Beyls 1991]. The emergent or subsymbolic paradigm seeks to “interpret rather than generate” [Blackwell and Bentley 2002], and is therefore usually associated with Nierhaus’s notion of genuine composition [Nierhaus 2009]. However, a caveat which authors choosing this path have encountered was pointed out by Miranda: the biggest difficulty when using non-musical processes for algorithmic composition is deciding how to translate the data stream into a representation which is musically meaningful [Miranda 2001].

2.2

Formal Computational Approaches

This section will explain the specific algorithmic approaches that have been applied to computer-aided composition. It will be seen that many of these approaches have strong mathematical similarities (as shown in figure 2.1), and may produce statistically equivalent results depending on how they are implemented. As such, the organisation of this section does not strictly separate the algorithms based purely on their mathematical or purported musical properties. It does however indicate the range of distinct approaches to be found in the algorithmic composition literature.

§2.2 Formal Computational Approaches

13

The topics covered are grouped roughly into those that compose music using a statistical or probabilistic model of a style or corpus (Markov models and artificial neural networks); those which are most frequently associated with the ‘expert system’ paradigm in terms of being driven by systems of generative rules and constraints (formal grammars, finite state automata, case-based reasoning and fuzzy logic); and those which map the data from an extra-musical process onto a musical parameter space (chaos, fractals, cellular automata and swarm algorithms). For the most part the first two categories may be thought of as encoding ‘implicit’ and ‘explicit’ musical knowledge respectively. Evolutionary algorithms do not fall neatly into this particular taxonomy because although they encode musical knowledge, they navigate the space of musical possibilities stochastically.

2.2.1

Markov Models

Markov models were the earliest established extra-musical approach to computeraided composition to be widely adopted. In a survey of the first three decades of algorithmic composition, Ames cited several examples of their use from the 1950s onwards by composers such as Lejarin Hiller and Iannis Xenakis [Ames 1987]. Cohen described a number of early applications of the probabilistic replication of musical styles, treating what are essentially Markov chains as a musical application of Information Theory. Cohen’s notion of composition being regarded as simply “selecting acceptable sequences from a random source” is a potential motivation for using the technique for style imitation, suggesting that “the degree of selectivity of the works of composers is . . . a parameter of their style” [Cohen 1962]. Their relative ease of implementation has perhaps also contributed to their popularity in computer music [Ames 1989]. A simple Markov model consists of a collection of states and a collection of transition probabilities for moving between states in discrete time steps [Ames 1989]. The probabilities of states leading to one another may be represented by a ‘transition matrix’. The state space is discrete, and in musical applications, finite. A Markov chain is obtained by selecting an initial state and then generating a sequence of states using the transition matrix. How this model is utilised in algorithmic composition differs between implementations. States can be used, for example, to represent individual pitches, chords or durations; or they may be used to represent individual Markov chains of length n, which is equivalent to enforcing a dependency on events n time steps into the past. A Markov model in which all transitions depend on the previous n transitions is an nth -order Markov model; these are commonly used to instil a measure of contextsensitivity and thus encode musical objects at the phrase or cadence level. States may also represent entire vectors of potentially interdependent musical parameters, something utilised by Xenakis in the form of ‘screens’ [Xenakis 1992]. The transition matrix may be either constructed by hand, or derived empirically by performing an automated analysis on a database of existing musical works. The latter amounts to encoding each work as a sequence of states, and determining the transition

14

Overview of Computer-aided Composition

probabilities by the relative tallies of each transition (analogous to the experiments carried out by A. A. Markov himself using Russian texts [Ames 1989]). These options correspond with Cohen’s labels of ‘synthetic’ and ‘analytic-synthetic’ [Cohen 1962]. Both approaches are present in the literature, and the choice has depended principally on whether the user is attempting to generate a particular aesthetic for an individual composition [Ames 1989] or performing style imitation, where the purpose is for the randomly generated output to inherit the generalised musical rules implicit in the corpus [Cohen 1962]. Examples of the use of Markov chains for algorithmic composition are numerous. Ames documented his use of the technique to develop works for monophonic solo instruments [Ames 1989]. In his program, the transition matrix is hand-crafted, and the entries define the probabilities of melodic intervals, note durations, articulations and registers. Hiller and Isaacson’s Experiment 4 from the Illiac Suite operated in much the same manner [Hiller and Isaacson 1959]. Cambouopoulos applied Markov chains to the construction of 16th century motet melodies in the style of the composer Palestrina [Cambouropoulos 1994]. His approach also used hand-crafted transition matrices for melodic intervals and note durations; these were developed through manual statistical analysis of Palestrina’s melodies. Other authors have used a data-driven approach: Biyikoglu ‘trained’ a Markov model using the statistical analysis of a corpus of Bach’s chorales to generate four-part harmonisations [Biyikoglu 2003], while Allan solved the same chorale harmonisation problem using Hidden Markov Models [Allan 2002]. Allan’s solution uses one Hidden Markov Model to generate chord ‘skeletons’ (the notes of the melody are treated as observations ‘emitted’ by hidden harmonic states), and two more to fill in the chords and provide ornamentation. It then uses constraint satisfaction procedures to prevent invalid chorales, and crossentropy measured against unseen examples from the chorale set as a quantitative validation method. The reported success of Markov models is varied. Allan concluded that coherent harmonisation can indeed be achieved via statistical examination of a corpus [Allan 2002], while in Ames’ assessment this often leads to “a garbled sense of the original style” [Ames 1989]. Biyikoglu suggested that Markov chains are not appropriate for modelling hierarchical relationships, but are capable of providing smooth harmonic changes [Biyikoglu 2003]. Cambouopoulos highlighted the potential for higher order chains to simulate a measure of musical context [Cambouropoulos 1994], however Baffioni et al. observed that chains of too high an order simply end up reproducing entire sections of the original corpus, and instead proposed a hierarchical organisation of separate Markov chains accounting for form, phrase and chord levels [Baffioni et al. 1981]. As Ames suggested, the fundamental problem with many of these models is that they provide an aural realisation of the probability distributions within a data set but cannot discern the methods behind its construction, and therefore serve as little more than “partial descriptions of non-random behaviour” [Ames 1989].

§2.2 Formal Computational Approaches

2.2.2

15

Artificial Neural Networks

Artificial neural networks (ANNs) are often used to investigate the notion of musical style, and have been successfully used to perform style and genre classification (see section 4.4.1). ANNs are well-suited to these tasks because they are particularly good at finding generalised statistical representations of their input data [Russell and Norvig 2003]. In algorithmic composition, they tend to be aimed squarely at style imitation for this reason. The original motivations for pursuing this ‘connectionist’ approach as an alternative to expert systems were summarised by Todd, who championed ANNs as a way to gracefully handle complex hidden associations within a data set, as well as numerous ‘exceptions’ to the established musical rules which would ¨ normally inflate the knowledge-base of an expert system [Todd 1989]. Hornel and Menzel commented on neural networks’ abilities to circumvent the problem of rule ¨ explosion inherent in building sophisticated expert systems for style imitation [Hornel and Menzel 1998]. ANNs are loosely modelled on the architecture of the brain [Russell and Norvig 2003]. Networks are built of simple computational units known as ‘perceptrons’, which are analogous to the function of individual biological neurons. A perceptron calculates a weighted aggregate of its inputs, subtracts a ‘threshold’ value and ‘fires’ by passing the result through a differentiable activation function such as a sigmoid or hyper-tangent. The most common practical implementation of a neural network is known as a ‘multi-layer perceptron’ (MLP). This normally consists of a layer of ‘hidden’ neurons connected to both a set of inputs representing the input dimensions of the training set, and a set of output neurons which represent the output dimensions. The basic function of a neural network is to learn associations between input vectors and target output vectors by adjusting randomly initialised weights along network connections. A popular method for doing this is ‘gradient descent back propagation’, in which the input vectors are fed forward through the network and the mean-squared error between the output and target vectors is gradually reduced (subject to a scalar ‘learning rate’) over some number of epochs using the derivative of the error function. In this way the weights come to form a statistical generalisation of the training set through repeated exposure to input vectors. In musical applications, the outputs are normally fed back into the inputs to form a ‘recurrent neural network’ (RNN), and a technique such as back propagation through time (BPTT) can then be used to model temporal relationships in the corpus [Mozer 1994]. Neurons which feed back into themselves may also be used to implement short term neural ‘memory’. To compose new music using an RNN, a trained network is simply seeded with a new input vector and the outputs are recorded for some number of iterations. Todd’s original system restricted the domain to monophonic melodies represented using the dimensions of pitch and duration [Todd 1989]. He combined two different network types — a three-layer RNN with individual neural feedback loops to model temporal melodic behaviour at the note level, and a standard MLP which, when trained, acted as a static mapping function from fixed input sequences to output sequences [Todd 1989]. Mozer implemented an RNN that learned and composed

16

Overview of Computer-aided Composition

single-voice melodies with accompaniment, called CONCERT [Mozer 1994]. It improved on Todd’s work in various ways, such as using a probabilistic interpretation of the network outputs, and more sophisticated data structures for musical repre¨ sentation. Mozer’s network inputs represented 49 pitches over four octaves. Hornel and Menzel described a neural network system called HARMONET with the ability to harmonise chorale melodies, and a counterpart system MELONET for composing ¨ melodies [Hornel and Menzel 1998]. Both of their approaches used a combination of ANNs for the ‘creative’ work and constraint-based evaluation for the ‘book-keeping’. ANNs have also been used as fitness evaluators in evolutionary algorithms as one way of alleviating both the inadequacy of objective musical fitness functions and the ‘fitness bottleneck’ caused by human intervention (see section 2.2.5). For instance, Spector and Alpern used a three-layer MLP trained on the repertoire of jazz saxophonist Charlie Parker which was used to classify members of a population as either ‘good’ or ‘bad’ [Spector and Alpern 1995]. The aesthetic products from ANNs are also reported as being mixed. Mozer’s results when attempting to compose in the style of Bach were reported to be ‘reasonable’, but his experiments on European folk-tunes were less successful [Mozer ¨ 1994]. Hornel and Menzel’s compositions using HARMONET and MELONET, on the other hand, were evaluated as ‘very competent’, and showed that ANNs could ¨ be used to imitate characteristics strongly associated with a composer’s style [Hornel and Menzel 1998]. Todd avoided a judgement of merit regarding his ANN-composed melodies, stating only that they were “more or less unpredictable and therefore musically interesting” [Todd 1989]. A common criticism of most ANN approaches is that they essentially learn the statistical equivalent of a set of complex Markov transition matrices, and are therefore only slightly more capable than Markov chains of modelling higher order musical structure [Mozer 1994]. Phon-Amnuaisuk points out that they learn only ‘unstructured knowledge’ [Phon-Amnuaisuk 2004]. Eck and Schmidhuber have offered a potential remedy to this problem by using ‘long short term memory’ (LSTM) to allow for some association of temporally distant events manifesting as medium-scale musical structure. Their method resulted in the ‘successful’ production of improvisations over fixed Bebop chord sequences [Eck and Schmidhuber 2002].

2.2.3

Generative Grammars and Finite State Automata

Algorithmic composition systems incorporating generative grammars are what are most commonly referred to as musical ‘expert systems’, because they presuppose an encoding of explicit domain-specific rules, irrespective of whether those rules are encoded by hand or extracted automatically from a corpus. The attraction of this method is that it is capable of encoding the established musical knowledge of musicological texts, and it also provides a way to generate coherent musical structure at multiple hierarchical levels, while at the same time allowing for a large space of complex sequences [Steedman 1984]. Many of the the generative grammar systems are informed by the work of Chomsky regarding linguistic syntax [Chomsky 1957], and later work by Lerdahl and Jackendoff [Lerdahl and Jackendoff 1983] which builds

§2.2 Formal Computational Approaches

17

upon the musicological analysis theories of Schenker [Schenker 1954]. The generative grammar approach bears strong similarities to the implementation of finite state automata (FSA), and both grammars and FSA have been shown to function identically to Markov chains in certain circumstances [Roads and Wieneke 1979; Pachet and Roy 2001]. Material obtained by applying the production rules of a generative grammar is most often filtered using a knowledge-base of constraints which define the legal musical properties of the system [Anders and Miranda 2011]. A generative grammar can be described as consisting of an alphabet of nonterminal tokens N, an alphabet of terminal tokens T, an initial root token Σ and a set of production or rewrite rules P of the form A → B, where A and B are token strings [Roads and Wieneke 1979]. A grammar G is represented formally by the tuple G = ( N, T, Σ, P), and music is generated by establishing a set of musical tokens such as pitches, rhythms or chord types, and designing a set of production rules that implement legal musical progressions. Chomsky’s taxonomy of type 0, 1, 2 and 3 grammars (‘free’, ‘context-free’, ‘context-sensitive’ and ‘finite state’) [Chomsky 1957] is relevant to music production. For instance, Roads and Weineke observed that grammar types 0 and 3 are inadequate for achieving structural coherence [Roads and Wieneke 1979]. Rader utilised stochastic grammars in an early implementation of a Classical style imitator [Rader 1974]. The system he devised was a ‘round’ generator, wherein each incarnation of the melody is constrained to consonantly harmonise with itself at regular temporal displacements. It used an extensive set of production rules with assigned probabilities, and a set of constraints. Domain knowledge was derived from traditional harmonic theory, in this case Walter Piston’s treatise Harmony [Piston 1987]. Holtzman described a system in which the production rules of multiple grammar types were implemented along with ‘meta-production’ rules [Holtzman 1981], thus constituting the knowledge and meta-knowledge of an expert system [Mingers 1986]. These were accompanied by common transformational operations such as inversion, retrograde and transposition, and used to reproduce a work by the composer Arnold Schoenberg [Holtzman 1981]. Steedman modelled jazz 12-bar blues chord sequences with context-free grammars [Steedman 1984], using an approach informed directly by the musicological work of Lerdahl and Jackendoff [Lerdahl and Jackendoff 1983]. ˇ produced what was, according to Pachet and Roy [Pachet and Roy 2001], the Ebcioglu ˇ 1988]. His first real solution to the four-part chorale harmonisation problem [Ebcioglu system implemented an exhaustive optimisation process using multiple automata and sets of constraints based on traditional harmonic rules for generating chord skeletons, pitches and rhythms from an initial melody. Storino et al. used a manually encoded generative grammar to compose pieces in the style of the Italian composer Legrenzi [Storino et al. 2007]. Both Zimmerman [Zimmermann 2001] and Hedelin [Hedelin 2008] have used grammars to generate large compositional structures which are then filled with chord skeletons using Riemann chord notation [Mickselsen 1977], before finally being fleshed out with note-level information — the aim being to bring form and construction closer to one another instead of relying on a single set of production rules to generate ‘incidental’ musical structure [Hedelin 2008]. Cope’s system Experiments in Musical Intelligence (EMI) uses a type of FSA called

18

Overview of Computer-aided Composition

an augmented transition network (ATN), which is combined with a ‘reflexive pattern matcher’ to form a data-driven expert system [Cope 1992]. The analysis of a manually encoded and annotated corpus of works is performed using a method purportedly informed by the work of Schenker [da Silva 2003]. This method is referred to by Cope as SPEAC, which is an acronym for the possible chord classifications ‘statement’, ‘preparation’, ‘extension’, ‘antecedent’ and ‘consequent’ depending on a chord’s makeup and context. A ‘signature dictionary’ of statistically significant recurring musical fragments of between 1 and 8 intervals is built using the pattern matcher [da Silva 2003]. To produce new works, the ATN implements a set of production rules designed to stochastically generate a new SPEAC sequence, and constraint systems are applied to determine the final pitch, duration and note velocity information. EMI has been used to compose thousands of works which closely mimic the styles of famous composers including Bach, Chopin, Beethoven, Bartok, and Cope himself. More recently, an ‘oeuvre’ of around one-thousand selected works in a wide range of styles produced by the system has been established as a style database itself, which Cope has used to interactively feed back into an updated system based on the same ‘recombination’ principles known as Emily Howell [Cope 2005]. Cope associates the notion of a prolonged style imitation feedback loop with his proposed definition of creativity, arguing that such a process is difficult to formally distinguish from the human creative process [Cope 2005]. In general, systems incorporating some form of generative grammar imbued with explicit musical knowledge have been found to give more convincing musical results for style imitation than the statistically oriented approaches of Markov chains and ANNs. Pachet and Roy concluded that the chorale harmonisation problem had essentially been ‘solved’ by expert systems [Pachet and Roy 2001]. The compositions produced by Cope’s programs have achieved notoriety for their quality [da Silva 2003]. Storino et al. found that grammar-based systems were frequently capable of successfully fooling audiences of musicians into believing that computer-composed works were in fact human-composed [Storino et al. 2007]. However, many of these approaches still suffer from problems common to expert systems generally, including the encoding of large enough knowledge bases [Coats 1988] and the potential for intractability due to combinatorial explosion [Pachet and Roy 2001]. Steedman noted that simple grammars will always produce correct musical syntax, but have a natural propensity to generate music with no semantic: the encoding of musical meaning is an extremely difficult problem [Steedman 1984]. Miranda has claimed that the biggest weakness of these systems, in the context of composing genuinely new music, is their innate inability to break rules [Miranda 2001].

2.2.4

Case-based Reasoning and Fuzzy Logic

Case-based reasoning (CBR) and fuzzy logic also fall within the expert system paradigm because they implement architectures that couple a knowledge-base with an inference engine to generate musical sequences [Sabater et al. 1998]. CBR systems rely on a database of previous valid musical ‘cases’ from which to infer new knowl-

§2.2 Formal Computational Approaches

19

edge, and are therefore inherently data-driven, even though they may further incorporate a set of immutable knowledge-engineered rules or constraints [Pereira et al. 1997]. A CBR system uses past experience to solve new problems by storing previous observations in a ‘case base’ and adapting them for use in new solutions when similar or identical problems are presented [Ribeiro et al. 2001]. Sabater et al. used case-based reasoning, supported by a set of musical rules, to generate melody harmonisation [Sabater et al. 1998]. The rules represent ‘general’ knowledge derived from traditional harmonic theory, while the cases in the database represent the ‘concrete’ knowledge of a musical corpus. Their system consists of a CBR engine with a case base, and a rule module which only suggests a solution when the CBR fails to find an example of a past solution for a particular scenario using a ‘na¨ıve’ search (in this case a note to be harmonised). Successful solutions to problems are added to the case base for future use. The system conforms to the traditional notion of an expert system which encodes domain knowledge, problem solving knowledge and meta-level knowledge [Connell and Powell 1990]. Ribeiro et al. implemented an interactive program called MuzaCazUza which uses a CBR system to generate melodic compositions [Ribeiro et al. 2001]. The case base is populated with works by Bach. In this system, case retrieval is done by using a metric based on Schoenberg’s ‘chart of regions’ [Schoenberg 1969] and an indexing system to compare a present case with a stored case. The case with the closest match is considered. After each retrieval phase, a musical transformation such as repetition, inversion, retrograde, transposition, or random mutation is applied by the user, and an ‘adaptation’ phase simply drags non-diatonic notes into their closest diatonic positions. The authors suggest continually feeding the results of a CBR system back into the case base, thus creating a model not unlike the one proposed by Cope [Cope 2005]. Pereira et al. used a similar system to Ribeiro et al., this time with a case base consisting of the works of the composer Seixas [Pereira et al. 1997]. Their CBR engine is modelled on cognitive aspects of creativity — ‘preparation’; that is, the loading of the problem and case base; ‘incubation’, which consists of CBR retrieval and ranking based on similarity metric; ‘illumination’, which is the adaptation of the retrieved case to the current composition; and ‘verification’, which in this case is the analysis by human experts. During the incubation stage, the standard ‘musically meaningful’ transformations of inversion, retrograde and transposition are employed to expand the system’s ability to generate new music. According to Sabater et al. the combination of rule and case-based reasoning methods is especially useful in situations where it is both difficult to find a large enough corpus, and inappropriate to work only with general rules [Sabater et al. 1998]. Pereira et al. believe that CBR systems contain a lot more scope for producing music that is different from the originals than musical grammars inferred from a corpus [Pereira et al. 1997]. At least one musical expert system based on fuzzy logic has been described in the literature. The system by Elsea [Elsea 1995] was implemented in Zicarelli’s Max environment [Zicarelli 2002]. The term ‘fuzzy logic’ is a potential misnomer, as the word ‘fuzzy’ refers not to the logic itself, but to the nature of the knowledge being

20

Overview of Computer-aided Composition

represented [Zadeh 1965]. The knowledge base in a fuzzy system distinguishes itself by being made up of ‘linguistic’ rules with meanings that cannot be expressed by ‘crisp’ boolean logic. For instance, the fuzzy rule “If there have been too many firsts in a row, then root or second” [Elsea 1995] is a linguistic expression guiding the inference system to avoid prolonged sequences of first inversion chords. Calculations based on this rule are made possible by assigning fractional ‘membership values’ to the quantities of successive first inversion chords that could to some degree be considered ‘too many’. The final decision of whether to transition to a root or second inversion chord is made using a translation from fuzzy membership values to corresponding fuzzy values in the decision space, which are then ‘defuzzified’ to a single value using an algorithm such as Mamdani or Sugeno [Hopgood 2011]. This process is deterministic and constitutes a precise mapping. Sophisticated fuzzy expert systems may suffer the same problems of knowledge-engineering, ‘rule explosion’ and computational complexity as crisp expert systems, but they are a lot more graceful when handling missing, inconsistent or incomplete knowledge [Zeng and Keane 2005] and are therefore potentially more effective at making musically meaningful inferences using small corpora.

