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Über dieses Buch

This volume comprises a selection of papers presented at the first International C- ference on Mathematics and Computation in Music – mcm2007. The conference took place at the Staatliches Institut für Musikforschung PK – National Institute for Music Research in Berlin during May 18–20, 2007 and was jointly organized by the National Institute for Music Research Berlin and the Society of Mathematics and Computation in Music. The papers were selected for the conference by the program committee and classfied into talks and posters. All papers underwent further selection, revision and elaboration for this book publication. The articles cover a research field which is heterogeneous with respect to content, scientific language and methodology. On one hand, this reflects the heterogeneity and richness of the musical subject domain itself. On the other hand, it exemplifies a t- sion which has been explicitly intended by both the organizers and the founders of the society, namely to support the integration of mathematical and computational - proaches to music theory, composition, analysis and performance. The subdivision into three parts reflects the original structure of the program. These parts are opened by invited papers and followed by talks and posters.



Mathematical Modeling and Computation in Music

Invited Lectures

Rhythm and Transforms, Perception and Mathematics

People commonly respond to music by keeping time, tapping to the beat or swaying to the pulse. Underlying such ordinary motions is an act of perception that is not easily reproduced in a computer program or automated by machine. This paper outlines the flow of ideas in

Rhythm and Transforms


Sethares 2007

), which creates a device that can “tap its foot” along with the music. Such a “beat finding machine” (illustrated in

Fig. 1

) has implication for music theory, on the design of sound processing electronics such as musical synthesizers, on the uses of drum machines in recording and performance, and on special effects devices. The beat finder provides a concrete basis for a discussion of the relationship between the mind’s processing of temporal information and the mathematical techniques used to describe and understand regularities in data. Extensive sound examples (

Sethares 2008

) demonstrate beatbased signal processing techniques, methods of musical (re)composition, and new kinds of musicological analysis.

Fig. 1.

A foot-tapping machine designed to mimic people’s ability to synchronize to complex rhythmic sound must “listen” to the sound, locate the underlying rhythmic pulse, anticipate when the next beat timepoint will occur, and then provide an output

William A. Sethares

Visible Humour — Seeing P.D.Q. Bach’s Musical Humour Devices in The Short-Tempered Clavier on the Spiral Array Space

We explore the use of the Spiral Array spatial visualization of tonal evolution through time for the visual analysis of P.D.Q. Bach’s

The Short-Tempered Clavier

. In particular, we analyze situations in which we can see some of the humour devices abstracted in an earlier study by David Huron. We conclude that although we can see a good number of Schickele’s humour devices — such as incongruent styles, musically improbable tonality and harmony shifts, and excessive repetition — we do not yet have sufficient information to form a robust computer-based method for detecting musical humour.

Elaine Chew, Alexandre François

Metalanguage and Representation

Category-Theoretic Consequences of Denotators as a Universal Data Format

The R




Music Software uses denotators as the universal data format. Denotators are objects (or points) in general spaces called forms. Denotators and forms constitute a general architecture of concepts, which itself is embedded in category theory. More precisely, the category in question is the category of set-valued presheaves over the modules. This embedding allows some new and important properties which are useful for modeling musical objects.

In this paper we discuss the consequences of using the limit and colimit constructions of category theory in the implementation of denotators in R





Gérard Milmeister

Normal Form, Successive Interval Arrays, Transformations and Set Classes: A Re-evaluation and Reintegration

Normal form is an ordering standard for pitch-class sets that facilitates finding structural relationships and properties through comparative analysis. However, the analytical process often fails to correctly identify important relationships or properties, because the normal form algorithm generates misaligned orderings for many pitchclass sets. For example, a comparative analysis of normal form INT1 (Morris 1987, 40–41, 107–109) relationships often indicates pitch-class sets are not inversionally related even though they are members of the same T




I type.