2.2.5

Evolutionary Algorithms

The term ‘evolutionary algorithms’ refers to a collection of techniques inspired primarily by Darwinian natural selection [Husbands et al. 2007]. Two of these techniques which have been investigated in the field of algorithmic composition are genetic algorithms, and to a lesser extent genetic programming. These algorithms implement sophisticated heuristics for converging on local optimal solutions in very large search spaces. The reason for their popularity in algorithmic composition is their ability to traverse diverse regions of a space of musical solutions stochastically. This is advantageous for musical optimisation problems like four-part harmonisation, because it renders them no longer computationally intractable compared to expert system soluˇ ˇ 1988]. Furthermore, with a stochastic approach comes tions like Ebcioglu’s [Ebcioglu the apparent implication that new music unhindered by generative rules is possible [Gartland-Jones and Copley 2003]. Thus, while in non-artistic fields genetic algorithms and genetic programming are usually used to solve optimisation problems, in music they are also commonly exploited for their ‘exploration’ abilities, and are sometimes claimed to be analogous to elements of the human composition process [Gartland-Jones 2002]. Genetic algorithms (GA) are a heuristic search technique in which candidate solutions are represented as a population of strings or ‘chromosomes’ [Burton and Vladimirova 1999]. Each ‘gene’ of the chromosome represents a dimension of the solution space. A stochastic search process is controlled by a selection procedure based on individual ‘fitness’ and ‘reproductive’ operators to obtain successive generations of a population, and ‘mutation’ operators to randomly introduce new genetic material into an existing population. The search runs for a fixed number of generations, or until the fittest individual is somehow deemed fit enough to

§2.2 Formal Computational Approaches

21

be the final solution. Reproductive operators typically implement ‘genetic crossover’ to merge a number of parents into an offspring, and mutation operators are used to modify individual genes or small sections of an offspring’s chromosome. In the simplest ‘traditional’ GA, individuals are represented by binary strings and genetic operators operate at the binary level, with crossover occurring at arbitrary points along the string and mutation operators causing random ‘bit flips’ [Engelbrecht 2007]. However, for algorithmic composition most authors have found it necessary to instill the evolutionary process with a measure of musical domain knowledge to radically enhance the process. In particular, chromosomes are used to represent musical information at a higher level of abstraction, and ‘musically meaningful’ mutation operators are chosen, including the transformational procedures of inversion, reversal and transposition [Burton and Vladimirova 1999]. Fitness evaluation is usually cited as the most problematic aspect of GAs. Gartland-Jones and Copely classified genetic algorithms by their use of either ‘automatic’ (using an objective function or an ANN trained on a corpus) or ‘interactive’ (requiring human inspection/listening) fitness functions [Gartland-Jones and Copley 2003]. The latter are often referred to as interactive genetic algorithms (IGAs) [Biles 2001]. Phon-Amnuaisuk et al. used a GA to create traditional four-part harmonies [PhonAmnuaisuk et al. 1999]. They relied on an objective knowledge-based fitness function for the evaluation of chromosomes. The chromosomes encoded short thematic passages, the mutation operators included ‘perturbation’, which nudges a note in a single voice up or down a semitone; ‘swapping’, where chords are altered by swapping two random voices; ‘re-chord’ which randomly modifies the chord type; ‘phrase-start’, which mutates a phrase to begin on a root chord; and ‘phrase-end’, which mutates a phrase to end on a root chord. The main reproductive procedure involved splicing the chromosome strings at a random crossover point. The fitness function was a cast of rules commonly listed in traditional voice-leading theories. Biles presented a genetic algorithm called GenJam for generating monophonic jazz solos [Biles 1994]. GenJam initialises individuals within a population of melodic passages. It performs musically meaningful mutations such as inversion, reversal, rotation and transposition. The fitness of each individual in a generation is determined by a human operator, and the best individuals are used as the parents of the following generation. According to Biles, this feedback process converges on solos which match the taste of the human operator [Biles 1994]. The main disadvantage of this method is that the reliance on human feedback for evaluating fitness manifests as a bottleneck which makes the convergence process orders of magnitude slower than using objective fitness functions. Biles has addressed this problem by using entire audiences instead of individual users [Biles and Eign 1995], using ANNs for fitness functions [Biles et al. 1996], and removing fitness evaluation altogether by drawing the initial population from an established database of superior specimens [Biles 2001]. Genetic programming (GP) is an extension to the GA paradigm in which the individuals in the population are not vectors representing points in a solution space, but hierarchical expressions representing mathematical functions or the code for entire algorithms [Burton and Vladimirova 1999]. GP individuals are normally represented

22

Overview of Computer-aided Composition

as expression tree structures; consequently the selection, reproduction and mutation mechanisms are designed specifically to operate on these structures [Engelbrecht 2007]. GP fitness functions are more commonly realised as error or ‘cost’ functions because they are very popular for solving symbolic regression problems, but aside from these differences GP and GA implementations are fundamentally the same. Laine and Kuuskankare [Laine and Kuuskankare 1994], for instance, generated an initial population of melodies using simple mathematical operators and trigonometric functions, then evolved the population by performing crossover and mutation on subtrees. Longer and more complex musical phrases result from the increasing complexity of the population generations. Puente et al. used a GP technique to evolve context-free grammars for producing melodies in the style of a corpus of works by several famous composers [Puente et al. 2002]. In this instance the fitness function was simply a statistical comparison between the population members and the melodies from the corpus. Burton and Vladimirova suggested that genetic techniques allow a greater scope of musical possibilities and often subjective ‘realism’ than other approaches such as ANNs, which are restricted by training data; expert systems, which are often restricted by computational complexity and knowledge-engineering issues; and purely stochastic generators which exhibit good unpredictability but ‘questionable musicality’ [Burton and Vladimirova 1999]. However, they and many other authors have acknowledged the perennial problem of designing effective fitness-evaluation methods that reduce the counter-productive dependence on human interaction — the ‘fitness bottleneck’ [Biles et al. 1996]. Additionally, many conundrums are ever-present in the tuning of genetic algorithm parameters, such as whether to implement ‘elitist’ selection policies that may converge too quickly to local optima, or policies that retain a high level of diversity and allow low-quality individuals to continue reproducing [Burton and Vladimirova 1999]. Phon-Amnuaisuk et al. discovered that despite the supposed advantages of using GAs for four-part harmonisation, a simple rule-based system was capable of achieving consistently better results as far as the GA’s fitness function was concerned [Phon-Amnuaisuk et al. 1999]. They attributed this to the GA’s lack of sufficient ‘meta-knowledge’, a natural trait for an expert system by virtue of the fact that the structure of the search process can be easily encoded in the program. They also noted the GA’s inability to guarantee globally optimal solutions (a caveat of stochastic search), and declared the GA model ill-suited to musical optimisation problems. Despite all this, both interactive and non-interactive GAs continue to be used successfully for tasks like jazz improvisation [Biles 2007] and the composition of thematic bridging sections between user-supplied ‘source’ and ‘target’ passages [Gartland-Jones 2002].

2.2.6

Chaos and Fractals

Approaches to algorithmic composition in the tightly related fields of chaos and fractals have been popular as alternatives to the expert-system paradigm because of their tendency to exhibit recurrent patterns or multi-layered self-similarity, while at the

§2.2 Formal Computational Approaches

23

same time being fundamentally unpredictable or complex [Harley 1995]. Both are linked to mathematical resultants of the behaviour of iterated function systems (IFS) and dynamical systems, and were introduced as an alternative explanations for complex natural phenomena such as weather systems and the shape of coastlines [Mandelbrot 1983]. According to Harley [Harley 1995], their applicability to music has been influenced by the work of Lerdahl and Jackendoff, who provided convincing models for analysing musical self-similarity [Lerdahl and Jackendoff 1983]; and Voss and Clarke, who demonstrated that some music contains patterns which can be described using 1/ f noise [Voss and Clarke 1978]. The non-musical, numerical data streams created by applying such algorithms are not usually termed ‘emergent behaviour’ because they are not generated by the interaction of a virtual environment of simple interacting units. However, they share the property of being able to generate complexity at the ‘macroscopic’ level from simplicity at the ‘microscopic level’ [Beyls 1991]. Furthermore, their successful conversion into musical information is at the mercy of the mapping problem noted by Miranda [Miranda 2001], a problem also faced by systems of emergent behaviour such as cellular automata and swarms. Chaotic systems were explored by Bidlack as a means of using simple algorithms for endowing computer generated music with ‘natural’ qualities — for instance, those which can be found relating to either organic processes or divergent mathematical phenomena [Bidlack 1992]. Bidlack noted that the resultant complexity had more potential in computer synthesis, but suggested that the technique could be useful for perturbing musical structure at various levels of hierarchy, in order to instill a system with a measure of unpredictability. Dodge described a ‘musical fractal’ algorithm utilising 1/ f noise, arguing along the lines of Voss and Clarke that 1/ f noise represents a close fit to many dynamic phenomena found in nature [Dodge 1988]. He drew the analogy between his recursively ‘time-filling’ process and Mandelbrot’s recursively ‘space-filling’ curves. The time-filling fractal form is seeded by an initial pitch sequence, which is then filled in by 1/ f noise and mapped to musical pitch, rhythm and amplitude. Harley produced an interactive algorithm that centres on a ‘generator’ which provides the output of a recursive logistic differential equation; a ‘mapping’ module which scales the output to a range specified by the user; a third module which provides statistical data on the generator’s output over specified timeframes to provide knowledge of high-level structures to the user; and a fourth module which the user controls to reorder the generator output in the process of translating it to musical parameters [Harley 1995]. These modules can be networked together in order to act as raw input or as input ‘biases’ for one another. There are several examples in the algorithmic composition literature of the use of Lindenmayer Systems (L-Systems) for generating fractal-like structures. L-Systems were originally introduced to model the cellular growth of plants [Lindenmayer 1968], and first explored for musical applications by Prusinkiewicz [Prusinkiewicz 1986]. L-Systems are deterministic and expressed almost identically to Chomsky’s grammars, with the crucial difference being that instead of production rules applying sequentially, they are applied concurrently; this is what allows self-similar substructures to quickly propagate through what are exponentially expanding strings. The work by

Overview of Computer-aided Composition

24

DuBois is a recent example of the use of L-systems for musical composition [DuBois 2003]. The author separated the process into string production and string parsing, and noted that choosing the mapping scheme to use for the latter stage was critical to the aesthetic qualities of the result. He described various mapping schemes, such as ‘event mapping’, where a pre-compositional process assigns the tokens in the resulting one-dimensional string to events like notes, rests and chords; and ‘spatial mapping’, where tokens represent distances in pitch from the preceding note, and can be used to create block chords or combined with event mapping to create melodies. An additional scheme involves ‘parametric mapping’ where tokens are not assigned to musical parameters directly, but to controllers affecting the mapping of subsequent tokens to musical events. Dubois used the intermediate output of musical notation which was then interpreted by professional performers [DuBois 2003]. These approaches have allowed for alternatives to the reliance on both implicit and explicit musical domain knowledge, while allowing for the successful generation of coherent self-similar structures; and many authors have espoused their use in algorithmic composition in a general sense because of their scope for creating genuinely new musical material. However, they all ultimately put the user in charge of completing the act of composition by inventing a meaningful mapping from the data stream to musical parameters, which from a musical standpoint is hardly any different to the ‘auralisation’ of actual natural phenomena such as seismic activity [Boyd 2011] or tree-ring patterns.2

2.2.7

Cellular Automata

Cellular automata (CA) provide a means for the generation of complex emergent structures from the local interaction of simple, usually orthogonally-interconnected units. They have become a popular paradigm for exploring the analogies between mathematical models and biological phenomena. The motivation for the use of CA in computer-aided composition is cited by Miranda as being an expert system’s hardwired inability to compose new musical styles [Miranda 2003]. Some types of CA bear a strong relationship to chaotic dynamic systems because they exhibit unpredictable behaviour at the macroscopic level despite being deterministic. This was formally identified by Wolfram, who devised a widely-referenced taxonomy for describing CA types [Wolfram 2002]. CA can also been described mathematically in terms of finite state automata [Neumann and Burks 1966], and ‘static’ L-Systems [DuBois 2003]. A CA consists of grid of cells which begin in an arbitrary initial configuration and update their states at every time-step during execution. At a given time-step, t, the new state of a cell is determined by the state of its orthogonal neighbours at time t − 1 using a set of evolution rules specified before run-time. Cell states are usually binary or ternary, and cell types are often classified using a ‘KxRy’ notation, where x refers to the number of immediate neighbours and y refers to the radius of influence. CA are also classified according to their number of possible evolution rules, which is a function of the number of possible cell states, the radius of cell influence and 2

http://traubeck.com/years/

§2.2 Formal Computational Approaches

25

the number of immediate neighbours. Wolfram’s taxonomy identified four different classes of CA behaviour [Wolfram 2002]: Type 1 ‘convergent’, in which a static uniform grid state is quickly reached; Type 2 ‘steady cycle’, in which stable repeating patterns quickly emerge; Type 3 ‘chaotic’, in which no stable patterns emerge and any apparent structures are transient; Type 4 ‘complex’, in which interesting patterns are perceivable but no stability occurs until after a large number of time steps. The mapping of a CA to a musical parameter space is non-trivial, and as important to the act of composition as choosing the rule set. Frequently the resulting patterns are mapped to pitches restricted to a certain scale, such as chromatic, pentatonic or diatonic [Millen 2004]. Miranda distinguished between simplistic mappings of grid cells to MIDI note numbers and the more sophisticated method of mapping structural changes in groups of cells to higher-level musical structures [Miranda 2003]. Bilotta et al. identified analogous mapping categories of ‘local’ and ‘global’ [Bilotta et al. 2001]. They also use ‘indirect’ methods of manipulating the structure of the information contained in the CA before translating it into music [Bilotta and Pantano 2002]. Resultant structures characterised by researchers as ’gliders’, ’beetles’, ’solitons’, ’spiders’ and ’beehives’ contain varying degrees of recognisable musical harmonies when mapped directly from cell states [Bilotta and Pantano 2002]. Miranda presented a CA system for algorithmic composition called CAMUS for mapping Conway’s Game of Life to a harmonic musical output using each cell’s coordinates [Miranda 2003]. Bilotta et al. described a series of musical works produced using a genetic algorithm to further evolve the musical information resulting from a mapping of a binary CA’s output to musical parameters [Bilotta et al. 2000]. They concluded that type 1 CA are good for rhythmic generation, types 2 and 4 are good for harmonic generation, and type 3 are less useful except with very simple initial conditions. CA also feature in several interactive compositional or improvisational tools. Millen presented such a system wherein the musical parameters that the cells map to can be altered by the user during performance in reaction to visual observation of the grid state [Millen 2004]. Dorin used boolean networks (BNs) instead of CA to produce complex polyrhythms [Dorin 2000]. BNs are one-dimensional configurations of binary state machines — that is, each unit performs a boolean operation using the inputs from its two neighbours. An autonomous, synchronous boolean network is a special case of a CA [Dorin 2000]. Dorin observed that it is rare for a BN’s stable pattern to be broken even when significantly perturbed in real-time, and that this makes them ideal for generating rhythmic material for live applications. Dorin also produced a CA mounted on the faces of a virtual cube called LIQUIPRISM, distinguishing it from the more common form of CA environment which models the surface of a torus [Dorin 2002]. A stochastic element is introduced by occasionally activating cells which have

26

Overview of Computer-aided Composition

been in ‘off’ states after substantial periods of inactivity. The mapping from the CA to music in any given time-step is done through a process of eliminating cells which are not moving from off to on and then selecting a maximum of two cells from each face. Each face maps to a MIDI channel being fed into a synthesiser. Miranda believes that CAs are appropriate tools for generating new material, but concedes that they seem better suited to synthesis than composition. In his estimation the musical results “tend to lack the cultural references that we normally rely on when appreciating music” [Miranda 2003]. Bilotta et al. noted that as a general rule, only a very small subset of the available rule sets give ‘appreciable’ musical results, but that certain configurations can generate ‘pleasant’ harmony [Bilotta et al. 2001]. Dorin has demonstrated that the combination of musical and visual output of CAs can manifest as effective multimedia art [Dorin 2002].

2.2.8

Swarm Algorithms

Some researchers have pursued music generation by modelling the interaction between simple agents in artificial swarms. This model is often promoted as a remedy to the ‘lack of expression’ inherent in knowledge-based systems [Blackwell and Bentley 2002]. The approach relies on the self-organisation of agents to form complex emergent spatial and temporal structures. Beyls’ view in the context of music generation was that “behaviour may be thought of as an alternative to knowledge” [Beyls 1990]. Although this principle is also fundamental to the use of cellular automata, swarm algorithms can instead be traced back to the work of Reynolds, who proposed the first algorithms for modelling the emergent geometrical organisation of birds and other animals [Reynolds 1987]. Swarm agents are therefore generally much more sophisticated than the cells in a CA, being instilled with mobility in 2D or 3D space, sets of goals, many possible ‘social’ interactions [Miranda 2003] or ‘personality traits’ [Bisig et al. 2011], and sometimes finite energy sources which must be replenished by the swarm environment [Beyls 1990]. However, the resulting data streams are generally still at the mercy of the mapping problem; that is, finding a meaningful translation from an extra-musical data stream to a musical parameter space [Miranda 2001]. Blackwell and Bentley’s composition system based on swarm behaviour is perhaps the most widely referenced [Blackwell 2007]. In a system called SWARMUSIC [Blackwell and Bentley 2002] the agents or ‘particles’ implement the simple behaviours of swarm attraction and repulsion within the environment of a 3D ‘box’. The authors argue that this style of behaviour constitutes a form of ‘swarm improvisation’, conceding that compositional structure generally cannot be achieved by such simple behaviour. A linear mapping occurs in three dimensions corresponding to the particles’ positions in the box from the perspective of a hypothetical viewer. These dimensions, which correspond to particles’ x, y and z coordinates, are MIDI duration, MIDI pitch and MIDI velocity. The default ranges of these parameters are constrained for the purpose of Blackwell and Bentley’s implementation. According to the authors, the purported success of the free improvisation system is due to its focus on swarm ‘collaboration’ and ‘expression’ – it develops its own ‘musical language’ rather than

§2.3 The Automated Schillinger System in Context

27

attempting to assume a pre-existing one [Blackwell and Bentley 2002]. Miranda described a system for producing music using a community of simple agents with ‘auditory, motor and cognitive skills’ who collectively ‘evolve’ a set of melodies, but without the use of a genetic algorithm [Miranda 2003]. This system is an example of a swarm approach that does not require the mapping from emergent structure to musical information. Miranda’s approach encodes melody using an abstract representation of pitch trajectories forming an overall contour. The contour elements dictate relative magnitudes of pitch changes, rather than the actual intervals. Agents are instilled with the goal of imitating what they ‘hear’, and so develop individual sets of (initially random) tunes by gauging the tunes’ success through reinforcement from other agents. Elements of tunes which are also exhibited by other members of the community are strengthened, and those elements which aren’t are eventually purged. In this way a communal musical ‘repertoire’ is established [Miranda 2003]. Bisig et al. [Bisig et al. 2011] discussed another example of a swarm approach to algorithmic composition, but this time they confronted what they term the ‘mapping challenge’ by proposing to shift the focus of musical creation from the mapping itself to the types of underlying structures created by the flocking simulation. Similar to Blackwell and Bentley’s system [Blackwell and Bentley 2002], in this simulation the neighbourhood forces of attraction and repulsion are implemented which determine the swarm’s behaviour. Agents are also endowed with ‘adaptive traits’ which change over time and affect their interaction with the rest of the swarm. The system’s architecture is split into three stages: the swarm itself, a module which interprets and codifies the behaviour of the swarm, and a musical engine which integrates elements of sample playback and granular synthesis. Different pieces are composed by changing the properties of the agents and their environment. Each composition is based on a ‘core idea’, such as the triggering of piano notes via swarm collisions, or the changing spatial distribution of agents to generate rhythms. The authors point out that the success of a swarm algorithm for generating music relies on the continual injection of human creativity in regard to the design of the mapping schemes and the design of the simple rules governing agent behaviour [Bisig et al. 2011].

2.3

The Automated Schillinger System in Context

The automated Schillinger System presented in chapter 3 of this thesis uses a set of generative and transformational procedures, each invoked sequentially and seeded with random numbers. It is not interactive and does not rely on a corpus of existing musical works. Although the generative procedures are necessarily rule-based, inasmuch as they are computable, the rules dictate the space of numerical patterns available at each stage of the composition process, rather than the space of legal musical combinations. Therefore, although the system clearly employs a form of implicit musical knowledge, whether or not it falls under the umbrella of style imitation is initially unclear. This question will be examined in detail in chapter 4. Furthermore, despite the fact that the system’s musical knowledge is essentially ‘engineered’, it may not be

28

Overview of Computer-aided Composition

ˇ [Ebcioglu ˇ 1988] entirely correct to label it an expert system in the manner of Ebcioglu or Cope [Cope 1987], due to the fact that it does not use a knowledge-base/inference engine architecture [Mingers 1986]. In figure 2.1 a dashed line has been placed around the automated Schillinger System, which tentatively includes it in the realm of musical expert systems. Schillinger’s system as a whole does not lend itself to the adaptation of any particular extra-musical computational approach listed in section 2.2, unlike other music theory treatises such as those by Piston [Piston 1987] and Hindemith [Hindemith 1945] which have been partially implemented using Markov chains [Rohrmeier 2011; Sorensen and Brown 2008]; or standard harmony texts which can be partially exˇ 1988] or GA fitness funcpressed as grammar-based optimisation problems [Ebcioglu tions [Phon-Amnuaisuk 2004]. Its automation therefore falls into Supper’s first category (algorithms which encode musical theory without the use of an established extra-musical approach), and partly into Supper’s second category (algorithms used as a direct manifestation of a composer’s expertise) [Supper 2001], due to the necessity for the programmer to define many aspects of the formal interfacing between Schillinger’s various theories. In the academic literature, the category into which the automated Schillinger System most readily falls is Ames’ definition of ‘bottomup processing’, which refers to the piecing together of ‘kernels’ of primary material into larger compositions using transformation procedures [Ames 1987]. The system presented in this thesis positions itself as a particular collection of algorithms for music generation which have not been previously considered as a single entity for implementation, despite the fact that many of them are commonly used individually, and are thus familiar to computer music researchers in a variety of contexts. As can be seen in figure 2.1, the automated Schillinger System sits within a class of algorithms that process some form of musical domain knowledge, but do not rely on a data-driven or interactive approach to derive that knowledge. This causes it to fall outside of the most common approaches used by computer-aided composition researchers, but nevertheless into categories acknowledged by both Ames [Ames 1987] and Supper [Supper 2001].

Chapter 3

Implementation of the Schillinger System

3.1

Introduction

This chapter details the construction of an ‘automated Schillinger System’ based solely on The Schillinger System of Musical Composition. The books of the Schillinger System which have been considered in the scope of this work are Theory of Rhythm, Theory of Pitch-scales, Variations of Music by Means of Geometrical Projection, and Theory of Melody. Together these theories have been adapted to produce a pair of separate modules, one for composing harmonic passages and another for composing melodic pieces. Both modules operate using the ‘push-button’ paradigm and thus require no interaction with the user during the composition process. Sections 3.2 to 3.5 of this chapter constitute a condensed summary of the first four books of Schillinger’s original text to the extent necessary to explain the fundamentals behind the current automated system. It will be seen that much of this content is problematic to realise as a computer implementation and requires the resolution of inconsistencies or inadequate definitions. Despite this, it is not the purpose of this chapter to critically evaluate the practical merit of Schillinger’s formalism, nor the mathematical or scientific correctness of any of Schillinger’s generalisations, all of which are matters of contention as noted in section 1.2.3. Section 3.6 documents the software architecture of the automated Schillinger System and describes how Schillinger’s separate theories have been linked together to form the harmonic and melodic modules. It also describes various additional algorithms which have been necessary to complete this task. The final section (3.7) lists the parts of books I–IV which have been omitted from the current system for various reasons as discussed there. The discussions of Schillinger’s procedures will not be accompanied by explicit references to his original text, however a listing of the most important functions constituting the automated Schillinger System can be found in appendix C, and this list may be used to refer directly back to Schillinger’s volumes if desired. 29

30

3.1.1

Implementation of the Schillinger System

A Brief Refresher

There are many musical terms used throughout this chapter that readers may not be familiar with, or that have different definitions in other disciplines. This section explains some terminology which should facilitate the discussion while minimising potential confusion. Many of these definitions are not rigorous in terms of their broader implications, but are nevertheless adequate in the current context. Pitch/Tone The fundamental frequency of a sound with respect to a discrete system of musical tuning, in this case the 12-tone equally-tempered system featured on a standard piano keyboard. Identity The name assigned to a pitch within a system of tuning. Semitone The smallest distance between any two pitches in the aforementioned tuning √ system, produced by raising or lowering a pitch’s frequency by a factor of 12 2. Interval The distance between two pitches measured in semitones. Octave 12 semitones; the interval at which two pitches share the same identity as a result of their frequencies differing by a factor of 2. Register A localised region of the pitch space, applied either as a general notion (for example ‘high’/‘middle’/‘low’) or as a specific range of pitches. Scale A group of pitches or intervals which serve as a basis for generating musical pitch material. Diatonic Relating to only the pitches belonging to a class of Western scales made up of seven tones. Chromatic The property of pitches of a musical passage or scale being separated by semitones, or containing alterations to diatonic pitches. Tonic The starting pitch in a scale, and/or the pitch that acts as the most important musical reference point for a given composition or passage. Root The starting pitch in a scale. Duration The length of time between the onset and conclusion of a sounding pitch, usually relative to some reference value or measurement. In this chapter the term ‘relative duration’ will be used specifically to refer to that which is relative to a minimum time-span of 1. Note Usually interchangeable with pitch and identity, but also used to mean a discrete unit of musical information possessing duration.

§3.1 Introduction

31

Rhythm A sequence of durations.1 Voice A sequence of single notes related in succession. Voice-leading The rules or procedures which apply when determining the movement of individual voices within the larger wholes of harmony and counterpoint. Polyrhythm Multiple differing rhythms occurring simultaneously. Texture A term encompassing various aspects of music such as its density in the temporal and spectral domains or its aesthetic ‘surface quality’. Attack The temporal point of onset of a sounding pitch. Dynamics Variations in loudness or intensity. Modulation The change or period of change from one tonic to another. MIDI Musical Instrument Digital Interface; the dominant protocol for passing symbolic musical information between both hardware and software synthesisers. In addition to these terms, this chapter uses a standard known as ‘Scientific Pitch Notation’2 , where a pitch’s label consists of its identity followed by its octave number. Pitches C4 –B4 lie in the octave above and including middle-C on a piano keyboard. It should also be noted that MIDI note values range from 0–127, with the value 60 being equivalent to C4 .3 The use of Schillinger’s terminology will be kept to a minimum, because not all of it is especially helpful in simplifying the expression of ideas. Many problems with Schillinger’s heavy use of jargon were quite vocally drawn attention to by Barbour [Barbour 1946] and Backus [Backus 1960]. Despite this, several of the terms are still useful because they serve as short-hand for certain data structures which will be referred to frequently. All instances of Schillinger’s terminology will be defined as needed.

3.1.2

The Impromptu Environment

The system is written in a programming environment called Impromptu, an interpreter with the advantage of built-in interfaces to MIDI and audio drivers. It also has the feature of being able to execute selected portions of the text buffer at the user’s behest, known as ‘live coding’ [Sorensen and Gardner 2010]; however, as the purpose of this program is to compose musical passages autonomously rather than facilitate real-time performances, this feature is not being exploited at present. The reason 1 This is an extremely simplistic notion of rhythm which only applies to the current version of the automated Schillinger System. 2 This standard has been in use since its adoption by the Acoustical Society of America in 1939. 3 MIDI notes 0 and 127 correspond to C and G respectively. These pitches exist well beyond the 9 -1 usable musical range.

Implementation of the Schillinger System

32

Impromptu has been used is that it allows for rapid development in the LISP-based language Scheme, which has been found by many authors in the field of algorithmic composition to be appropriate for representing musical information. The built-in MIDI interface also allows for instant musical feedback and hence much faster debugging of functions operating in the musical domain. Other algorithmic composition environments such as SuperCollider4 or Max [Zicarelli 2002] would have been equally appropriate for developing the automated Schillinger System. The system outlined in this chapter manipulates two dimensions of musical information at the symbolic level (pitch and duration), which are able to be conveniently mapped to both MIDI data streams and musical notation. In the Impromptu environment, the LISP-style list format is used for coding. Many instances of list notation will accordingly be used throughout this chapter for illustrative purposes. Pitch is represented as MIDI note numbers. Duration is represented as both ‘relative durations’ during the composition process (defined in section 3.1.1), and at the output stage by durations numerically equivalent to those displayed in standard musical notation.

3.2

Theory of Rhythm

Schillinger’s Theory of Rhythm provides procedures which are mostly used to generate and manipulate sequences of relative durations. In this chapter Schillinger’s term ‘rhythmic resultant’ will be used to refer to a sequence of relative durations produced by a rhythmic procedure. Depending on the context, a rhythmic resultant will be treated as either a rhythm to be assigned to a pitch sequence, or a pattern with which to apply change at any structural level.

3.2.1

Rhythms from Interference Patterns

The ‘interference’ between any number of lists of integers is generated by treating the integers as temporal durations, superimposing the lists and forming a single new list out of the onsets of every duration. A small example is included in figure 3.1 to accompany this explanation. 3

3 2

2 2

1

2 1

2

Figure 3.1: The interference pattern generated from two lists. The top two lists (3 3) and (2 2 2) produce the resultant pattern (2 1 1 2).

The situation in the figure is expressed as follows: interference-pattern((3 3) (2 2 2)) = (2 1 1 2) 4

supercollider.sourceforge.net

§3.2 Theory of Rhythm

33

A particular space of symmetrical rhythmic resultants called ‘primary resultants’ is formed by the interference between two integers, where each integer’s duration repeats until the point where they both synchronise. The aforementioned figure 3.1 is the generation of a primary resultant using arguments 2 and 3. primary-resultant(2 3) = (2 1 1 2) A ‘secondary resultant’ is generated by recursively calculating the interference pattern between a primary resultant and the same resultant offset by the larger of its two initial parameters, until it has a total duration of the square of the larger parameter. This is visualised in figure 3.2. (3)

2

2

1

1

2

1

1

1

1

2

1

1

2

2 1

Figure 3.2: Secondary resultant of integers 2 and 3

secondary-resultant(2 3) = (2 1 1 1 1 1 2) The term ‘tertiary resultant’ will be used to refer to either one of a pair of rhythmic resultants which form a polyrhythm – one rhythm existing as the ‘lead’ and one as the ‘accompaniment’. In the current system the lead and accompaniment resultants are treated as separate entities (see section 3.7). This function accepts three integers instead of two, but otherwise uses the same interference method as for a primary resultant. The ‘lead’ resultant is the pattern formed by all three integers, while the ‘accompaniment’ is formed by the interference of their respective complementary factors with respect to a lowest common multiple. In line with Schillinger’s suggestion, the three-integer parameter lists for the tertiary resultant generator are limited to integers which belong to the same summation (Fibonacci) series. tertiary-resultant-lead(2 3 5) = (2 1 1 1 1 2 1 1 2 2 1 1 2 2 1 1 2 1 1 1 1 2) tertiary-resultant-accompaniment(2 3 5) = (6 4 2 3 3 2 4 6) Three trivial ways of combining primary and secondary resultants to form modest self-contained rhythmic patterns are mentioned, each of which utilises a single pair of parameters. They are listed using Schillinger’s terms below: Balance: a concatenation of the secondary resultant, the primary resultant, and the relative duration equivalent to the larger of the two parameters; Expand: a concatenation of the primary resultant and the secondary resultant;

Implementation of the Schillinger System

34

Contract: a concatenation of the secondary resultant and the primary resultant. res-combo-balance(2 3) = (2 1 1 1 1 1 2 2 1 1 2 3) res-combo-expand(2 3) = (2 1 1 2 2 1 1 1 1 1 2) res-combo-contract(2 3) = (2 1 1 1 1 1 2 2 1 1 2)

3.2.2

Synchronisation of Multiple Patterns

Material may be obtained by the synchronisation of a rhythmic resultant with a sequence of arbitrary elements, the latter of which may represent pitch values or higherlevel structural elements. There are two procedures used in this implementation which fall under this umbrella. The first procedure combines elements from the cyclic repetitions of each sequence until both sequences end simultaneously. The result of this is a pair of sequences each containing a number of elements equal to the lowest common multiple of the lengths of the two inputs. Figure 3.3 contains a musical example to illustrate the concept in visual terms.

5 5 555 5 )

5 5 5 E5 5

5 5 5 5 5 5 G 5 5E5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Figure 3.3: The synchronisation of a duration pattern with a pitch sequence. Each pitch is paired with a duration, in a cyclic fashion, until both sequences end simultaneously.

The second procedure interprets the rhythmic resultant as a sequence of coefficients C of length m, and synchronises it with an arbitrary sequence of elements E of length n such that element ei mod n is appended to the result ci mod n times. This continues until the last elements in both C and E are processed simultaneously. The results of this procedure are often used as parameter vectors for input to other procedures. In the following example, the element ‘0’ is repeated three times, ‘1’ is repeated twice, and so on. coefficient-sync((3 2 1) (0 1)) = (0 0 0 1 1 0 1 1 1 0 0 1)

3.2.3

Extending Rhythmic Material Using Permutations

Schillinger provides a small set of methods for building longer and more complex rhythmic patterns from the variations of short and simple ones. The predominant method of achieving variation throughout Schillinger’s system is by permutation. The ‘circular permutations’ of a sequence are a subset the complete permutations of that sequence, formed by iteratively moving the last element of a sequence to the head or vice versa; for example:

§3.3 Theory of Pitch Scales

35

circular-permutations(2 1 1) = ((2 1 1) (1 2 1) (1 1 2)) For most purposes the circular permutations are recommended by Schillinger because they retain substructures present in the original material. An example below shows the use of circular permutations to build a longer duration sequence — a ‘continuity’ to use Schillinger’s term — from a shorter one. general-continuity(2 1 1) = (2 1 1 1 2 1 1 1 2) Three further methods of deriving new patterns through circular permutations can apply to sequences which are already the required total duration. In the first two instances, the sequences are assumed to be primary, secondary or tertiary resultants. • Split the sequence into into a set, S, of n groups of equal total duration, such that n > 1 and is the smallest factor among the integers used to generate the original sequence. Select from the circular permutations of S. • Split the sequence into a set of groups, S, where each group is of total duration n and n is the larger of the integers used to generate the original sequence. Select from the circular permutations of S. • Select from the circular permutations of the original sequence.

3.2.4

Rhythms from Algebraic Expansion

A space of non-symmetrical rhythmic resultants can be obtained by a method of algebraic expansion. A relative duration sequence D of length n and total duration d is raised to a power x using a brute-force method with no intermediate summations. The resultant is a sequence of n x terms with a total duration d x . For example: (2 1 1)2 = (4 2 2 2 1 1 2 1 1) An additional important part of this procedure, as far as Schillinger is concerned, is overlaying the resultants of all powers 0 . . . x to form a texturally rich polyrhythm. This is done by multiplying the elements of each resultant Di (for i < x) by a scalar n x−i to make each resultant the same total duration. As mentioned in section 3.7, polyrhythms are not within the scope of this work; instead the resultants are treated individually.

3.3

Theory of Pitch Scales

Schillinger’s Theory of Pitch-scales contains both scale generation techniques and harmony generation techniques. Schillinger’s long and detailed theories of harmony have not been considered in the current scope of the work due to time constraints; instead, the harmony generator that is discussed in section 3.6.2 derives its initial chord progressions from the procedures in section 3.3.4.

36

Implementation of the Schillinger System

In this section, and throughout the rest of the chapter, the term ‘scale’ will be used to refer to a sequence of intervals, while ‘pitch-scale’ will refer to a sequence of pitches instantiated from a scale using a tonic pitch. Algorithmic composition researchers tend to prefer one representation over the other depending on the nature of the problem being attempted; the automated Schillinger System uses both of these representations, each depending on the requirements of the procedure at hand. A scale is variously converted into a ‘local’ pitch-scale for some purposes and a ‘full’ pitch-scale for others: the local pitch-scale will contain one more pitch than the number of intervals in the scale, while the full pitch-scale is the enumeration of a scale over the entire span of the valid pitch range (in this case, MIDI note values 0–127).

3.3.1

Flat and Symmetric Scales

The first group of scales will be known as ‘flat’ scales. A flat scale is a list of intervals with no sub-lists. Such a scale is defined by Schillinger as having a range of less than one octave – that is, a maximum range of 11 semitones – and a number of intervals between 1 and 6. Aside from these two constraints, randomly generated flat scales are uniformly distributed over the space within the octave. So-called ‘Western’ scales are a subset of the six-interval scales which, Schillinger argues, can be shown to be built from ‘tetrads’ — four-note combinations implying three-interval scales. The three-interval scales from which Western scales may be built are (2 2 1), (2 1 2), (1 2 2) and (1 3 1). An arbitrary Western scale is built by joining two of these sub-scales with a centre interval of 2, and subsequently removing the last interval (the last interval produces a repetition of the tonic at the octave which is not necessary for its completeness). For example, the scale known as harmonic minor is formed like so: (2 1 2) (2) (1 3 1) −→ (2 1 2 2 1 3) In this implementation, a bit passed to the flat scale generator specifies whether to restrict six-interval scales to Western scales or not (see the parameter settings in section 3.6.3). A ‘symmetric’ scale consists of a group of identical sub-scales spaced at equal intervals over a specified number of roots which are relative to an arbitrary tonic. These scales may span one or more octaves. They are represented by a three element set consisting of a flat scale, the number of roots and the interval between the roots. Though it is not the place to go into detail about the implications of twelve-tone equal temperament, it is enough to state that a number of roots equal to a factor of twelve is required for the scale to be both mappable onto the tuning system in question and repeat at some number of octaves while remaining ‘symmetric’. The possible forms of symmetric scale are listed in table 3.1. In all nine cases the maximum range of the sub-scales is one semitone less than the root interval, and the range is allowed to be zero. A symmetric scale is generated by randomly selecting one of the nine types, and then selecting a random flat scale of the appropriate maximum range to be the sub-scale associated with each root. In many

§3.3 Theory of Pitch Scales

37

Table 3.1: Symmetric Scale Properties

Roots

Total Range

Root Interval

Total Range

Root Interval

2 3 4 6 12

12 (1 8ve) 12 (1 8ve) 12 (1 8ve) 12 (1 8ve) 12 (1 8ve)

6 4 3 2 1

24 (2 8ves) 36 (3 8ves) 60 (5 8ves) 132 (11 8ves)

8 9 10 11

cases a symmetric scale must be ‘flattened’ by concatenating a number of sub-scales equal to the number of roots, each appended with an appropriate interval to fill in the space between each sub-scale and its following root. Symmetric scales contain much more information than flat scales. How this information is used by the harmonic and melodic modules is discussed in sections 3.3.4 and 3.5.2 respectively.