The normal form ordering also obscures important structural properties, such as symmetry, for many pitch-class sets. In these problematic cases, a comparative analysis of normal form orderings cannot produce the relevant structural information without the aid of supplemental ad hoc operations. Creating jury-rigged add-on procedures seems to be the accepted solution to working around these informational inconsistencies, whereas modifying the algorithm to eliminate them is an approach that has not been pursued in the literature. In this paper, I will introduce a new normal form conceptualization and a new algorithm that corrects the problems inherent in John Rahn’s (Rahn 1980, 31–39) normal form algorithm.


Ciro Scotto

Melodic, Motivic and Metric Levels of Description

A Model of Musical Motifs

This paper presents a model of musical motifs for composition. It defines the relation between a motif’s music representation, its distinctive features, and how these features may be varied. Motifs can also depend on non-motivic musical conditions (e.g., harmonic, melodic, or rhythmic rules). The model was implemented as a constraint satisfaction problem.

Torsten Anders

Melodic Clustering within Motivic Spaces: Visualization in OpenMusic and Application to Schumann’s Träumerei

Based on the concepts of motive contour, gestalt and motive similarity, our model of motivic structure yields topological motivic spaces of a composition in which open neighborhoods correspond to groupings of similar motives. In Buteau 2006 we presented a model extension of an earlier approach in order to integrate the concept of melodic clustering in motivic spaces, demonstrated an application to the soprano voice of Schumann’s


, and provided a comparison with a human-made segmentation (clustering) analysis (

Repp 1992

) and a machine learning approach (

Cambouropoulos and Widmer 2000

). In this short paper, we present our novel dynamic visualization of melodic clustering in


software, and extend our initial analysis of


to multi-voice clustering.

Chantal Buteau, John Vipperman

Topological Features of the Two-Voice Inventions

The similarity neighbourhood model is a mathematical model making use of statistical, semiotical and computational approaches to perform melodic analysis of given music pieces. This paper is dedicated to the investigation of topological features and conditions in connection with the model on the one hand and concrete analyses on the other. Therefore, checking the topological features of the model as well as the analysis results is a good practice not only for theoretical but also for practical reasons. The topological features of the similarity neighbourhood model are investigated from a theoretical viewpoint, in order to figure out under which conditions the collection of the results yielded by the model define a topology. These topological features are then tested practically on the two-voice inventions. These investigations and tests have shown that the similarity neighbourhood model defines a topology not for all cases, but depending on the analysed musical piece.

Kamil Adiloĝlu, Klaus Obermayer

Comparing Computational Approaches to Rhythmic and Melodic Similarity in Folksong Research

In this paper we compare computational approaches to rhythmic and melodic similarity in order to find relevant features characterizing similarity in a large collection of Dutch folksongs. Similarity rankings based on Transportation Distances are compared to an approach of rhythmic similarity based on Inner Metric Analysis proposed in this paper. The comparison between the two models demonstrates the important impact of rhythmic organization on melodic similarity.

Anja Volk, Jörg Garbers, Peter van Kranenburg, Frans Wiering, Louis Grijp, Remco C. Veltkamp

Harmonic Levels of Description

Automatic Modulation Finding Using Convex Sets of Notes

Key finding algorithms, designed to determine the local key of segments in a piece of music, usually have difficulties at the locations where modulations occur. A specifically designed program to indicate modulations in a piece of music is presented in this paper. It was previously shown that the major and minor diatonic scale, as well as the diatonic chords, form convex sets when represented in the Euler lattice (

Honingh and Bod 2005

). Therefore, a non-convex set within a piece of music may indicate that this specific set is not part of a diatonic scale, which could indicate a modulation in the music. A program has been developed that finds modulations in a piece of music by localizing nonconvex sets. The program was tested on the first five preludes and fugues in a major key from the first book of Bach’s Well-tempered Clavier. It has been shown that the algorithm works best for modulations that involve many chromatic notes.

Aline Honingh

On Pitch and Chord Stability in Folk Song Variation Retrieval

In this paper we develop methods for computer aided folk song variation research. We examine notions and examples of stability for pitches and implied chords for a group of melodic variants. To do this we employ metrical accent levels, simple alignment techniques and visualization techniques. We explore how one can use insight into stability of a known set of variants to query for additional variants.