3.3.2

Tonal Expansions

The tonal expansion of a pitch-scale increases the total interval range of the pitch sequence while retaining the pitch identities (that is, the same notes in potentially different registers). The expansion of order zero is defined as the original setting of a pitch-scale; or more precisely, one in which its total interval content could not be reduced while retaining all the pitch identities. The first-order expansion of a pitch-scale is generated by cycling through the pitches and selecting every second pitch from the tonic (pitches 1, 3, 5 and so on), skipping over repeated pitches. The pitches in the new sequence are register-adjusted so that the sequence increases in pitch. An example tonal expansion is given below and is visualised in figure 3.4. 0th order:

(c d e f g a)

1st order:

(c e g d f a)

Order 0 expansion (original)



      

     

Order 1 expansion

Figure 3.4: The tonal expansion of a pitch-scale.

The ith -order tonal expansion is therefore attained by selecting every (i + 1)th pitch in the 0th -order pitch-scale and transposing them into order of increasing pitch in the same manner as above. To perform the tonal expansion of an arbitrary melodic sequence, the original pitch-scale S of the melodic pitches must be known. After performing an ith -order tonal expansion on S to obtain S0 , a scale ‘translation’ function maps the pitches in

38

Implementation of the Schillinger System

the sequence from S to their corresponding positions in S0 . Pitches in the original sequence that are not in S are left unmodified. The tonal expansion of a scale, as opposed to a pitch-scale, is necessary in many instances. In this case an arbitrary tonic is set, the scale is converted into a local pitchscale, the above expansion procedure is performed, and the resulting pitch-scale is converted back into a flat scale.

3.3.3

Nearest-Tone voice-leading

Nearest-tone voice-leading aims to minimise the total interval movement between each voice from one chord to the next in a harmonic passage. This procedure is suggested by Schillinger in lieu of the specific voice-leading techniques he introduces in later theories of harmony. It is applied in many places in his text, but only informally, such that many of the demonstrations do not represent ‘optimal’ solutions. For this implementation, it will be assumed that the aim of nearest-tone voice-leading is in fact to produce chord progressions with optimised minimum voice movement. An example is given here of optimal nearest-tone voice-leading between two fourvoice chords A and B. Chord A consists of fixed pitches, while the pitches in chord B can be octave-transposed (that is, moved ±12x semitones, x > 0) and reordered. A = (72 67 64 45) B = (72 56 55 51) The total interval movement between voices of the unmodified pair of chords is 12 — this is the result that must be minimised. The interval resulting from aligning a note bi with a j is found by transposing bi to a register such that |bi 0 − a j | ≤ 6. The algorithm implemented in this system first generates an interval matrix M representing the ideal alignments between all possible pairs of pitches in A and B, where both chords consist of n voices.   0 |b0 0 − a0 | . . . |bn 0 − a0 | 5    .. .. M( A, B) =  = . .  4 |b0 0 − an | . . . |bn 0 − an | 3 

4 1 4 1

5 0 3 2

 3 4   1  6

The optimal voice-leading combination can be found by converting the matrix M into a graph with adjacent rows and columns fully interconnected, in which nodes represent costs; and tracing a shortest path between either pair of opposite sides with the constraint that no row or column can be visited twice (this would imply re-using a pitch from B). This is shown in figure 3.5. Unfortunately, for the general case the greedy solution for this problem is usually sub-optimal, so the algorithm uses a recursive depth-first search with back-tracking and pruning to guarantee an optimal path. The optimal nodes visited during the search correspond to the voice-leading intervals created from the best alignment of the two chords: thus, tracing the resulting path

§3.3 Theory of Pitch Scales

39

0

4

5

3

5

1

0

4

4

4

3

1

3

1

2

6

Figure 3.5: Nearest-tone voice-leading search graph. The dotted line represents the suboptimal greedy solution; the solid line is the optimal solution found by back-tracking.

through the graph from one side to the other gives the pairs of pitches from A and B that should be aligned to each other using octave transposition. In this example, the optimal voice movement is found by substituting B, through subsequent reordering and octave-transposition of its original elements, with the chord (72 67 63 44). This gives a total interval movement of 2. The result is visualised in figure 3.6.

    

 A

   B



  

  





Figure 3.6: The result of performing nearest-tone voice-leading with a fixed chord A and adjustable chord B

The computational complexity of nearest-tone voice-leading for a sequence of m chords with n voices is O(mn!). However, in practice it runs significantly faster than the worst case scenario which would be equivalent to a brute-force approach. The potential for troublesome execution times for lengthy harmonic passages is offset by the fact that n is usually small. n ≤ 7 is used in the current system, and this encompasses a large range of potential harmonic textures and densities.

Implementation of the Schillinger System

40

3.3.4

Deriving Simple Harmonic Progressions From Symmetric Scales

The Schillinger System provides two ways of deriving harmonic progressions from pitch-scales. This section will outline both procedures as they have been implemented and discuss the problem of choosing between them automatically. The first procedure converts a pitch-scale into a progression of chords which are n-note aggregates of the pitch-scale units, where m is the number of notes in the pitchscale and 2 ≤ n ≤ m. The number of chords in the series will always be equal to m. Chords with roots towards the top of the pitch-scale must inherit pitches beyond the pitch-scale’s range, octave-transposed from below. An example is given in figure 3.7 for the symmetric scale ((5) 3 8) with C4 as the tonic and n = 3. The Ti brackets denote the roots and their sub-scales.

' ' ' ' ' G ' ' T0

T1

T2

=

' ' ' ' ' ' ' ' ' '  E ' ' E ' ' ' ' ' E'

Figure 3.7: Procedure 1: Extraction of n = 3 triadic harmony from a symmetric scale

Using this chord progression a ’hybrid harmony’ consisting of n + 1 voices is formed by adding a bass line centred an octave below the tonic. The bass line consists of the notes of the pitch-scale with their total interval range contracted through octave transpositions. The upper parts of the hybrid harmony are then processed using nearest-tone voice-leading (see figure 3.9). T0

G

'

T1

'

'

'

T2

'

'

'

' G ' '  '

' E' E'

T3

' E' '

' ' '

)

' ' EE' ' ' ' ' ' E' ' ' '

Figure 3.8: Procedure 2: Extraction of sub-scale tonal expansions from symmetric scale

The second procedure for deriving harmony from a symmetric pitch-scale is to take the 1st -order tonal expansions of each sub-scale, and to treat the resulting pitch sequences as chords. When the second procedure is used, the number of voices n is necessarily equal to the number of roots in the symmetric scale. An example is given in figure 3.8 for the symmetric scale ((2 3 2) 4 9) with C4 as the tonic. The Ti brackets denote the roots and their sub-scales. As tonal expansions near the top of the scale often get quite high above the musical staves, they have been transposed to the same register for the figure 3.8 example; this does not occur in the system.

§3.4 Variations of Music by Means of Geometrical Progression

41

No bass line is added in the second procedure to form a hybrid harmony. The harmonic progression is processed using nearest-tone voice-leading as before. After this processing the harmonies from each procedure appear as in figure 3.9.



Procedure 1 ('hybrid harmony'):

                 

       

        

Procedure 2:

   

   

Figure 3.9: Results of initial harmonic procedures after nearest-tone voice leading

Deciding between the two procedures is not clear-cut. Schillinger states that when the original setting of symmetrical pitch-scale is ‘acoustically acceptable’, it is appropriate to use procedure 1; while a lack of acoustic acceptability should invoke procedure 2. This term is not defined by Schillinger, so in order to automate the decision the terminology is interpreted to mean “containing sufficiently large intervals, on average, to avoid resulting cluster chords”. This implementation defines an acoustically acceptable symmetric scale to be one possessing both mean and mode intervals of ≥ 3 semitones when converted to a flat scale. The tendency is then for sub-scales with many close intervals to be expanded. Whether a scale is ‘acoustically acceptable’ or not has little bearing on how much consonance or dissonance a harmonic passage will contain after it has been processed further using the method in section 3.4.2. Moreover, it is generally not the nature of Schillinger’s system to discriminate between consonant and dissonant harmonies, because this undermines his holistic approach to musical style. Determining this property automatically without any the use of any kind of musical sensibility inserts a seemingly haphazard constraint.

3.4

Variations of Music by Means of Geometrical Progression

Schillinger’s geometrical variations correspond partially with aspects of other musical theories, such as Schoenberg’s ‘serial’ technique [Rufer 1965]. They are also similar to the operators employed in a variety of algorithmic composition approaches in the academic literature, such as mutation operators in genetic algorithms [Biles 1994].

3.4.1

Geometric Inversion and Expansion

The inversion I of a single pitch x occurs with respect to a pivot note p. The inversion of an entire chord simply maps each chord pitch in this way. The pivot in the example in figure 3.10 is C5 . I ( x, p) = p − ( x − p)

Implementation of the Schillinger System

42

' ' G '

Original:

=

' ' '

Inversion:

Figure 3.10: Pitch inversion

The value of the pivot in almost all cases in Schillinger’s text is either chosen arbitrarily or fixed as the tonic pitch of the passage being inverted. This implementation always uses the tonic as the pivot. In the case of a sequence of pitches or chords constituting a melodic or harmonic sequence, the pitches can either be inverted inplace with the above formula, or in the temporal domain by reversing both the pitch sequence and its associated duration sequence. The taxonomy of Schillinger’s ‘geometrical inversions’ follows in table 3.2. The common names for equivalents used in other musical theories are added for reference. Type 1 2 3 4

Description No modification Reversal of both pitch sequence and rhythm sequence Inversion of individual pitches followed by a type 2 reversal Inversion of pitches only

Common terminology — Retrograde (R) Retrograde Inversion (RI) Inversion (I)

Table 3.2: Schillinger’s inversion taxonomy

The expansion of material can occur in either the durational dimension or the pitch dimension, as is the case with geometrical inversions. The nth order expansion E of a single note x with respect to a pivot p is given by the formula below, and the expansion of a single chord is mapped in the same way as shown by the example in figure 3.11, where p = C4 and n = 2. E( x, n, p) = p + n( x − p) Original:

G ' ' '

=

Expansion: ' E ' '

Figure 3.11: Pitch expansion

As with inversion, the pivot value is sometimes chosen arbitrarily by Schillinger but is usually the tonic pitch. This implementation always uses the tonic pitch as the pivot. Note that the 1st order expansion maps to the original chord, while the 0th order expansion projects every pitch onto the pivot. The same formula is used to expand the pitches of harmonic or melodic material. Expansion in the temporal domain is performed by simply multiplying a sequence of durations by a scalar.

§3.4 Variations of Music by Means of Geometrical Progression

3.4.2

43

Splicing Harmonies Using Inversion

Schillinger provides what will be referred to henceforth as a ‘splicing’ procedure. It first generates a vector of inversion types by synchronising a rhythmic resultant with a list of possible inversion types (see the second procedure in section 3.2.2), then uses the vector to select and concatenate the inversions of chords from an initial chord sequence. Counters keep track of the points in the initial chord sequence that the procedure is up to. Type 1 and 4 inversions cause a counter to move forwards, while type 2 and 3 inversions cause a counter to move backwards. This process continues until the resulting sequence is the same length as the original. Once this splicing procedure is complete, the voices in the starting chord are shuffled, and an application of the nearest-tone voice-leading algorithm produces the final harmonic passage. The shuffling rearranges the vertical structure of the chord without changing the chord’s identity or the identities of any of the individual pitches. It has the effect of increasing the potential for different harmonic textures and densities. Figures 3.12–3.14 show an example of this process, using an initial chord sequence C and a vector V, generated from the rhythmic resultant R and type sequence T. The pivot used for the inversion is C4 , which is the tonic pitch closest to the center of the first chord of C. R = (2 1 1 2) T = (3 4) V = coefficient-sync(R, T) = (3 3 4 3 4 4)

     

   

Chord: 1

2

    3

    4

   

    5

6

   

   

   

7

8

9

   

10

   

       

11

12

13

Figure 3.12: Original chord sequence C Inversion type:

3

         

Chord: 13

12

4

3

13

12

4

3

4

                             13

1

12

11

4

                  3

9

4

6

3

7

6

Figure 3.13: Spliced chord sequence using C and V

Inversions of harmonic sequences can be musically analysed in terms of their relationship to the tonic: in simple tonal music, for instance, the inverted tonic chord

Implementation of the Schillinger System

44

          

        

   

          

   

               

Figure 3.14: Nearest-tone voice leading applied to sequence in figure 3.13

is equivalent to the subdominant with the opposite major/minor identity; and the inverted tonic-relative chord is equivalent to either the counter-tonic or the secondary dominant depending on whether the tonality is major or minor. No further musical detail will be entered into, but it is appropriate to point out that inverting segments of a chord progression usually adds complexity in a way that can be considered musically meaningful – it does not simply jumble the base material; nor is it a technique limited in practicality to 20th Century atonal music [Rufer 1965].

3.5

Theory of Melody

Melodic theories are far scarcer in musicological literature than harmonic theories, as ¨ ¨ observed by Hornel and Menzel [Hornel and Menzel 1998]. At around the same time that Schillinger was teaching his method in New York, the composer Paul Hindemith commented on the ‘astounding’ fact that “. . . instruction in composition has never developed a theory of melody” [Hindemith 1945]. Schillinger’s attempt to formalise melody was therefore quite unusual. In a nutshell, his method for melodic generation is to develop an abstract melodic contour, superimpose a rhythmic pattern and pitch-scale on the contour to obtain a melodic fragment, and then concatenate various manipulations of the fragment into a larger melodic composition with characteristics of musical form. As will be seen, his theory contains many uncertainties and complications for implementation.

3.5.1

The Axes of Melody

Schillinger’s concept of musical contours refers to combinations of linear segments, each with a specified pitch range and total duration. There is evidence presented in [Kohonen 1989] that the use of this idea dates back to at least 1719 with the composer Vogt. There is also a reference to the far more recent work of Myhill in [Ames 1987] which used a similar technique in the context of computer-aided composition. Schillinger describes melodies as sequences of pitch with duration in relation to a ‘primary axis’. This primary axis is, in fact, related to an extant statistical mode of a particular passage or section (that is, the most commonly occurring pitch identity) and, for the most part, reduces to the tonic. As this thesis is concerned with generating music, rather than analysing it, this definition is not especially relevant here. However, Schillinger’s Theory of Melody also uses the term more generally to mean an arbitrary ‘zero crossing’ pitch in a melodic contour; that is, the point of equilibrium

§3.5 Theory of Melody

45

that all melodic movements occur in relation to. The term ‘secondary axis’ is used to refer to an individual segment of a melodic contour. The type of a secondary axis dictates the general direction of its melodic trajectory in terms of its change in pitch relative to the primary axis. Henceforth the term secondary axis will be referred to as simply ‘axis’, and a contour formed by multiple axes will be referred to as a ‘system of axes’. Axes which move away from the primary axis are referred to by Schillinger as ‘unbalancing’, while those which move towards it are known as ‘balancing’. The unbalancing axes are thought of as implementing musical ‘tension’; the balancing axes musical ‘release’. Schillinger’s axis types are most easily represented using the taxonomy outlined in figure 3.15. The variable p shown in the diagram is known as the ‘pitch basis’, which is the default height of an axis in semitones.

Unbalancing

Balancing

Stationary

2p 6

7 5

p 1

2 0

primary axis 4

3 10

-p 9

8

-2p

Figure 3.15: Taxonomy of axis types

Combination axis types are possible, as demonstrated in figure 3.16. These can be expressed using any of the axis types in figure 3.15 with the proviso, inferred from Schillinger’s examples, that a combination does not contain both an unbalancing and a balancing axis. The melodic contour can then ‘oscillate’ between the axes using some pattern of alternation, allowing for more elaborate contours. Exactly how to oscillate between the axes in an axis combination is a concept that is expressed only informally by Schillinger. As with his other ‘forms of motion’, which will be discussed in section 3.5.3, they are presented using hand-drawn continuous trajectories which the composer is expected to convert to a discrete representation using their own judgement. In this implementation, the pattern of alternation is included in the representation of the axes: for example, when mapping a discrete pitchscale to the axis (1 4 (2 1)), two pitches map to axis type 1, followed by one pitch on axis type 4, and continuing cyclically. The treatment of these axis type combinations is otherwise the same as for the individual axis types, as described in sections 3.5.2 and 3.5.3. Finally, each axis is accompanied by a ‘pitch ratio’ P and a ‘time ratio’ T, which

Implementation of the Schillinger System

46

2p p primary axis (0 6) -p (1 4)

(2 0 3)

-2p

Figure 3.16: Examples of axis combinations, which the contour alternates between.

act as coefficients for the pitch basis p and a time basis t. These parameters affect the speed of changes in pitch, and the total interval range over which they occur. Figure 3.17 illustrates this. The variable t is a relative duration that can be thought of as analogous to the numerator of the music’s time-signature. t

2p p

t

t

Type = 1 P=2 T=1

primary axis Type = (4 9) -p P = -1 T=2 -2p

Figure 3.17: Time and pitch ratios, T and P, applied to axes, which alter their default rate and total range of change in pitch.

3.5.2

Superimposition of Rhythm and Pitch on Axes

The process for converting a melodic axis into a pitch sequence is split into two procedures. First, the points of intersection between the axis and the rhythmic attack points are established by multiplying the aggregate duration from the start of the axis by the axis gradient for each point. The gradient of an axis is the ratio between the product of the pitch coefficient and pitch basis, and the product of the time coefficient and time basis. The vertical components of the resulting intersection points can be interpreted as frequencies belonging to an arbitrary tuning system. An example using a system of three axes is shown in figure 3.18, in which the time basis is t = 4 and the pitch basis is p = 5. Schillinger gives no particular indication of how to arrive at these values, but it

§3.5 Theory of Melody

47

can be inferred from his examples that p should be half the total interval range of the chosen scale (rounded upwards) and t can be chosen at random from an appropriate range (see section 3.6.3).

2t 2

t

1 1 1 1

2

1

2

t 1 1

3

2p

Primary Axis

p

Figure 3.18: Superimposing rhythmic resultants onto an axis system. The duration attacks are projected onto the axis to produce a set of intersection points.