Jörg Garbers, Anja Volk, Peter van Kranenburg, Frans Wiering, Louis P. Grijp, Remco C. Veltkamp

Bayesian Model Selection for Harmonic Labelling

We present a simple model based on Dirichlet distributions for pitch-class proportions within chords, motivated by the task of generating ‘lead sheets’ (sequences of chord labels) from symbolic musical data. Using this chord model, we demonstrate the use of Bayesian Model Selection to choose an appropriate span of musical time for labelling at all points in time throughout a song. We show how to infer parameters for our models from labelled ground-truth data, use these parameters to elicit details of the ground truth labelling procedure itself, and examine the performance of our system on a test corpus (giving 75% correct windowing decisions from optimal parameters). The performance characteristics of our system suggest that pitch class proportions alone do not capture all the information used in generating the ground-truth labels. We demonstrate that additional features can be seamlessly incorporated into our framework, and suggest particular features which would be likely to improve performance of our system for this task.

Christophe Rhodes, David Lewis, Daniel Müllensiefen

The Flow of Harmony as a Dynamical System

When analysing the evolution of harmony within a composition, one can distinguish between two parts: on the one hand a dynamical system and on the other hand the composer. The dynamical system summarises the rules of harmony, whereas the composer intervenes at certain points, e.g. by choosing new initial values. This viewpoint is helpful for the musical analysis of a composition and will be exemplified by analysing the first movement of Beethoven’s first symphony.

Peter Gies

Tonal Implications of Harmonic and Melodic Tn-Types

Music composed of


(in the psychoacoustical sense of

sounds that have pitch

) can never be completely atonal (

Reti 1958

). Consider any quasi-random selection of tones from the chromatic scale, played either simultaneously or successively. Most such sets generate associations with musically familiar pitch-time patterns and corresponding tonal stability relationships (

Auhagen 1994

). A pattern of pitch can imply a tonal centre simply because it reminds us of a tonal passage: it has

tonal implications

that depend on the intervals among the pitch classes (pcs) in the set.


The only clear exceptions to this rule are trivial: the null set (cardinality = 0)


and the entire chromatic aggregate (cardinality = 12). Since every interval, sonority and melodic fragment has tonal implications, even the so-called “atonal” music of composers such as Ferneyhough, Ligeti and Nono is full of fleeting tonal references: at any given moment during a performance, some pitches are more likely than other pitches to function as psychological points of reference. In the following, I will use the terms “tonal” and “atonal” in this broad, psychological sense.

Richard Parncutt

Computational Models in Music Psychology

Calculating Tonal Fusion by the Generalized Coincidence Function

Models of pitch perception in the time domain suggest that the perception of pitch is extracted from neuronal pulse series by networks for periodicity detection. A neuronal mechanism for periodicity detection in the auditory system has been found in the

inferior colliculus

(Langner 1983). The present paper proposes a mathematical model to compute the degree of coincidence in the periodicity detection mechanism for musical intervals represented by pulse series. The purpose of this model is to study the logical structure of coincidence and to define a measure value for the degree of coincidence.

The model is purely mathematical but has a strong relation to physiological data presented by Langner. As the sensation of consonance depends mostly on pitch, frequency is the only parameter to be regarded in the model. The integration of other parameters and the adaptation to further physiological data should be easy but still lies ahead.

The model is a mathematical basis for a concept of consonance based on pitch perception models in the time domain. In contrast to the concept of the

sensory consonance

it does not refer to the percept of roughness, which nevertheless is important for the perceived pleasantness of consonances.