The second procedure maps the vertical components of the intersection points onto discrete pitches within the standard Western tuning system, which is equivalent to the MIDI pitch space (twelve-tone equal temperament). In the case of flat scales, this requires the selected local scale to be instantiated as a full pitch-scale across the range of MIDI pitches using the tonic as the primary axis. The diagram in figure 3.19 shows an example of the intersection points from figure 3.18 in relation to the intervals of the flat scale (2 1 2 2 1 2). For symmetric scales the superimposition method is similar, except that each subscale root is taken into account. First, the sub-scales are rearranged through octavetransposition, such that original distance r between roots is reduced to 12 − r and the sub-scale whose root is the tonic is positioned in the middle of the other sub-scales. The melodic axes are then partitioned and shifted vertically, if required, so that segments within distance p above the primary axis are assigned the primary axis, segments within distance 2p above the primary axis are assigned the root 12 − r above, and so on. Segments within distance p below the primary axis are assigned the root 12 − r below, and so on. Separate pitch-scales for each root are then superimposed on their respective axes; from thereon the rest of the process for melodic generation is identical. It can be seen in figure 3.19 that most of the intersection points do not fall neatly in

Implementation of the Schillinger System

48

2 2

Local Scale

Full Scale

1

a6 a5

2 a4

2

a3

1

a2

2 2

a1

Primary Axis

2 1

Figure 3.19: Superimposition of the flat scale (2 1 2 2 1 2) on the system from figure 3.18. Vertical components of the intersection points must be adjusted to align with the pitches in the given scale on the left.

line with the discrete pitches of the scale. Schillinger stops short of providing rules for resolving each situation, focussing instead on the notions of ‘ascribed’ motion (moving to the ‘outside’ of the axes), ‘inscribed’ motion (moving to the ‘inside’ of the axes) and various forms of discrete oscillatory motion; and leaving it the composer to exercise musical judgement. Consequently, the examples in Schillinger’s text do not follow any ostensible rules consistently enough to be extended to general cases. This is understandable from the outset, given his philosophy of reducing the presence of potential stylistic constraints in his system, but it does mean that automatically resolving the intersection points to scale pitches manifests as a significant obstacle in adapting the framework to computer implementation. This problem will be addressed in detail in section 3.5.3.

3.5.3

Types of Motion Around the Axes

This section outlines one possible algorithm for mapping the vertical components of the intersection points, found using the procedure shown in figure 3.18, onto the pitches of a discrete pitch-scale. The most difficult part of the Theory of Melody to formally adapt is Schillinger’s notion of fine-grained oscillatory melodic motion relative to the axes. This is primarily because the different types of motion tend to be defined using hand-drawn continuous curves, which are generally intended to be converted to a discrete representation

§3.5 Theory of Melody

49

using a composer’s musical judgement. To complicate matters further, Schillinger incorrectly defines the motions of sine and cosine, as Backus also noted [Backus 1960]. Despite this, it is possible to derive a concrete framework that implements the types of oscillation Schillinger intended to represent. They can be reduced to ‘inscribed’ motion, ‘ascribed’ motion, ‘alternating’ motion and ‘revolving’ motion. Inscribed and ascribed motion require intersection points to be dragged to scale pitches that are on the side of the axis closest to and furthest from the primary axis, respectively. Alternating motion requires a continuous crossing of the axis, as shown in figure 3.20(a), while revolving motion is supposed to follow a more ‘sine-like’ crossing of the axis as shown in figure 3.20(b). 1

1

0

0

-1

(a)

-1

(b)

Figure 3.20: Alternating and revolving motion types about an axis, represented here as zero

Although Schillinger’s definitions for these four motion types appear to be presented in clear terms, the precise rules for applying them to axes with non-zero gradients can only be inferred through demonstration, and unfortunately the definitions often contradict his use of them in the provided examples. Therefore it has been necessary for this author to devise an appropriate algorithm from scratch in order to allow the system to function (see algorithm 3.1 below). Two principles were adhered to in an attempt to avoid imparting too much of the author’s aesthetic influence on the system. Firstly, the algorithm is tuned to reproduce Schillinger’s examples as closely as possible on average; and secondly, it is designed to tend away from sequences of repeated notes. The latter decision is based on a general compositional principle that was judged not to be inherently style-specific. For implementation, the types of motion can be sufficiently encoded using the following parameters. • bias := inscribed (-1) | ascribed (1) • alternating := true | false • revolve := down (-1) | none (0) | up (1) A ‘motion type’ is assigned to every axis as a (bias alternating revolving) tuple. The bias switches polarity at every intersection point if the alternating bit is set to true. Revolving motion is applied in a constant fashion regardless of the current bias setting — if the revolve field is set to −1, the melody moves down in pitch, while if set to 1 the melody moves up in pitch. Only when the revolve field is zero is bias applied. If the revolve field is initialised to zero, no revolving motion occurs at all; whereas initialising it to non-zero causes different results depending on

50

Implementation of the Schillinger System

Algorithm 3.1 Resolve a sequence of intersection points X to pitches P, using pitchscale S, motion parameters bias, alternating and revolve, and the axis gradient. for all xi in X do if revolve = 1 then pi = above( S, pi−1 ) nextrev = −1 revolve = 0 else if revolve = −1 then pi = below( S, pi−1 ) nextrev = 1 revolve = 0 else if bias and gradient are same polarity then xi = xi +1 end if if xi falls exactly on a pitch-scale note then pi = xi else if xi is equidistant from below( S, xi ) and above( S, xi ) then if bias = 1 then pi = above( S, xi ) w = below( S, xi ) else pi = below( S, xi ) w = above( S, xi ) end if else if above( S, xi ) is closer to xi than below( S, xi ) then pi = above( S, xi ) w = below( S, xi ) else pi = below( S, xi ) w = above( S, xi ) end if if pi = pi−1 then pi = w end if end if revolve = nextrev end if if alternating then bias = −bias end if end for N.B. when pi−1 or xi+1 exceed the range of i, the pitches corresponding exactly to the start and end-points of the axis are assigned instead.

§3.5 Theory of Melody

51

its initial polarity. This means that, in total, the parameters allow for twelve different forms of motion. In algorithm 3.1, the functions below(S, a) and above(S, a) are assumed to return the closest pitch from S which is below or above the point a. To illustrate how this algorithm applies to the scenario shown previously in figure 3.19, all twelve motion combinations are documented in table 3.3 and figure 3.21 as they pertain to the first axis in that scenario, with the primary axis instantiated as C4 . Bias Alternating Revolve Type Label Cross-point a1 60.00 a2 62.50 a3 63.75 a4 65.00 a5 66.25 a6 67.50 Figure 3.21 references

-1

1

-1

T 0

1

-1

F 0

1

-1

T 0

1

-1

F 0

1

60 63 65 67 65 70 A

60 63 65 67 65 70 B

60 63 62 67 68 70 C

60 62 63 65 63 67 D

60 62 63 65 67 68 E

60 62 60 65 67 68 F

62 63 65 65 63 67 G

62 63 65 65 67 68 H

62 63 62 65 67 68 I

62 63 65 67 65 70 J

62 63 65 67 68 70 K

62 63 62 67 68 70 L

Table 3.3: Resolution of points in figure 3.19 using the possible motion combinations, with C4 as the primary axis

A: D: G: J:

             

B: E:

       H:        K:

       C:        F:        I:       

L:

                           

Figure 3.21: Musical representation of the results in table 3.3

Figure 3.22 shows the result of applying the motion type (-1 false 0) to every axis in the figure 3.19 scenario.



 

  













Figure 3.22: Resolution of figure 3.19 using motion type (-1 false 0)

As a final case in point, Schillinger’s retrofitting of the opening of a composition by J.S. Bach to a pair of axes to demonstrate the theory’s efficacy is compared with the same pair of axes processed using the automated Schillinger System. This serves to

Implementation of the Schillinger System

52

illustrate some of the issues that have been mentioned. The comparison can be found in table 3.4 and figure 3.23. Table 3.4: Modelling Bach: Schillinger’s representation and this system’s equivalent

Axis 1

2

Parameter Axis type Rhythm Motion Axis type Rhythm Motion Scale

Schillinger’s text ‘ 0a ’ (-2 2 2 2 2 2) ‘sine with increasing amplitude’ ‘b’ (2 1 1 1 1 1 1 1 1 1 1) ‘sine+cos with constant amplitude’ (2 2 1 2 2 2)

This system (1 0 (1)) (-2 2 2 2 2 2) (-1 false 0) 2 (2 1 1 1 1 1 1 1 1 1 1) (1 false -1) (2 2 1 2 2 2)

Figure 3.23: Modelling Bach: comparison between Schillinger (left) and the this system (right)

The fact that the automated Schillinger System comes close to replicating the passage from Bach is not intended to be a measure of its success. In fact, it raises the question of whether Schillinger’s system (and, by extension, the automated system) is really capable of generating music independent of style, or if it has simply been modelled off existing music using a different methodology to the treatises which Schillinger hoped to supersede. In order to examine this question properly, it is necessary to collect data on the stylistic properties of the system’s output. The experiments designed to do this can be found in chapter 4 of this thesis. In any case, the parameters and the algorithm presented in this section provide a concrete specification of axis-relative motion which this author believes successfully encapsulates the ideas Schillinger expressed informally.

3.5.4

Building Melodic Compositions

A system of axes which has been converted to a sequence of pitches with an associated sequence of relative durations forms a melody. Depending on the stochastic parameters which have been used to generate it, the melody may be reasonably musically self-contained, or it may constitute a short melodic fragment. In both cases this melody is used as the basic material for building a complete melodic composition. This is done by appending the initial melody with a series of modifications of either

§3.5 Theory of Melody

53

the melody or its individual axes. Schillinger suggests that these modifications can be any combination of the following: • Tonal expansion • Circular permutation • Geometrical inversion (types 1–4) • Geometrical expansion The procedure which builds the melody takes a vector representing the sequences of axes to use, and four vectors representing the respective sequences of modifications. As usual, Schillinger provides no formal guidelines for generating these vectors other than implying that the original melody should feature unmodified at the beginning and with minimal modification at the end of the composition. This basic constraint has been implemented, as well as some other constraints which have been informed by Schillinger’s examples. In all instances below, L is the nominal length of the final composition. • The axis vector A is defined as { a0 , a1 , . . . , a L }; 0 ≤ ai ≤ n, where n is the number of axes constituting the initial melody, and zero is used to denote the full initial melody comprising all the axes in their initial order. The axis terms a1 . . . a L−1 are selected randomly with 10 percent weighting given to a value of zero and 90 percent distributed evenly among rest, while the term a0 is restricted to zero and a L is restricted evenly to either zero or the last axis in the system. These simple constraints tend to generate melodic ‘expositions’ followed by sequences of ‘developments’, and also tend to enforce similarity between the opening and closing sections of compositions. • The permutation vector P is defined as { p0 , p1 , . . . , p L }; 0 ≤ pi < length( ai ). As per Schillinger’s recommendation, the permutations are restricted to circular permutations in order to maintain the basic interval structure of the sequence. Terms p0 and p L are restricted to zero, while terms p1 . . . p L−1 are uniformly random. The permutation of an axis applies to its pitches but not its durations. • The tonal expansion vector S is defined as {s0 , s1 , . . . , s L }; si ∈ {0, 1}. The terms refer to orders of tonal expansion as explained in section 3.3.2, and their probabilities are weighted equally. Higher orders are avoided because their intervals quickly become enormous, and ‘collapsing’ the pitches (as used for geometrical expansions — see below) loses the original shape of the melody, which is not intended by Schillinger in this case. s0 and s L are restricted to zero. • The inversion vector I is defined as { j0 , j1 , . . . , j L }; 1 ≤ ji ≤ 4; that is, a selection from the taxonomy of inversions presented in section 3.4, with 20 percent weighting given to type 1 (no inversion) and 80 percent distributed uniformly among the rest. The term j0 is restricted to zero, while j L is restricted evenly to type 1 or 4.

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• The expansion vector E is defined as {e0 , e1 , . . . , e L }; ei ∈ {1, 2, 3, 5, 7}. The terms refer to the orders of expansion as in section 3.4. Orders 4 and 6 are omitted (upon Schillinger’s recommendation) because they do nothing more than reduce the space of pitches to a subset of order 2. These expansions frequently extend far beyond the range of the piano, so they are routinely ‘collapsed’ back to the register of the starting note of the sequence through octave transpositions. Geometric expansions are used sparingly because they modify the original material to the greatest extent. Thus a weighting of 60 percent is assigned to order 1 (no expansion), with 40 percent distributed uniformly among the rest. e0 and e L are restricted to order 1. A melodic composition can then be expressed as the sequence { M0 , M1 , . . . , ML }, where Mi is built using the formula below. The order of operations has been inferred from the examination of Schillinger’s examples. expgeometric (permute (invert (exptonal (ai ), si ), ji ), pi ), ei )

3.6

Structure of the Automated Schillinger System

All of the procedures described up to this point exist independently as a set of compositional ‘building blocks’, and as such they cannot be used to compose music without being interfaced in some way. Although Schillinger’s theories regularly reference one another, in the first four books there are no formalised higher level procedures for creating compositions from scratch. This section outlines the software solution that has been devised by this author to encompass all four theories in a fully automated system which can compose self-contained, single-voice melodic compositions, and multi-voice harmonic passages. To orient the reader, a basic overview of the system’s architecture is contained in figure 3.24. On the following page the reader will find a more comprehensive call graph of the automated Schillinger System. This graph refers to all of the individual procedures necessary to summarise system’s architecture. The points in the system that the user interfaces with can be found in the bottom left and top right corners (‘compose harmony’ and ‘compose melody’). Red boxes surround the groups of procedures that are either associated with or directly implement Schillinger’s theories in books I–IV.

Contract Pitch Range

Splice Harmony

Compose Harmony

Re-voice Starting Chord

Invert Harmony

GEOMETRIC VARIATIONS

Expand Voice

Invert Voice

Automated Schillinger System: Call Graph

Acoustically Acceptable?

Adjust Register

Symmetric to Sub-scales

Scale to Basic Harmony

THEORY OF PITCH SCALES

Generate Symm. Harmony

Nearest-tone Voice Leading

Hybrid Harmony

Scale Tonal Expansion

Random Symm. Scale

Scale Translator

Random Scale

Random Flat Scale

Build Melody

Algebraic Exp.

THEORY OF RHYTHM

Coeff. / Group Synchronisation

Self-contained Rhythm

Interference Pattern

Tertiary Res.

Secondary Res.

Primary Res.

Convert Basis

Group Attacks

Superimpose Pitch/Rhythm

Generate Build Params.

Random Axis System

THEORY OF MELODY

Permutation Generator

Rhy. Continuity

Resultant Group by Pairs

Random Res. From Basis

Group Durations

Generate Rhythm

Generate Secondary Axes

Compose Melody

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56

Automated Schillinger System Theory of Rhythm Harmonic Module

Theory of Pitch Scales Geometric Variations

Melodic Module

Theory of Melody

Impromptu

Figure 3.24: Basic overview of the structure of the automated Schillinger System

The following sections describe the higher level procedures that were necessary to complete the automated system. As far as Schillinger’s system itself is concerned, they are entirely arbitrary manifestations of this author’s interpretation of the formalism as a whole. This is somewhat problematic, and even though every effort has been made to impart as little aesthetic influence as possible through these procedures, such influence is difficult to perceive in the system’s output and the lack of it cannot be guaranteed.

3.6.1

Rhythm Generators

Despite the abilities of the rhythmic procedures in section 3.2 to generate a vast space of content, one lingering aspect of the Theory of Rhythm that remains largely undefined by Schillinger is how to select from it; this is left entirely to the composer’s musical taste. In lieu of any formal procedures, the current section describes this author’s necessary solution for providing rhythmic resultants to the harmonic and melodic modules. As mentioned above, this solution is quite arbitrary — it has been designed to incorporate as much of the content produced by his procedures as possible. The schematic in figure 3.25 shows how the automatic Schillinger System’s two rhythm generators are structured. Calling functions make one of the following requests, in which t is the time basis and T is the time ratio. • Rhythm generator 1: generate-rhythm(t, T) • Rhythm generator 2: random-resultant(t) The functionality of each part is listed below. • The primary, secondary and tertiary resultants are produced using pairs or trios of integers as described in section 3.2.1.

§3.6 Structure of the Automated Schillinger System

1. Generate Rhythm

Group Durations

Primary Resultant Interference Pattern

Permutation Generator

Rhythmic Continuity

Algebraic Expansion

57

2. Random Resultant

Secondary Resultant

Tertiary Resultant Resultant Group by Pairs

Self-contained Rhythm

Figure 3.25: Call graph showing the structure of the rhythm generators

• Rhythm generator 2 (‘random resultant’) selects between the three kinds of resultants with equal probability. In line with Schillinger’s suggestion the inputs for the tertiary resultant function are confined to trios of integers ≤ 9 drawn from the same Fibonacci sequence. Primary and secondary resultant inputs are also confined to an enumerated set of possible pairs at Schillinger’s behest, with all integers i such that i ≤ 9. In all cases one of these integers is fixed as t. • The function which generates random resultant combos in the manner shown in 3.2.1 does so by randomly generating both a primary and secondary resultant using t, with the same constraints as rhythm generator 2. • The permutation generator returns a random circular permutation of its input. • The ‘Self-contained rhythm’ function first extracts a random sub-group G of duration t from a resultant R provided by rhythm generator 2. It then collects a random resultant combo using t, algebraic expansions of G using powers 2 and 3, and continuity patterns of all variation types listed in section 3.2.3 generated from both R and G. Finally, it randomly selects a resultant from the subset of the collection possessing total durations less than T × t. • Rhythm generator 1 randomly selects from a ‘self-contained’ rhythm, a random (t × T )-duration sub-group of a resultant R provided by rhythm generator 2, and a random t-duration sub-group of R concatenated to a recursive call to rhythm generator 1 with arguments t and T − 1. Rhythm generator 1 is used by the melodic module to randomly generate rhythmic resultants of specific total durations that are then superimposed onto axes as described

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in section 3.5. Rhythm generator 2 supplies only randomly selected symmetrical resultants of arbitrary total duration. These are used by the harmonic module to splice harmonic inversions together as shown in section 3.4, by the melodic module to determine the pattern of alternation between the individual axes in a combination axis, and also by rhythm generator 1. The rhythmic generators do not attempt to assess the inherent quality of a resultant or its applicability to the context it is required in. Instead, they make the assumption that all rhythms which satisfy the constraints t and T imposed by the caller are equally viable (and by implication, that Schillinger’s rhythmic procedures are doing something musically meaningful). Thus, in effect the rhythmic generator does nothing more than impose a probability distribution across the space of all possible resultants of a given total duration, as a side-effect of the generative procedures it has at its disposal. To illustrate the point, figure 3.26 shows the relative frequency of all possible resultants that are encompassed by the time basis t = 4, with T = 1. 0.35

Probability of occurence

0.3 0.25 0.2 0.15 0.1 0.05

) (4

1) (3

3) (1

2) (2

1) (2

1

1) (1

2

2) 1 (1

(1

1

1

1)

0

Rhythmic resultants

Figure 3.26: The probability distribution imposed by rhythm generator 1 across the space of rhythmic resultants for t = 4 and T = 1.

Degazio pointed out that Schillinger’s method of treating rhythmic cells as multilevel structural generators could be used to produce fractal structures [Degazio 1988]. This possibility has not been pursued in the current scope of work because the harmonic and melodic modules contain only very limited opportunities to incorporate such structures. Additionally, given that the current thesis is concerned with adapting Schillinger’s system as a music-generating entity in itself, the application of Degazio’s ideas would likely fall outside of this goal.

3.6.2

Harmonic and Melodic Modules

The harmonic module uses rhythm generator 2 , the procedures pertaining to symmetric pitch-scales and the geometric variation procedures to build a harmonic passage.

§3.6 Structure of the Automated Schillinger System

59

Virtually all of the required functionality for this process has already been discussed in sections 3.3 and 3.4; the module merely controls the data flow during the composition process. Figure 3.27 contains a visual representation of the module’s operation. The constraints applied during composition can be found in section 3.6.3. Random Symmetric Scale

NO

3.3.1

YES

3.3.4

Acoustically acceptable?

Symm. Scale to Sub-scales

3.3.4 Tonal Expansions

3.3.2

Symm. Scale to Chords

Contract Pitch Range

3.3.4

3.3.4

Hybrid Harmony

3.3.4

Rhythmic Generator 2

3.6.1 3.2

Harmony Splicer

3.4.2

Geometric Inversions

3.4.1

Nearest-tone Voice Leading

3.3.3 Output

Figure 3.27: Harmonic module data flow, including relevant section numbers pertaining to this chapter.

The melodic module incorporates all four of Schillinger’s theories that have been examined in previous sections. The composition process is visualised in figure 3.28. As with the harmonic module, the melodic module controls the data flow during this process, thereby acting as an interface between Schillinger’s theories. However, so far the process for generating a melody is only well defined if the axis system is already known (as was the case in for examples in section 3.5). Unfortunately Schillinger provides no explicit method for generating axis systems, so this author has provided two further procedures to accomplish this task.

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60

Generate Axis System

Rhythm Generator 1

3.6.1 3.2

3.6.2 3.5.1

Generate Secondary Axes

3.6.2

Random Flat or Symmetric Scale

3.3.1

Superimpose Rhythm and Pitch onto Axes

3.5.2 3.5.3 3.3.2 Tonal Expansions (Scale Translator)

Geometric Variations Generate Build Parameters

3.5.4

3.4 Build Melody

3.5.4

Permutation Generator

3.2.3

Output

Figure 3.28: Melodic module data flow, including relevant section numbers pertaining to this chapter.

The first produces a set of axis parameters: a sequence of axis types, a sequence of time ratios, a sequence of pitch ratios, a time basis t and a ‘degree of motion’. Currently, the axis types are influenced by the user in the form of ‘stimulus’ list such as the following: (u b u b) A value of u indicates an ‘unbalanced’ axis, while b indicates a ‘balanced’ axis. These values are used to choose axis types (or combinations of axis types) at random from the taxonomy in figure 3.15. The time basis, time ratio and pitch ratio associated with each axis are chosen at random from the ranges documented in section 3.6.3. The degree of motion is a concept included by the author to ensure that rather than

§3.6 Structure of the Automated Schillinger System

61

the oscillatory motion types of each axis being selected at randomly from the twelve possible types (see section 3.5.3), a relatively consistent amount of either angular or smooth step-wise movement is applied from axis to axis. A degree of motion is selected at random from the range [1, 5]. The meaning of these options is described below. The second procedure in the chain, as observed in figure 3.28, is necessary to provide a system of axes which can then undergo the superimposition process. Each axis output by this procedure consists of the corresponding axis type and pitch ratio P generated by the first procedure; a rhythmic resultant provided by rhythm generator 1 of total duration t × T; and a motion type of the form (bias, alternating, revolve). Table 3.5 shows how the motion type is influenced using the degree of motion by applying different probabilities to the individual parameters of the motion type tuple. The u and b options in the bias column apply when the axis type is respectively unbalancing or balancing. Informally, the degrees range from guaranteed smooth motion to guaranteed oscillatory motion with frequent melodic leaps. Table 3.5: Probabilities of motion type parameters for different degrees of motion

Degree 1 2 3 4 5

Bias -1 1 b: 1 u: 0 b: 1 u: 0 b: 0.7 u: 0.3 0.5

1 0 b: 0 u: 1 b: 0 u: 1 b: 0.3 u: 0.7 0.5

Alternating T F 0 1

Revolve -1 0 1 0 1 0

0

1

0

1

0

0.2

0.8

0.25

0.5

0.25

0.5

0.5

0.25

0.5

0.25

1

0

0.5

0

0.5

Once a melodic composition is generated, the module converts the resulting sequence of relative durations (accompanying the pitch sequence) into a standard form appropriate to be mapped to musical notation, by dividing each relative duration by the power of 2 closest to the time basis.

3.6.3

Parameter Settings

The table in this section (table 3.6) contains the parameter ranges that are wired into the ‘push-button’ version of the automated Schillinger System. Within the specified ranges, the actual values chosen are uniformly random for each execution of a module. The table does not include constraints that are in place according to Schillinger’s explicit recommendations, thereby contributing directly to the modelling of the theories. This information is meant to complement the constraints introduced by the author as part of the process of adapting the procedures, such as those that were mentioned in sections 3.5.3, 3.5.4 and 3.6.2. Generally speaking, the settings have been chosen with the view to coaxing forth a representative cross-section of the system’s

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Table 3.6: Parameter settings used by the author for the ‘push-button’ system

Section Harmonic module

Melodic module

Parameter No. symmetric sub-scale intervals Restrict 7-tone scales to Western Tonic note Time basis for splicing Possible inversion types No. flat scale intervals No. symmetric sub-scale intervals Flat scale range Restrict 7-tone scales to Western Tonic note Time basis for rhythm Nominal length Pitch ratio Time ratio

Range/setting [1, 6] false [C3 , C5 ] [3, 9] [1, 4] [1, 7] [2, 6] [5, 12] true [C3 , C5 ] [3, 9] [5, 9] [1, 2] [1, 4]

musical capability without requiring an enormous quantity of output and analysis. Specifically, each parameter has been given its current setting by the author for any one of three reasons: • To avoid unreasonably long computation times in the Impromptu environment; • To reduce the presence of clusters in the output possessing particular anomalous characteristics, such as harmonies that contain only a single repeated chord, melodies with physically implausible intervals or music centered in extreme registers; • To implement musically logical lower or upper bounds that are not mentioned by Schillinger but are necessary to prevent output which is completely trivial, such as one-note harmonies or melodies5 ; or music that is absurdly long. In future work, specific combinations of these parameters may be established that serve as reliable prescriptions for stylistic or aesthetic properties in the music’s output. They could also be made individually controllable by the user as part of a graphical or command-line interface. So far, the author has not been able to identify individual parameters that have a noticeable or measurable effect on the final output in terms of its style.

3.7

Parts of Schillinger’s Theories Not Utilised

The content of books V–XII of the Schillinger System has not been used in either of the modules due to the restricted scope of this thesis. Additionally, several aspects 5

This is not to suggest that single-note melodies cannot be musically interesting. In this case however, they will certainly be trivial.

§3.7 Parts of Schillinger’s Theories Not Utilised

63

of books I–IV have also been omitted from the project for various reasons. These are listed below to help give a clear idea of the extent and limitation of the current work, and also as a reference for future work. • The use of tertiary generators, variation techniques and algebraic expansions for producing poly-rhythmic textures has not been included because the system does not currently incorporate a notion of polyphony. Polyphony is central to the construction of more complex compositions, requiring the context of books V–XII. • The application of resultants and synchronisation to ‘instrumental forms’ is omitted because it pertains to instrumentation and orchestration, which are discussed in later books. • Rests are not incorporated into the rhythmic generator for want of a more sophisticated method determining their placement. Schillinger offers minimal advice on the placement of rests. • Rhythmic accents are not incorporated because they are only covered extremely briefly and fall partly into the realm of Schillinger’s Theory of Dynamics. • Schillinger’s ‘evolution of rhythm styles’ is omitted because it consists primarily of an analytical discussion with reference to popular musical styles of his time of writing, rather than any explicit generative procedures. • The discussion of ‘rhythms of variable velocities’ is relevant to the field of expressive performance rather than to algorithmic composition as such. The problem of expressive performance is mentioned in chapter 4 of this thesis. • The use of synchronisation to produce simple looping melodic forms from pitchscales has not been incorporated into the melodic module because it does not fit with the melodic axis paradigm, which is what the current melodic module is built around. As it is presented, it also produces absolute rhythmic monotony, which has been avoided for this system’s melodies. • Schillinger’s ‘evolution of pitch-scale families’ refers to the use of interference, subdivision, circular permutation and transposition to build a set of supposedly related scales which may bring unity to a longer form piece. As both modules in this system are focussed on smaller compositions, this concept has been abandoned for the present time. • The concept of ‘melodic modulation’, as discussed in the Theory of Pitch-scales; that is, concatenating the synchronised melodic forms mentioned above into longer sequences using multiple pitch-scales with pivot sequences at the connection points, has not so far been incorporated into the melodic module. Again this is due to it being largely incongruous with the axis paradigm. Schillinger’s method of identifying and reusing motifs using this concept should also be noted.

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Implementation of the Schillinger System

• Producing melodic continuity from symmetric pitch-scale ‘contractions’ has been omitted for the same reasons as above. • The accompaniment of the simple harmonic procedures in section 3.3.4 with melodic forms derived from the same pitch-scale has been omitted from the current implementation, because without significant human intervention it places too many restrictions on the current method for harmonic generation used in the harmonic module. • The concatenation of short melodies into longer melodies using only geometrical inversions has been avoided as a technique in itself, because the equivalent functionality exists in the melody builder as part of the somewhat more sophisticated melodic module. • Geometrical expansions in the temporal domain have been left out of the melodic module for the time being because they produce quite drastic incongruities in what are currently short-form compositions. It may be more appropriate to include this once more explicit concepts of form and higher-level structure have been incorporated from later books. • The geometrical expansion of harmonies is not currently performed, √ because it 12 has the effect of simply√projecting a chord √ progression from √the 2 tuning sys√ tem into whole-tone ( 6 2), diminished ( 4 2), augmented ( 3 2) and tritone ( 2 2) systems. This technique was deemed unnecessarily limiting for short harmonic passages, but could be viable in the context of longer compositions. • No attempt has been made to automate Schillinger’s notion of musical semantics because it is mostly in the form of philosophical discussion. The section on climax and resistance in relation to a ‘psychological dial’ is particularly noteworthy because in the past it has been referred to by successful film composers [Degazio 1988]. As explained in section 3.6.2, the user is currently in control of ‘seeding’ the melodic module with a set of abstract axis types, but no explicit musical meaning is drawn from their combination when building a composition. • Schillinger’s application of melodic trajectories to generate short embellishments has not been used in the current system, but is fairly amenable to being added in the short term. • The very brief discussion on melodic modulation in the context of axis systems is omitted because it was felt that it would be better considered in the future alongside Schillinger’s other discussions of melodic modulation in the context of pitch-scales. • Finally, the use of ‘organic forms’ (melodic motifs or entire passages generated using number sequences related to the Fibonacci series) in melody generation

§3.8 Discussion

65

has been omitted due to time constraints. These motifs could easily be incorporated into melodic compositions by giving the melody builder the opportunity to select them either as a possible variation or an alternative initial sequence. This requires the composition’s pitch-scale to be derived from the motif. To summarise, the elements of Schillinger’s theories listed above have mostly been left out either due to time constraints or because they are too heavily related to theories in books V-XII to warrant further investigation without the additional context. All of the items stand to be revisited in future work.

3.8

Discussion

The construction of an algorithmic composition system based entirely on Schillinger’s theories has presented several hurdles. In particular, none of the first four books of the Schillinger System under consideration contain the means for formally interfacing each collection of procedures, and even some of the procedures which are amenable to computer realisation require significant reinterpretation to make this plausible. In both cases the author has been obliged to devise and implement algorithms not present in Schillinger’s theories, and it is possible that this has influenced the aesthetic characteristics of the system’s output in ways that are difficult to detect, something undesirable but unavoidable. Nevertheless, this chapter has shown that the bulk of the material in these books can in fact be adapted to computer implementation. As far as the author can ascertain this is the first system of its kind to be formally documented. Two modules have been presented that automatically compose harmonies and melodies using Schillinger’s theories in a non-interactive ‘push-button’ paradigm. These modules have been described in detail, and the points in the system’s operation where constraints on the output space are enforced have been documented. Of particular note is a new formal definition of Schillinger’s ‘forms of motion’ in section 3.5.3, which allows for generation of melodies using the informal framework he provided in the Theory of Melody. This was followed by a comparison between the formal and informal procedures in the context of music by J. S. Bach, which has raised further pertinent questions about the nature of the automated Schillinger System’s output with regard to musical style. As it stands, this chapter’s content also provides a valuable resource for others wishing to approach Schillinger’s first four theories of composition, because it contains concise explanations of the majority of their generative procedures. Up to this point the automated Schillinger System has been discussed in terms of its procedures, but not in terms of the quality or stylistic diversity of the music it is capable of producing. This is another matter entirely which will be explored extensively in chapter 4 as a means of critically evaluating the system.

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Chapter 4

Results and Evaluation

4.1

Introduction

An algorithmic composition system is of no use if it does not produce musically meaningful output. In a survey of the first three decades of computer-assisted composition, Ames acknowledged the evaluation of output to be a highly problematic but essential aspect of this research [Ames 1989]. Miranda has frequently noted the difficulty of verifying musical output without intervening human subjectivity [Miranda 2001; Miranda 2003]. Section 4.2 will briefly survey the most common methods of assessment employed by authors who have viewed it necessary to go beyond a cursory personal judgement. In the sections thereafter, informed by past methods of evaluation, two experimental methods will be described that have been used to gain some insight into the aesthetic and stylistic characteristics of the output from the automated Schillinger System. The first experiment draws on the burgeoning field of musical information retrieval (MIR); in particular, automated genre classification. Section 4.4 presents a method for measuring the style and diversity of MIDI output using MIR-oriented machine learning software, and the corresponding results. The second experiment is a listening survey involving expert participants, which provides a useful collection of both quantitative and qualitative data from which to develop robust conclusions regarding the subjective properties of a representative group of samples of the system’s output. Section 4.5 describes the details of the listening survey and presents the results from it. Section 4.6 summarises and discusses the implications of the results of both experiments.

4.2

Common Methods of Evaluation

In describing the genetic algorithm-based improvisation system GenJam, Biles claimed that solos begin to yield “pleasing” results after five generations and “reasonable” results after ten generations [Biles 1994]. Johnson-Laird referred to the results of a constraint-satisfaction composition system as “simplistic but pleasing” [JohnsonLaird 1991]. Johanson and Poli, referring to a system using genetic programming, gave the concluding statement that “almost all of the generated individuals were 67

Results and Evaluation

68

pleasant to listen to” [Johanson and Poli 1998]. This kind of cursory subjective judgement by authors in the published literature is common. There is no suggestion being made here that these judgements are necessarily unjustified, but they are fundamentally unscientific, prone to bias and therefore unsatisfactory [Wiggins et al. 1993]. The formal assessment of the validity of musical passages has often been attempted using objective functions, mostly in the context of genetic algorithms where it is necessary to sort population members by fitness. These objective functions typically calculate a ‘penalty’ score based on how many and what kinds of rules in a knowledge base are broken [Phon-Amnuaisuk et al. 1999], or perform a statistical comparison to a corpus of musical exemplars [Puente et al. 2002]. Unfortunately these methods are limited to musical problems with well-defined, widely documented aesthetic constraints — namely traditional chorale harmonisation.1 Pearce and Wiggins have discussed more advanced frameworks intended to replace subjective judgements with extensive musical analysis, but they too can only operate within specific stylistic boundaries [Pearce and Wiggins 2001]. It is desirable to move beyond this kind of evaluation. For this reason some authors have undertaken more rigorous evaluations of output by involving one or more ‘musical experts’. Phon-Amnuaisuk engaged a senior musicology lecturer to mark computer output using the same criteria as first-year students of harmony [PhonAmnuaisuk et al. 1999]. Hild et al. used “an audience of music professionals” who ranked the output of the system HARMONET to be on the level of an improvising organist [Hild et al. 1991]. Periera et al. used “expert musicologists” to give a panel-style evaluation using criteria such as musical interest and musical reasoning [Pereira et al. 1997]. Storino et al. have concentrated on whether or not humans are able distinguish human-composed music of a particular style from similar computer-composed music in controlled experiments [Storino et al. 2007]. In the human experiments above where the focus is not on fooling participants with style imitation but rather seeking a genuine appraisal of merit, none of the methods or results are presented in the literature except anecdotally, and there is little evidence that they are particularly rigorous. This thesis will take the concept of assessing musical merit one step further by performing a far more in-depth survey of expert human participants using carefully designed criteria. The details of this study comprise section 4.5.

4.3

Automated Schillinger System Output

Before the details of the experiments designed for evaluation are presented, it is important to make clear exactly what is being evaluated. The automated Schillinger System does not output audio data; instead it generates symbolic data constituting pitch and duration information in the form of LISP 1

Even within this apparently well-defined problem space, the use of objective functions to guide musical quality is highly questionable, given that the exemplars of four-part chorale writing routinely break the rules of harmony that have supposedly been derived from them [Radicioni and Esposito 2006].

§4.3 Automated Schillinger System Output

69

data structures (discussed briefly in section 3.1.2). This has two implications: firstly, a process must take place in order to convert the symbolic data into audio, and secondly, such a process will necessarily add information pertaining to musical dimensions other than pitch and duration. The simplest solution is to map the pitch and duration information to raw MIDI output, using default values for the other musical dimensions (primarily tempo, timbre and note velocity). This method was used during development because it allowed instant feedback; the provision of audio and MIDI interfaces is one of the advantages of writing Scheme in Impromptu. The plain pitch and duration data is sufficient for this chapter’s genre classification experiment, however the audio generated for instant feedback is only adequate for verifying the correctness of the program. In order to assess the musical merit of pitch and duration data, this data needs to be heard in the context of a fully embodied parameter set in order to avoid biasing or distracting the listener by the lack of variation in the dimensions which aren’t controlled by the system. This is especially important when listeners undertaking the evaluation are musical experts with limited or no experience in computer-aided composition. This issue has been identified by several authors working in the field of automated musical performance [Widmer and Goebl 2004; Arcos et al. 1998]. Kirke et al. provided a comprehensive survey of the approaches taken towards simulating the human performance of musical data sets [Kirke and Miranda 2009]. The goal of this field of research is to extend the realm of computer generated parameters to the total symbolic parameter space of music, which would ultimately enable software to give expressive renditions of computer-generated compositions instead of just ’robotic’ ones. In particular, it focuses on the context-sensitive prediction of tempo and note velocity information. The computational approaches include ’expert’ non-learning performance systems, regression methods, neural networks, case-based reasoning systems, statistical graphical models, and evolutionary models. Although automated expressive performance is clearly beyond the scope of this thesis, it is still necessary for the music to be presented to a human audience in the form of expressive performances. Such an approach using human performers has been used extensively by Cope, for similar reasons related to bias as listed above [da Silva 2003]. In this case however, to avoid the inconvenience of obtaining professional performances from multiple instrumentalists, a high quality digital sound library has been used to provide the timbres for a series of performances recorded by the author using sequencing software. These sequences are subsequently rendered to audio. Figure 4.1 gives a visualisation of the entire process, which incorporates the open-source musical engraving software LilyPond to produce the intermediate output of standard musical notation. (LilyPond is also used to generate the MIDI files to be used for genre classification.) The reader, should they wish to briefly become listener, is directed to the audio samples on the CD accompanying the hard copy of this document. The samples are also available online.2 2

To access the MP3 files online, follow the hyper-links in the electronic copy of this document contained in table 4.3, located in section 4.5.1.

Results and Evaluation

70

MIDI Files

Schillinger System

Classifier

Sound Library

LilyPond

PDF

Author's Performance

Sequencer

Audio

Human Audience

Figure 4.1: Conversion process from list representation to audio

4.4

Assessing Stylistic Diversity

Both Schillinger and the editors of his published volumes make various claims to the effect that in its capacity as a formalism designed for human composers, the essence of the Schillinger system is independent of any overbearing stylistic framework. The foreword by Henry Cowell, a distinguished composer and contemporary of Schillinger [Quist 2002], suggests that Schillinger’s system is capable of generating music in any style [Schillinger 1978]. The reasoning behind these views is that rather than encoding explicit style-specific musical knowledge like many other music theory treatises, the Schillinger System encodes implicit musical knowledge in the form of procedures which, for the most part, can be expressed mathematically (see chapter 3). Given that the procedures have been adapted and implemented in the form of a computer system, the notions of style and diversity must be investigated; not simply to assess the credibility of the claims (it is not the express purpose of this section of the thesis to either validate or debunk them), but more importantly to determine whether or not the automated system could actually be used for generating material in a variety of musical contexts. It is for this reason that the active research field of genre classification has been employed. The goal of using a classifier is two-fold: to find out which musical categories are assigned to the output of the automated Schillinger System, and to find out whether the output contains a notable degree of statistical diversity — something that would manifest as the frequent assignment of several different genres. If the classifier were to give statistically significant results, then it would be meaningful to compare them to the assertions regarding style and diversity collected from participants in the listening survey (see section 4.5.4). Section 4.4.1 will give an overview of the field of automatic genre/style classification. This will serve as justification for the choice of software used to perform the experiment outlined in sections 4.4.3, 4.4.4 and 4.4.5. The results will be presented and discussed in section 4.4.6.