Martin Ebeling

Predicting Music Therapy Clients’ Type of Mental Disorder Using Computational Feature Extraction and Statistical Modelling Techniques


. Previous work has shown that improvisations produced by clients during clinical music therapy sessions are amenable to computational analysis. For example, it has been shown that the perception of emotion in such improvisations is related to certain musical features, such as note density, tonal clarity, and note velocity. Other work has identified relationships between an individual’s level of mental retardation and features such as amount of silence, integration of tempo with the therapist, and amount of dissonance. The present study further develops this work by attempting to predict music therapy clients’ type of mental disorder, as clinically diagnosed, from their improvisatory material.


. To predict type of mental disorder from computationally-extracted musical features of music therapy improvisations.


. Two hundred and sixteen music therapy improvisations, obtained from seven music therapists’ regular sessions with their clients, were collected in MIDI format. A total of fifty clients contributed musical material, and these clients were divided into three groups according to their clinical diagnosis: RET (mentally retarded), DEV (developmental disorder), and NEU (neurological disorder). The improvisations were subjected to a musical feature extraction procedure in which 43 musical features were automatically extracted in the MATLAB computing environment. These features were then entered into a discriminant function analysis as predictors of type of mental disorder.


. The analysis produced two significant discriminant functions, and the emergent model correctly classified 80% of clients. Significant predictor variables fell into three main categories: those relating to pitch, those relating to temporal aspects, and those relating to tonal clarity and dissonance.


. The present study suggests that an individual’s type of mental disorder can be predicted from a statistical analysis based upon the computational extraction of detailed musical features from their improvisatory material. As such, it offers further evidence for the usefulness of computational music analysis in music therapy.

Geoff Luck, Olivier Lartillot, Jaakko Erkkilä, Petri Toiviainen, Kari Riikkilä

Nonlinear Dynamics, the Missing Fundamental, and Harmony

We review the historical and current theories of musical pitch perception, and their relationship to the intriguing phenomenon of residue pitch. We discuss the nonlinear dynamics of forced oscillators, and the role played by the Fibonacci numbers and the golden mean in the organization of frequency locking in oscillators. We show how a model of the perception of musical pitch may be constructed from the dynamics of oscillators with three interacting frequencies. We then present a mathematical construction, based on the golden mean, that generates meaningful musical scales of different numbers of notes. We demonstrate that these numbers coincide with the number of notes that an equal-tempered scale must have in order to optimize its approximation to the currently used harmonic musical intervals. Scales with particular harmonic properties and with more notes than the twelve-note scale now used in Western music can be generated. These scales may be rooted in objective phenomena taking place in the nonlinearities of our perceptual and nervous systems. We conclude with a discussion of how residue pitch perception may be the basis of musical harmony.

Julyan H. E. Cartwright, Diego L. González, Oreste Piro

Computational Models for Musical Instruments

Dynamic Excitation Impulse Modification as a Foundation of a Synthesis and Analysis System for Wind Instrument Sounds

The Variophon is a wind synthesizer that was developed at the Musicological Institute of the University of Cologne in the 1970/80ies and was at that time based on a completely new synthesis principle: the pulse forming process. The central idea of that principle is that every wind instrument sound can basically be traced back to its excitation pulses, which independently of the fundamental always act upon the same principles. In a recent project, supported by the German Research Foundation (DFG), the synthesis method of excitation impulse modification has been transferred to a digital platform.

The aim of the software-based modelling is twofold: creating an experimental system for analyzing and synthesizing (wind) instrument sounds, as well as building a synthesizer, that would be an alternative to comparable wind instrument synthesis applications. On the one hand this sound synthesis technique accounts for the place where the sound is generated, on the other hand only a single breath controller is required to produce all the sound-nuances that are possible on a real instrument.

First of all the analogue circuits of the different instrument modules of the Variophon will be mapped onto a digital representation by means of the analogue circuit simulation software LTSpice. After the original algorithms have been analysed, the Digital Variophon will be rebuilt in the modular Reaktor environment by Native Instruments (NI). Finally the experimental system will be validated by means of a prototypical perception experiment.

Michael Oehler, Christoph Reuter

Non-linear Circles and the Triple Harp: Creating a Microtonal Harp

With the increased usage of microtones by contemporary composers, research is needed into the various systems currently in use along with other potential ways of constructing musical temperaments. Considerations need to be made regarding the performance practicalities of such systems, especially when used on conventional orchestral instruments with limited means of producing unconventional intervals. The harp is an instrument which is confined to equal temperament and therefore is problematic for composers wishing to use alternative tuning systems. This article looks at the possibility of creating a microtonal harp using an older version of the instrument and describes a new tuning system proposed by Sturman (

Sturman 2005

) using a non-linear circle equation. The article outlines the details of this tuning system and how it fits with the triple harp. It then explains some of the practicalities involved in the process of composition using this unusual tuning in conjunction with the traditional instrument.

Eleri Angharad Pound

Comparative Computational Analysis

Applying Inner Metric Analysis to 20th Century Compositions

This paper compares metric analyses of three pieces by Skrjabin, Webern and Xenakis using

Inner Metric Analysis

. Inner Metric Analysis assigns to each note of a piece a metric weight. The analysis is based on the detection of regular pulses created by the onsets of notes. Metrically strict pieces, such as Renaissance madrigals or ragtime pieces, often result in metric weight profiles that correspond to the accent schema of the notated bar lines. Hence these pieces are characterised as being

metrically coherent

since the notes generate a metrical structure that is synchronous with the abstract grid of the bar lines. Compositions of the 20th century very often do not follow such a strict metricity. The metric profiles therefore give interesting insights into the time organisation of these pieces far beyond the notated bar lines. Furthermore, we apply a processive approach to meter in order to study how the metric structure evolves over time while listening to the music.

Anja Volk

Tracking Features with Comparison Sets in Scriabin’s Study op. 65/3

Comparison set analysis is a method with which, for instance, formal divisions of musical compositions can easily be perceived. Its applications are, to a certain extent, comparable to methods used in pattern matching. The purpose in comparison set analysis is to evaluate the prevalence of a chosen musical property through a musical piece. In comparison set analysis the basic musical units like pitches, pitch classes or durations are segmented into overlapping sets and these segments are then compared with the (pre)selected comparison set(s). The results, which are largely defined statistically, can be presented in different forms of graphs showing trends, mean points and, in the case of multiparametric comparison set analysis, connections between the parameters analysed. In this paper, I will demonstrate the method by analysing some piano pieces by Scriabin. Both pitch-class sets and rhythm sets are applied.

Atte Tenkanen

Computer Aided Analysis of Xenakis-Keren

The similarity neighbourhood model is a computer aided mathematical approach to the paradigmatic analysis of the melodic content of a piece of music. It makes use of statistical, semiotical and computational approaches to perform an exhaustive search on melodies in given music pieces. It has been designed to help music theoreticians to perform melodic analysis of given music pieces. This paper, presents the analysis of Keren, which is a solo trombone piece of Xenakis, based on the results obtained from the similarity neighbourhood model. The similarity neighbourhood model identifies melodic similarities depending on the contour similarities of melodic segments. Keren does not contain similar melodic segments in the sense of the Baroque or classical period of western music. However this paper shows that the results of the similarity neighbourhood model are interpreted in a way considering where similarities exist, so that they contribute to a music-theoretical analysis of the concerned piece.

Kamil Adiloĝlu, G. Ada Tanir

Automated Extraction of Motivic Patterns and Application to the Analysis of Debussy’s Syrinx

A methodology for automated extraction of repeated patterns in discrete time series data is presented, dedicated to the discovery of musical motives in symbolic music representations. The basic principle of the approach consists in a search for

closed patterns

in a multi-dimensional parametric space, comprising various features related to melodic and rhythmic aspects, which can be organized into note-based and interval-based descriptions. The pattern description is further reduced through a lossless pruning of the sequence description. This requires in particular a detailed estimation of the specificity relations between patterns. For instance, a pattern is more specific than its suffix, and a melodic-rhythmic pattern is more specific than its rhythmic component. A notion of cyclic pattern is introduced, enabling an adapted filtering of a different form of combinatorial redundancy caused by successive repetitions of patterns. The use of cyclic patterns implies a necessary chronological scanning of the musical sequence. The resulting algorithm offers compact motivic analyses of simple monodies. As an illustration of the analytic capabilities of the computational system, a complete analysis of Debussy’s


is presented.