§4.4 Assessing Stylistic Diversity

4.4.1

71

Overview of Automated Genre Classification

Automatically classifying musical genre or style by examining an file’s audio or symbolic (usually MIDI format) musical content has applications primarily in musical information retrieval and cognitive science. In the former case, the goal is to automate the human task of assigning genres to tracks in musical databases to facilitate searching, browsing and recommendation. In the latter, the goal is to discover the processes behind the human cognition of musical style, and often to try and determine how composer styles are manifested statistically or structurally. The computational approaches for each discipline have tended to be slightly different in the literature. MIR research focuses predominantly on statistical feature extraction and standard machine learning techniques. Style cognition research has a longer history, and has seen emphasis on grammatical and probabilistic models in additional to statistical feature extraction. Scaringella et al. [Scaringella et al. 2006] provide a comprehensive survey of automatic genre classification, pointing out that it is an extremely non-trivial problem not only for technical reasons, but also due to many endemic problems with genre definitions themselves. One of these problems is the lack of a consistent semantic basis: labelling can derive from geographical origins (Latin), historical periods (Classical), instrumentation (Orchestral), composition techniques (Musique Concr´ete), subcultures (Jazz), or from terms which are coined arbitrarily in the media or by artists (Dubstep). Issues of scalability arise whenever new genres emerge from combinations of old ones. Pachet and Cazaly noted the utter lack of consensus on genre taxonomies among researchers and popular musical databases [Pachet and Cazaly 2000]. These problems cannot be ignored when designing classifiers. Scaringella argues that attempting to derive genre from audio requires the assumption that it is as much an intrinsic attribute of a title as tempo, which is “definitely questionable” [Scaringella et al. 2006]. Dannenberg et al. commented that higher-level musical intent appears “chaotic and unstructured” when viewed as low-level data streams [Dannenberg et al. 1997]. On the other hand, one particular study seems to provide good motivation for this line of research: Gjerdingen and Perrott found that humans with variable musical backgrounds were able to correctly categorise musical snippets of only 250ms in 53 percent of cases, and snippets of 3 seconds in 72 percent of cases [Gjerdingen and Perrott 2008]. This result is convincing evidence that even untrained humans have an innate ability to recognise style from a small amount of data, which implies that the data must contain some measurable characteristics which make that possible. Therefore, in MIR the importance to date has been on the extraction of meaningful statistical features from short frames of audio data. Statistical features extracted from audio fall into the broad categories of temporal, spectral, perceptual and energy content [Scaringella et al. 2006]. The precise feature extraction algorithms are numerous and need not be discussed here. Feature patterns are used to train models based on unsupervised clustering algorithms or supervised learning algorithms. In both cases the resulting model of pattern separation is used as the basis for the classification of new patterns extracted from unlabelled pieces of mu-

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Results and Evaluation

sic. Various authors have reported success with an array of different algorithms and feature sets, for both audio and symbolic data [Scaringella et al. 2006]. The advantage of symbolic data is that reliably discerning musical statistics such as pitch and chord relationships is easily accomplished; a disadvantage is the shortage of important spectral information. Chai and Vercoe classified symbolic encodings of monophonic folk melodies as being Irish, German or Austrian using Hidden Markov Models, with an accuracy approaching 80 percent [Chai and Vercoe 2001]. The classification of symbolically encoded folk songs was also addressed by Bod, using probabilistic grammars to achieve 85 percent accuracy [Bod 2001]. Shan and Kuo trained a genre classifier using both MIDI harmonies and melodies [Shan and Kuo 2003]; they used a method combining a priori pattern finding with heuristics, which achieved an accuracy of 84 percent using just melodic features. Keirnan used self-organising maps to successfully partition audio into three classes representing the composers Friederick, Quantz and Bach [Kiernan 2000]. Ruppin et al. [Ruppin and Yeshurun 2006] used the K-nearest neighbour algorithm to classify MIDI files as either Classical, Pop or Classical Japanese, with 85 percent accuracy. Kosina used K-nearest-neighbours to classify audio as Metal, Dance or Classical with 88 percent accuracy [Kosina 2002]. Xu et al. distinguished between Pop, Classical, Jazz and Rock audio using support-vector machines, with 96 percent accuracy [Xu et al. 2003]. Among the most comprehensive and successful work in MIR to date is that by McKay, who used a learning ensemble consisting of neural network and K-nearest-neighbour classifiers trained on MIDI files using 111 features and audio using 26 features, each weighted by sensitivity using a genetic algorithm. This system achieved a 9-genre classification accuracy of 98 percent [McKay 2010]. The majority of authors agree that improvement can be made by increasing the sophistication of the feature sets, but evidently there is still no widely accepted algorithm for making even extremely broad classifications. Some authors have deduced that the relatively small size of the datasets may be to blame — both McKay and Ponce ´ et al. have concluded that song databases much larger than those currently de Leon in use are the key to assessing the real worth of particular combinations of feature sets ´ et al. 2004]. McKay also advoand learning algorithms [McKay 2010; Ponce de Leon cates the training of classifiers on both audio and symbolic features simultaneously. This requires perfect MIDI transcriptions of audio files, a rare commodity that will continue to rely on highly skilled human labour until significant advances are made in the field of automated polyphonic transcription [McKay 2010]. The recent release of a million-song feature-set for public use [Bertin-Mahieux et al. 2011] is likely to instigate the next generation of MIR research and a significant raising of the bar in the near future. In the meantime, it must be stressed that the assignment of genre labels to the automated Schillinger System’s output will be flawed to an extent; the purpose of the experiment is simply to determine whether the output’s statistical characteristics point more towards certain styles than others, and whether the output contains a notable degree of diversity.

§4.4 Assessing Stylistic Diversity

4.4.2

73

Choice of Software

As described in section 4.3, the output of the automated Schillinger System requires conversion to audio for the human participants in the listening survey; however, only MIDI files were able to be used for the purpose of automated classification. The main reason for this is that the method for encoding audio from symbolic musical data in figure 4.1 is time-consuming, and it was desirable to classify a large number of compositions in order to obtain statistically significant results. The use of MIDI files meant that symbolic classification software was required. Classification software designed specifically for MIR research is currently difficult to come by. Fortunately McKay has developed a suite for precisely this purpose called jMIR [McKay 2010] which may be used for both symbolic and audio files, and a predecessor called Bodhidharma [McKay 2004] which was designed specifically for working with MIDI files and is equivalent to using jMIR in symbolic mode. Bodhidharma was responsible for the winning entry at the 2005 MIREX music classification conference [Mckay and Fujinaga 2005]. It extracts up to 111 selectable features, uses a hierarchical taxonomy of 9 root genres and 38 leaf genres, and uses a learning ensemble consisting of artificial neural network and K-nearest-neighbour classifiers [McKay 2004]. Furthermore, it is accompanied by a sizable training set of 950 MIDI files (referred to henceforth as the Bodhidharma set) intended for use with the hierarchical taxonomy. It is therefore arguably the best possible means for performing a classification experiment on MIDI data currently publicly available. Other options for MIDI feature extraction and analysis such as Humdrum [Huron 2002] and The MIDI Toolbox [Eerola and Toiviainen 2004] were examined, but proved to be not as comprehensive as Bodhidharma.

4.4.3

Classification Experiment

The goals of this experiment can be summarised as follows: 1. To find out which genres are automatically assigned to the Schillinger output; 2. To see if those assignments are significantly different for the outputs of the harmonic and melodic modules; 3. To test the hypothesis that the output from the Schillinger system is stylistically diverse. The method used is outlined below: 1. Automatically generate sets of melodies and harmonies using the automated Schillinger System; 2. Establish appropriate configurations for training a classifier for each set; 3. Train separate classifiers on the Bodhidharma set using the two configurations;

Results and Evaluation

74

4. Present the Schillinger sets to their respective classifiers to obtain genre labels; 5. Analyse the distribution of genre labels to satisfy each goal above.

4.4.4

Preparation of MIDI files

The current version of the automated Schillinger System is effectively engineered as a ‘push-button’ solution consisting of separate modules for generating harmonies and melodies. The combinations of parameters controlling these modules, specified by the author, have been listed in section 3.6.3. The melodic module accepts as input a vector of high-level axis types (see section 3.6.2). One-hundred MIDI melodies were generated using the input (u u b b) — that is, a sequence of two ‘unbalancing’ axes followed by two ‘balancing’ axes. This set will be referred to as the 100M set. The harmonic module is fully automated. Onehundred harmonies were generated which will be referred to as the 100H set.3 Harmonies were encoded as MIDI files using one voice per track, in order to improve the performance of the feature extractor for pitch-class and textural features [McKay 2010]. Ideally, to properly control the experiment the Bodhidharma set should be modified to be in exactly the same format as the system’s output. This would mean creating one Bodhidharma set with all non-melodic tracks removed and another with all nonharmonic tracks removed, with appropriate rhythmic quantization applied to all note events. There are both practical and technical reasons why this could not be done in the required time-frame: • Distinguishing between melodic and harmonic tracks is very problematic in some genres, despite being simple in others (those with lead vocals, for instance); • Melodic content contributes to harmonic content, and the music’s functional context can easily change when the melody is absent; • The two issues above mean that automating such a process could not give reliable results without implementing a complex set of algorithms for musical analysis. Such an implementation would be inordinately time-consuming within the scope of the thesis, as would the manual modifications otherwise required; • With so much information missing, the classifier’s ability to train successfully on the Bodhidharma set may end up being too weak. If this were the case then it might give credence to the notion that statistically similar harmonies or melodies can be adapted to multiple genres, but it could just as easily lead to meaningless classification results. 3 The first constraint in table 3.6 restricts harmonies to between 2 and 7 voices. This is deliberate, because anything thicker than 7 voices causes the nearest-tone voice leading algorithm to have an unreasonable execution time due to its computational complexity and the fact that Impromptu is an interpreter. See section 3.3.3.

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75

Thus, a potentially less-than-ideal situation was settled upon to ensure the experiment was at least feasible.

4.4.5

Classifier Configuration

Bodhidharma’s strength as a MIR utility lies in the carefully designed set of 111 statistical features that are extracted from the MIDI data. These features are split into groups pertaining to instrumentation, texture, rhythm, dynamics, pitch, melody and chords. The complete list can be found in [McKay 2004]. In order to focus as closely as possible on the parameters controlled by the automated Schillinger System, the classifiers for the 100H and 100M sets were trained with certain features switched off for practical considerations, as shown below in table 4.1. For instance, it would not make sense to include the instrumentation features in the training patterns when every single member of the 100H and 100M sets uses the single default instrument of grand piano. McKay and Fujinaga pointed out that instrumentation features can have strong classification ability on their own [Mckay and Fujinaga 2005], therefore it is necessary to remove the possibility of all 200 samples being assigned a genre which is strongly defined by the presence of grand piano. Similar reasoning lies behind the ignoring of features relating to dynamics and, in the case of the block harmonies of the 100H set, rhythm.4 Table 4.1: Classification Experiments Using a 38-leaf Hierarchical Taxonomy

Feature Types Instrumentation Texture Rhythm Dynamics Pitch Statistics Melody Chords Root success rate Leaf success rate

Default

100M

100H

on on on on on on on 84.7 58.3

off on on off on on on 67.0 43.9

off on off off on on on 80.0 57.0

Table 4.2, found below, lists the parameter settings which have the most impact on the execution time and the classification accuracy for the training set. As Bodhidharma is flexible enough to allow training sessions which may run for impractical amounts of time (in the order of several CPU-weeks), it was necessary to make several compromises. The final configuration was slightly more liberal than one used by McKay which was deemed successful in [McKay 2004]. Using this configuration, the various combinations of extracted features lead to the root and leaf classification success rates on the training set found above in table 4.1. It should be noted that using 4

In fact, Bodhidharma contains a bug that causes division by zero during the extraction of certain rhythmic features from MIDI sequences in which note events are perfectly quantized and regularly spaced — so the decision was further enforced by circumstance.

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Results and Evaluation

a hierarchical taxonomy tends to hinder the assignment of correct root categories — when trained with Bodhidharma’s regular flat taxonomy, root success rates are generally above 95 percent [McKay 2004]. The leaf success rates for the training set, while not spectacular, are still impressive compared to the expected success rate of 2.63 percent for pure chance (that is, the random assignment of leaf genres), and hence should be adequate for gaining an insight into the characteristics of the 100M and 100H sets. Table 4.2: Bodhidharma configuration

Preference Training/test split Cross validation Weight multi-dimensional features Flat classifier ensemble Hierarchical classifier ensemble Round robin ensembles Max GA generations Max change in GA error before abort Max NN epochs Max change in NN error before abort Certainty threshold

4.4.6

Setting 80:20 NO YES YES YES NO 105 10e-5 2000 10e-7 0.25

Classification Results

The classifier was trained on the Bodhidharma set. The resultant training time for the configuration described in 4.4.5 was roughly 300 minutes. The 100M and 100H sets were then fed to the classifier to obtain genre labels. The assignment of genres for the two sets is presented in figures 4.2 and 4.3. In the case where multiple outputs of the neural network fired above the certainty threshold, multiple genres were assigned. This provision is widely considered to be representative of how genres are assigned by humans [Scaringella et al. 2006; McKay 2010], and is the reason for the relative genre assignments in the graphs summing to more than 100 percent. In figures 4.2 and 4.3, clustering is apparent in the broader genres of Jazz, Rhythm and Blues, and Western Classical. Many genres have not been assigned at all. There is also a significant difference between the assignment of harmonies and melodies. 100M was classified as 67 percent Jazz, 16 percent Rhythm and Blues and 82 percent Western Classical. Conversely, a convincing 100 percent of the 100H set is deemed to be Western Classical with only 4 percent being assigned Jazz. These figures are apparently strong evidence that the output of the automated Schillinger System does in fact have salient statistical properties which are suggestive of particular styles, and that the melodic module has more diverse output than the harmonic module. These results will be discussed further in section 4.6, in the context of the data from the listening survey.

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§4.4 Assessing Stylistic Diversity 77

100% Leaf Classifier Results

100H 100M

80%

70%

60%

50%

40%

30%

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10%

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Figure 4.2: Leaf genres assigned to 200 samples from a 38-leaf hierarchical taxonomy

100% Root Classifier Results

100H 100M

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Figure 4.3: Root genres assigned to 200 samples from a 38-leaf hierarchical taxonomy

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4.5

Results and Evaluation

Assessing Musical Merit

Section 4.2 mentioned the inadequacy of informal assessments of computer-generated compositions. As the automated Schillinger System does not make any attempt to imitate a particular style, there is no objective function of any complexity that will be able to give an indication of the inherent quality of the compositions. Hence, there is a motivation to evaluate the system using a group of expert listeners, using a more rigorous and repeatable method than is typically found in the academic literature. The aim of the experiment is to gather and process subjective data as objectively as possible to correctly identify consensus or variation in collective opinion. The following sections will outline the details of a listening experiment that has provided strong indications about the intrinsic musical merit of the material generated by the automated Schillinger System. Sections 4.5.1 and 4.5.2 will discuss the survey and the audio samples, and justify the decisions that went into their preparation. Section 4.5.3 will present the quantitative results from the sections of the survey involving Likert scales, and section 4.5.4 will present the qualitative results obtained from participants’ written responses using a process of analysis borrowed from the field of Grounded Theory.

4.5.1

Listening Survey Design

Unlike the situation in section 4.4.3 in which a batch of 200 output samples could be presented to a classifier, only a handful of samples can be presented to an audience. A group of three melodies and three harmonies was split evenly between the main system modules of harmony and melody. In order for this group of six samples to be properly representative of the range of output from the automated Schillinger System, a selection process was necessary, because it is possible for the system to produce a string of pieces utilising collections of parameters that effectively form clusters in terms of their resulting interval distributions or rhythmic content. In the literature, Holtzman acknowledged the necessity of selecting from the output in this way. “Ultimately, a composer must choose which generated utterances to use, how to interpret the data generated by the machine, and so on. The composer may be seen as a selector” [Holtzman 1981]. Cope selected exemplars from his system’s output to constitute a final representative collection [Cope 2005]. The decision to render the selected output as audio performances, to provide the listeners with a complete musical context, was informed by several authors who have faced the same decision. Hiller commented that “performance is, without a doubt, the best test of the results” [Hiller 1981]. Gartland-Jones followed a similar philosophy in a festival installation of an interactive GA system, where output was performed on guitar: “Recording the output on a real instrument enabled the perceived musicality of the fragments to be brought out, and provides additional musical dimensions” [Gartland-Jones 2002]. DuBois is of the same view – that the product of his L-System is an intermediate state requiring joint interpretation by the composer and performer to render it fit for consumption [DuBois 2003].

§4.5 Assessing Musical Merit

Title Harmony #1 Harmony #2 Harmony #3 Melody #1 Melody #2 Melody #3

Instrumentation Rhodes piano Orchestra Grand piano Clarinet Grand piano Violin

View (Appendix A) A.1 A.2 A.3 A.4 A.5 A.6

79

Media URL Listen Listen Listen Listen Listen Listen

Table 4.3: Output samples used in the listening survey

It should be noted that these issues are not relevant for all computer music systems. This includes those with output that is physically impossible to perform and those which are interactive during performance [Blackwell 2007; Biles 2007]. The method for generating performances from the system’s output for the listening survey was described earlier in section 4.3. To prevent the listeners from becoming bored of and potentially biased against the timbre of a single instrument, a variety of instruments was used. Table 4.3 lists the instruments used for each sample. These titles correspond with tracks 1–6 on the CD accompanying this thesis. The table also contains hyper-links for listening to the audio files online. The survey was designed in consultation with Jim Cotter, a senior lecturer in composition at the Australian National University (ANU). The survey preamble encourages participants to provide entirely subjective opinions, and to judge musical merit against their own musical experiences instead of attempting to compare the samples to other computer-aided composition software. For each audio sample, listeners were asked register their opinion of four different aspects of the music on a Likert scale, as well as to provide written opinions on what intrigued or bored them. Likert scales are widely used in many fields of research within the humanities; they are used to rank opinion strength and valence as shown in figure 4.4. Their symmetry allows for a respondent to express impartiality. Five labels were used, with four extra nodes interspersed so that participants would feel free to register opinions between the labels. -4 -3 -2 -1 0 +1 +2 +3 +4 O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Very negative

Negative

Neutral

Positive

Very Positive

Figure 4.4: Likert scale example

The Likert scales for each audio sample represented the dimensions gut reaction, interestingness, logic and predictability. The final page of the survey registered two further dimensions — diversity and uniqueness. Each term may be mostly self-explanatory, however they were deliberately not defined or clarified for the participants prior to the commencement of the experiment. Instead, it was intended for them to decide for themselves precisely what to listen for, rather than add the distraction of trying to reconcile worded definitions with what they were hearing. Explanations of the dimensions encompassed by the survey are itemised below.