Olivier Lartillot

Pitch Symmetry and Invariants in Webern’s Sehr Schnell from Variations Op.27

We use the Argus algorithm as outlined in (

Chew 2005



) to measure (single or clustered) pitch changes in Webern’s

Sehr schnell

, the second piece in his

Variations Op.27

. In previous analyses employing the Argus algorithm, the computational results have been used to determine points of statistically significant change, which correspond to key or section changes. Instead of focussing only on the peaks (points of significant change), this paper considers symmetries and invariants revealed by the numerical results, paying particular attention to the stationary points as reflected by the zero values on the graph. These zero points signify places with identical or symmetric mappings of the pitch(es) in consecutive time windows. We analyze the results for small window sizes of one, two, and three eighth notes. The findings give rise to a pitch geometry map inside the Spiral Array, centered on the radius through pitch A, that explains Webern’s pitch choices in

Sehr schnell


Elaine Chew

Computational Analysis Workshop: Comparing Four Approaches to Melodic Analysis

We compare four computational approaches of melodic analysis according to diverse approach aspects: input type (monophonic or polyphonic), pattern identification type (strict or similar), analysis segmentation, aim of approach, motivic pattern representation, and type of result representations. The considered four computational approaches are the following: a similarity neighbourhood approach by Adiloglu (

Adiloglu and Obermayer 2006a



), a multiple viewpoint representation and discovery approach by Anagnostopoulou (

Anagnostopoulou, Share and Conklin 2006

), a topological approach by Buteau (2005), and an approach based on multidimensional closed pattern mining by Lartillot (

Lartillot and Toiviainen 2007


Chantal Buteau, Kamil Adiloĝlu, Olivier Lartillot, Christina Anagnostopoulou


Computer-Aided Investigation of Chord Vocabularies: Statistical Fingerprints of Mozart and Schubert

We introduce and demonstrate an original software tool for determination of some style fingerprint (from the point of view of harmony) of two world famous composers — W. A. Mozart and F. Schubert. The new version of the ANALYSIS software (previously CACH, Ferkova 1982) for automatic analysis of classical harmonic structures provides a powerful extension of its previous music data processing.

Eva Ferková, Milan Zdímal, Peter Sidlík

The Irrelative System in Tonal Harmony

This paper is addressed to music theorists and musicologists specialising in harmony topics. It presents a computational approach to the investigation of tonal structure in musical pieces. With the use of this analytical system, the quantitative prevalence of chords classified by ranges of a given key in a musical piece can be determined.

Miroslaw Majchrzak

Mathematical Approaches to Music Analysis and Composition

Invited Lectures

Mathematics and the Twelve-Tone System: Past, Present, and Future

Certainly the first major encounter of non-trivial mathematics and non-trivial music was in the conception and development of the twelve-tone system from the 1920s to the present. Although the twelve-tone system was formulated by Arnold Schoenberg, it was Milton Babbitt whose ample but non-professional background in mathematics made it possible for him to identify the links between the music of the Second-Viennese school and a formal treatment of the system. To be sure, there were also important inroads in Europe as well,1 but these were not often marked by the clarity and rigor introduced by Babbitt in his series of seminal articles from 1955 to 1973 (

Babbitt 1955








Robert Morris

Approaching Musical Actions

So, an improvisation has been going on for some time, but its impetus is dying out, at first in a good way, all getting more quiet in a nice contrast to what has gone before, but soon, in fact now, we need a new idea, of course (inescapably) related to what we have been playing already, but one that will have a fresh effect and that can carry us into a fertile territory that will in some way complement what has gone before. I gather up into my mind and intuition some threads that have been woven into everything else so far, and form a tentative image of some new pattern to weave, and I act. The act is the public manifestation of my inner representation of my projection of the music we have played onto the screen of the future. The other musicians respond to this new musical context with their own representations, projections and actions, in a spreading web of new musical relations, represented individually and to some extent variously in each musician, and manifested publicly in our shared acoustic space, which serves as our blackboard — the space in which we communicate to each other.