80

Results and Evaluation

• Peoples’ gut reactions were recorded so that a measure could be obtained of whether the group actually enjoyed what they were listening to on a fundamentally aesthetic level. This question was placed at the top of each page to increase the likelihood of it being answered first in a spontaneous way. This kind of measure is obviously important if the point of a composition system is to produce music that people like. • Interestingness is, broadly speaking, a measure of how well the music holds peoples’ attention, and as far as composition as an art-form is concerned, a measure of success. Miranda concluded that while computers can compose music, rulebased systems seldom produce interesting music [Miranda 2001]. Given that the automated Schillinger System is rule-based, it is clearly important to find out if it can produce interesting music or not. • Logic was chosen as a subjective measure because several authors or their audiences have commented on the fact that despite computer compositions being ‘pleasing’ or ‘acceptable’, they are often criticised for lacking logical progression, development or higher-level structure [Pereira et al. 1997; Mozer 1994]. Although logic in terms of musical structural coherence can, to some extent, be measured quantitatively by searching for multilevel self-similarity in the manner of Lerdahl and Jackendoff [Lerdahl and Jackendoff 1983], it is still an important element to test subjectively because it has more than one possible interpretation. • Predictability was used to roughly measure the ‘surprise’ factor (or lack thereof) which can either contribute to or detract from the other three elements. It is conceived as a subjective measure of information content, thus bearing some relation to work by Cohen [Cohen 1962] and Pinkerton [Pinkerton 1956]; and also to Schillinger’s notion of the ‘psychological dial’ which has occasionally been referred to by film composers [Degazio 1988]. The neutral position on the Likert scale indicates a balance between predictable and unpredictable musical events in the minds of the listeners. It was expected that each listener’s ideal balance would lie at this position even if their respective tastes for unpredictability differed wildly. For this reason the extreme points of the scale were labelled ‘too predictable’ and ‘too unpredictable’ so that the relationship to musical merit could be more easily inferred. • Diversity was intended to collect data to compare to the results of the automatic classification system, and aid in interrogating the notion that Schillinger’s system is somehow neutral in a stylistic sense. It also helped in assessing how the system’s output might apply to different musical contexts in practice. • Uniqueness was intended to gauge how different the music was to that which the audience had heard in the past. This question was included in order to add perspective to the interpretation of the other answers. For instance, if the

§4.5 Assessing Musical Merit

81

group were to claim that they had essentially ‘heard it all before’, this might add credibility to positive or negative consensus in other questions. • The survey’s final question was whether, as composers, the participants could imagine using the system themselves to generate raw musical material. The answers to this question may indicate whether a more advanced interactive version of the system would be adopted for experimentation if it were made available to the wider composition community. The complete survey has been included in this document in Appendix B for reference.

4.5.2

Listening Experiment

A total of 28 survey participants ranging from first-year undergraduates to postgraduates and lecturers were recruited from the composition department at the ANU School of Music. Composers in particular were chosen because they are trained to possess a strong ear for multiple levels of musical structure, they tend to have an extremely diverse range of musical tastes and listening experiences, and they may also be able to perceive the construction of the samples in terms of their knowledge of compositional techniques. The survey procedure was approved by the ANU’s Human Research Ethics Committee.5 Undergraduates were requested to note their composition enrollment level (how far through their major they were). This field was marked ‘N/A’ by post-graduates. Each audio sample was played twice over loudspeakers while participants filled in the survey questions. Each first playing was followed by a 30-second pause and each second playing by a 60-second pause. Participants were then given time to fill in the section of general opinions regarding the group of compositions as a whole.

4.5.3

Quantitative Analysis and Results

This section describes box plot summaries of the data collected from the Likert scales for the six samples, found below in figure 4.5. The plots represent the dimensions gut reaction, interestingness, logic and predictability for the three harmony samples (H1H3) and the three melody samples (M1–M3). Boxes represent interquartile ranges (the ‘middle 50 percent’ of opinion), diamonds indicate arithmetic means, red bars indicate medians, and ‘whiskers’ (the dashed lines) indicate extremes of opinion. There are no outliers. Given that each scale contains five labels with extra nodes in between them, the range for each dimension is [−4, 4]. No participant marked in-between any of the nine nodes, so only integers were recorded. Two of the participants wrote comments on the general opinions page instead of answering the Likert scales. These answers were transferred verbatim into the text fields on that page to ensure any qualitative data was not lost, and the opinions were converted into reasonable estimates on the Likert scale of what these persons were thinking. 5

Ethics protocol no. 2008/237

Results and Evaluation

82

Gut Reaction H1 H2

(a)

H3 M1 M2 M3 −4

−3

−2

−1

0

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

Interestingness

(b)

H1 H2 H3 M1 M2 M3 −4

−3

−2

−1

0 Logic

(c)

H1 H2 H3 M1 M2 M3 −4

−3

−2

−1

0 Predictability

H1

(d)

H2 H3 M1 M2 M3 −4

−3

−2

−1

0

Figure 4.5: Box-plots representing individual samples

The gut reaction mean results in figure 4.5(a) range from exactly neutral for sample H2 to 1.43 for sample M2, which is tending towards the value of ‘like’ on the Likert scale. For all samples except H2, the interquartile box lies on the positive side of neutral. H2 appears to have polarised the audience the most, with the mean, median and interquartile box lying exactly on or centered around zero. The overall response for interestingness, shown in 4.5(b), was unequivocally positive, with all means lying on

§4.5 Assessing Musical Merit

83

or above 1 and almost all of the interquartile data being above zero. The noticeably smaller interquartile boxes indicate a greater consensus of opinion. In figure 4.5(c), the unanimous perception of logic within M2 is striking. There is a greater range of means between samples (-0.86 to 2.14) and less consensus on each individual sample, indicated by most of the interquartile boxes being wider. In figure 4.5(d), the interquartile boxes for predictability are also generally wider, although the general perception is closer to neutral (a good balance between predictability and unpredictability). Samples H1 and in particular, H2, were perceived unanimously as too unpredictable. It is notable that samples H3 and M2, which have the highest means for gut reaction and logic, also have the two lowest means for predictability (suggesting they were the most predictable). Sample H2, which was the least liked according to its gut reaction, was also considered the most interesting (by a slight margin), the least logical and the most unpredictable. The figure 4.5 plots suggest that overall, people enjoyed what they heard, and found it somewhat interesting and logical; but that each individual sample certainly polarised the audience to a degree, as indicated by the width of the interquartile boxes and the extent of the whiskers. The opinions of logic and predictability also appear to have differed significantly between samples, compared to the measures of gut reaction and interestingness. Sample Aggregates Gut Reaction

(a)

Interestingness Logic Predictability −4

−3

−2

−1

0

1

2

3

4

2

3

4

General Opinions Diversity Interestingness

(b)

Logic Predictability Uniqueness −4

−3

−2

−1

0

1

Figure 4.6: Box-plots representing overall opinion

The box plots in figure 4.6 give further promising indications of the intrinsic merit of the samples. Plot 4.6(a) was calculated by aggregating the data across all six samples for each dimension; hence, it shows overall an extreme range of opinion, but it also shows that the average opinions on gut reaction, interestingness and logic were positive and predictability was close to ideal. Plot 4.6(b) represents the final page of

Results and Evaluation

84

the survey which collected participants’ overall opinions of the set of samples after listening was concluded. Once again, there is the suggestion of an overall positive reaction for the measures which were used for each sample. It is interesting to note the strong correspondence between figures 4.6(a) and 4.6(b) for interestingness, logic and predictability. This indicates that opinions changed very little on average between the listening phase and the final page of the survey. The opinion of diversity is positive, which is supportive of the idea that the automated Schillinger System may at least be useful in a variety of stylistic contexts. The only strongly negative measure is that of uniqueness, which is an assertion that the audience did not encounter anything especially unfamiliar. Table 4.4: Kruskal-Wallis variance measure p for each dimension across all 6 samples

Dimension Gut Reaction Interest Logic Predictability

Mean 0.84 1.26 0.71 0.28

Median 1 2 1 0

Std. Dev. 1.71 1.55 1.94 1.76

p 0.0125 0.9605 > > > > > > > > >





s = (2 1 2 2 1 2) p = 5 t = 4 axis1 = (1 rhythm(t, 2) 2 (-1 false 0)) axis2 = (2 rhythm(t, 1) 1 (-1 false 0)) axis3 = (3 rhythm(t, 1) 1 (-1 false 0)) axes = (axis1 axis2 axis3) M = superimpose(s, C4, p, t, axes) C = buildParams(axes, 8) pdf(buildMelody(s, M, C))

    

9

  







  

 

     

      

 

 

    

    

Figure 5.1: The potential functioning of a terminal-based interactive Schillinger System

inconsistency lamented by [Barbour 1946], is enticing. As far as the author has been able to ascertain, no publication exists to serve this purpose. This resource would be particularly valuable to composers interested in Schillinger’s theories, as well as other developers of composition algorithms who might wish to program their own models of Schillinger’s procedures. There is ongoing activity within the Schillinger Society1 with the aim of encouraging a wider exploration and adoption of Schillinger’s work. This has been bolstered in recent years by online courses dedicated to the teaching of Schillinger’s methods.2 Moreover, the recent release of Mc Clanahan’s four-part harmonisation program based on Schillinger’s Special Theory of Harmony and further activity on the Schillinger CHI Project website3 seem to indicate a recent surge of enthusiasm around possible computer implementations of the Schillinger System. Future development of the work presented in this thesis could form a significant contribution to this movement.

1 2 3

www.schillingersociety.com See http://www.schillingersociety.com/moodle/ and http://www.ssm.uk.net/index.php http://schillinger.destinymanifestation.com/

Appendix A

Samples of Output

The system’s output, subsequent to being processed by LilyPond, consists of MIDI files and the corresponding musical notation in PDF format. This section contains the six example pieces used for the listening survey. Table 4.3 lists the instrumentation that was used to render each performance, and includes hyper-links for listening online.

A.1

Harmony #1

% $"! "! &$ ! "! A.2

11

! #! ! !

!!

!!

! "!

"! ! "! #!

"! !

"! ! ! !

" !!

! !

!!

" !! " !! ! ! ! !

" !!

! !

#! !

#! ! "! "!

Harmony #2

& $ !!

! !

! "!

" !!

# !!

" !!

!!

!!

" !!

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"! ! "!

!! "!

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! !

# " !! "!

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"!

# !!

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!!

" !!

" !!

" !!

& ! ! % !

" " !! !

"! "!

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" " !! !

" " !! !

&

!

99

"! ! "! "! "! !

"!

" !!

!!

" !!

! "! #!

!!

" !!

#!

!

Samples of Output

100

A.3

Harmony #3

% $ " !! "! #$ ! 11

21

% " !! # ! !

" !! ! !

% ! ! # "! !

! ! "! !

A.4

!! !!

!! !!

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!! !!

!! !!

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" " !! "! "!

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!

!

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!

!

!

!

!

!

!! "" !

" !! "!

" !! !

" !! "!

" !! " !

" !! !

" !!!

" !!!

Melody #1

"! "! ! ! ! "! & ! "! ! " & " ! ! ! $ " ! # 86 ! ! ! ! % ! ! ! ! " ! $ ! ! ! !% ! " ! ! ! $ ! % "! ! ! ! "! 8 & & & & !$ ! ! ! ! ! ! ! ! $ ! ! $ ! ! " ! ! ! ! ! '! ! ! !! # ! ! ! ! 15

& "! "! ! # ! ! ! ! ! ! ! ! ! " !% ! ! ! ! ! " ! "!$

& ! ! !$

! ! " !& ! %

§A.5 Melody #2

A.5

101

Melody #2

!" $ % !# " !# " # # # !#

# !# # # # !# # # # # "

!# # !# # " "

!# # # # # # !# $ " !# # # # !# # 16 !# " # # # !# # !# !" &# $ # !" " !# # & ! # # & &" # # # & & # " 23 &# " # !# !# " # !# # &# '# !# !" !# !# # " $ !" !# # !" # !# # "

9

A.6

# !# !# # !" # " !# # #

Melody #3

% & " # # # "# # # "# ! $ # " ! # # # " ! " "# " # ! # # # # # " # " # # $ # # $# 9

17

25

% # "! '# # "# # "# "# # $# # # # "! % " # "# !

"! # "# ! "!

# " ! # ! " # ' # "! # # " # ' # $ # # $ # # "#

# % # # "# # # " # " # $# "# !

"# " ! # # $ # !

33

% "# ! "# $! # "# "# # "# '# # "# !

41

% "# # "# '# # # "# "# "# $# $# # # # "# # #

48

56

# "# '# $# # $! #

# # $# # # # "# # "#

"# ! "# # # "# # $# # "# #

"# ! # "! $! # "! # "# '# "# # !

# # # "# "# # # $# #

% # "# " # # $# # " # "# " ! $# # "# " # ' ! # " ! # # "# " ! " # ! $ # $ # # " # " # % "# '# # "# # $# "!

"! "# # $! "!

"# # $# # $# ! # # "# !

"# "! #

102

Samples of Output

Appendix B

Listening Survey

The survey document that was used by participants is included for reference.

103

Listening Survey You are being asked to evaluate six samples of the output of a computer-automated composition system. Answer on the basis of what you feel to be the intrinsic musical merit of each individual piece from your expert musical experience. The goal is not to compare the examples with each other, to a human composer, or to any other composition software that you may be familiar with. Your evaluation should draw on your appreciation of music and the art of composition. Each sample will be played twice. The samples consist of three homophonic harmonies and three monophonic melodies. For each sample you will be asked to register four opinions: your gut reaction, your evaluation of its interestingness, your evaluation of its overall musical logic, and your evaluation of how predictable it was. There is also a general section at the end of the survey with several more questions relating to the group of pieces as a whole. Ideally your answers should be carefully considered subjective opinions. You are not expected to analyse any of the samples in terms of music theory. Indicate your answers by marking in the appropriate circle on each scale, for example: O––––––o––––––✓––––––o––––––O––––––o––––––O––––––o––––––O Really dislike

Dislike

Neutral

Like

Really like

Please consider writing free-form answers to questions in the spaces provided. These can be as long or as short as you like, containing prose, keywords, etc – I want to know exactly what you are thinking. You are allowed to leave individual answers blank if you wish, and you are free to opt out of this experiment completely if you are uncomfortable with any aspect of it. Optional: please indicate which COMP Level (1-6) you are presently studying: ______ (Write 'N/A' if this does not apply to you)

Matt Rankin 29/03/12

Sample #1: “Harmony #1” Gut reaction: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Really dislike

Dislike

Neutral

Like

Really like

Harmonic Interest: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Uninteresting

Not Very Interesting

Neutral

Fairly Interesting

Very Interesting

Harmonic Logic: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Illogical

Not Very Logical

Neutral

Fairly Logical

Very Logical

Predictability: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Too Predictable

Fairly Predictable

What aspects intrigued you, if any:

What aspects bored you, if any:

Balanced

Fairly Unpredictable

Too Unpredictable

Sample #2: “Harmony #2” Gut reaction: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Really dislike

Dislike

Neutral

Like

Really like

Harmonic Interest: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Uninteresting

Not Very Interesting

Neutral

Fairly Interesting

Very Interesting

Harmonic Logic: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Illogical

Not Very Logical

Neutral

Fairly Logical

Very Logical

Predictability: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Too Predictable

Fairly Predictable

What aspects intrigued you, if any:

What aspects bored you, if any:

Balanced

Fairly Unpredictable

Too Unpredictable

Sample #3: “Harmony #3” Gut reaction: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Really dislike

Dislike

Neutral

Like

Really like

Harmonic Interest: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Uninteresting

Not Very Interesting

Neutral

Fairly Interesting

Very Interesting

Harmonic Logic: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Illogical

Not Very Logical

Neutral

Fairly Logical

Very Logical

Predictability: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Too Predictable

Fairly Predictable

What aspects intrigued you, if any:

What aspects bored you, if any:

Balanced

Fairly Unpredictable

Too Unpredictable

Sample #4: “Melody #1” Gut reaction: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Really dislike

Dislike

Neutral

Like

Really like

Melodic Interest: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Uninteresting

Not Very Interesting

Neutral

Fairly Interesting

Very Interesting

Melodic Logic: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Illogical

Not Very Logical

Neutral

Fairly Logical

Very Logical

Predictability: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Too Predictable

Fairly Predictable

What aspects intrigued you, if any:

What aspects bored you, if any:

Balanced

Fairly Unpredictable

Too Unpredictable

Sample #5: “Melody #2” Gut reaction: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Really dislike

Dislike

Neutral

Like

Really like

Melodic Interest: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Uninteresting

Not Very Interesting

Neutral

Fairly Interesting

Very Interesting

Melodic Logic: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Illogical

Not Very Logical

Neutral

Fairly Logical

Very Logical

Predictability: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Too Predictable

Fairly Predictable

What aspects intrigued you, if any:

What aspects bored you, if any:

Balanced

Fairly Unpredictable

Too Unpredictable

Sample #6: “Melody #3” Gut reaction: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Really dislike

Dislike

Neutral

Like

Really like

Melodic Interest: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Uninteresting

Not Very Interesting

Neutral

Fairly Interesting

Very Interesting

Melodic Logic: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Illogical

Not Very Logical

Neutral

Fairly Logical

Very Logical

Predictability: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Too Predictable

Fairly Predictable

What aspects intrigued you, if any:

What aspects bored you, if any:

Balanced

Fairly Unpredictable

Too Unpredictable

General Opinions

Rate the overall diversity of the material: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Very Similar

Fairly Similar

Neutral

Fairly Diverse

Very Diverse

Rate the overall interestingness of the material: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Uninteresting

Not Very Interesting

Neutral

Fairly Interesting

Very Interesting

Rate the overall musical logic of the material: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Completely Illogical

Not Very Logical

Neutral

Fairly Logical

Very Logical

Rate the overall predictability of the material: O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O Too Predictable

Fairly Predictable

Balanced

Fairly Unpredictable

Too Unpredictable

How different is it to music you've heard before? O––––––o––––––O––––––o––––––O––––––o––––––O––––––o––––––O No Different

A Bit Different

Somewhat Different

Fairly Different

Very Different

Would you categorise it as belonging to any particular musical style or genre?

Were there any particular recurring features you found enjoyable or irritating?

Based on what you have heard today, could you imagine using this software as a compositional tool for your own purposes? (Please circle: Yes

/ No

/ Maybe )

Thanks for participating!

112

Listening Survey

Appendix C

Function List

Not every function in the automated Schillinger System is included here; there are dozens more which are concerned with auxiliary and standard musical operations, as well as interfacing with Lilypond (see section 4.3). The listing is limited to those which are related specifically to the implementation of Schillinger’s methods and those which were necessary to interface the methods in a sensible fashion. Refer to chapter 3 for details, and the call graph in section 3.6 for an overview of the system’s structure.1 The listing may also help to give some idea of the functions that would be available to the user in the proposed command-line interface mentioned in section 5.2. References back to Schillinger’s published volumes are included to aid further investigation.

C.1

Rhythmic Resultants — Book I: Ch. 2, 4, 5, 6, 12

interference_pattern primary_resultant secondary_resultant tertiary_resultant resultant_combo algebraic_expansion

C.2

Rhythmic Variations — Book I: Ch. 9, 10, 11

permutations_straight permutations_circular continuity_rhythmic general_homogeneous_continuity 1

Note that the graph in section 3.6 is a representation that has been further condensed to focus on the most important aspects of the system’s architecture. Not every function listed here is present on the diagram.

113

Function List

114

C.3

Rhythmic Grouping and Synchronisation — Book I: Ch. 3, 8

coefficient_sync group_duration group_attacks

C.4

Rhythmic Generators

random_resultant_from_basis random_combo_from_basis random_tertiary_resultant_from_basis self_contained_rhythms multiple_within_time_ratio subdivide_basis generate_rhythm convert_basis

C.5

Scale Generation — Book II: Ch. 2, 5, 7, 8

flat_scale flat_7_tone_scale scale_tonal_expansion symmetric_scale_small symmetric_scale_large random_scale

C.6

Scale Conversions — Book II: Ch. 5, 9

scale->pitch_scale scale->full_pitch_scale pitch_scale->scale symmetric_scale->scale symmetric_scale->pitch_scales symmetric-scale? extend_flat_scale scale_translate

C.7

Harmony from Pitch Scales — Book II: Ch. 5, 9

acoustically_acceptable? sub_chords

§C.8 Geometric Variations — Book III: Ch. 1, 2

sub_chords_of_scale nearest_tone_voice_leading range adjust_voice_register adjust_harmony_register

C.8

Geometric Variations — Book III: Ch. 1, 2

invert_voice invert_chord invert_harmony revoice_starting_chord generate_spliced_harmony compose_harmony expand_voice expand_chord expand_harmony contract_pitch_range

C.9

Melodic Functions — Book IV: Ch. 3, 4, 5, 6, 7

random_axis_system generate_secondary_axes partition_axis_system adjust_axis_to_pitch_scale superimpose_pitch_rhythm_on_secondary_axes generate_continuity_parameters build_melody compose_melody

115

116

Function List

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