John Rahn

A Transformational Space for Elliott Carter’s Recent Complement-Union Music

Elliott Carter’s recent music exploits a special combinatorial property of the all-trichord hexachord. I show how this property can be reconceived in terms of interesting and analytically significant musical transformations: three involutions on the pitch-class aggregate which constitute a Klein four-group, and which have a natural interpretation as the symmetry group on a particular 12-vertex geometrical structure. Accordingly the opening of Carter’s

Figment II

for solo cello can be analyzed transformationally as a complete traversal of this structure by just a few, striking, characteristic gestures.

John Roeder

Mathematical Approaches to Composition


I was pleased to learn that it would be possible to make a little exhibition of drawings here in Berlin, as well as presenting my lecture, because the


I’m working on now are as much visual as aural, and I think these structures need to be seen as well as heard. Of course, those who are simply reading this text will only see the drawings that are included here, but hopefully this will be enough to convey the general idea.

Tom Johnson

From Mathematica to Live Performance: Mapping Simple Programs to Music

This paper focuses on selected simple programs used to model generative processes for basic elements of music material such as rhythm, pitch and texture, as well as large-scale works of music. After presenting decisions on sound mapping procedures, I’ll introduce the system

NKM, A New Kind of Music

, designed by Peter Overmann, director of software technology for the


programming environment.


is a system controlled by

cellular automata

(CA), modeling a number of processes in nature. The CA presented in the paper belong to a group of

elementary rules

that encapsulate four classes of complexity, from simple to universally complex, conceived by Stephen Wolfram and presented in his book

A New Kind of Science

. All of the examples were generated in


, the software by Wolfram Research Inc. Mathematical basis for the examples can be found in the book

A New Kind of Science.


Katarina Miljkovic

Nonlinear Dynamics of Networks: Applications to Mathematical Music Theory

Algebraic approaches to modelling and the theory of dynamical systems are important aspects of theories of mathematics and music. Group-theoretic approaches have been used for some time in models of pitch-class, tuning and interval etc.More recent approaches by (


) and others strikingly extend this algebraic formulation into the realm of modules and categories. And the theory of dynamical systems has found musical applications in both algorithmic music creation (for example in the compositions of Agostino Di Scipio), and the physical modelling of musical instruments (in the work of Xavier Rodet and others at IRCAM).

Jonathan Owen Clark

Mathematical Approaches to Musical Analysis and Performance

Form, Transformation and Climax in Ruth Crawford Seeger’s String Quartet, Mvmt. 3

The paper reckons the progression of permuted registral states in Ruth Crawford Seeger’s String Quartet as a walk through a Cayley graph of the symmetric group, S4. The graph privileges the 2-cycles that exchange adjacent voices in the four-voice quartet texture, imposing a metric upon the group elements that allows one to distinguish permutations that are contextually close or ‘smooth’ from those are distant or ‘agitated.’

Edward Gollin

A Local Maximum Phrase Detection Method for Analyzing Phrasing Strategies in Expressive Performances

This paper proposes a Local Maximum Phrase Detection (LMPD) method for the analysis of phrasing strategies in expressive performances. The LMPD method systematically extracts a quantitative representation of phrasing strategy by equating the occurrence of a local maximum in the loudness curve with the occurrence of a phrase or sub-phrase. We further define mathematical descriptors for phrase strength and volatility, and phrase typicality, for comparing phrasing strategies among performances. Phrase strength measures the prominence or clarity of a phrase, and the volatility is defined as the standard deviation of the phrase strengths within a performance. Phrase typicality quantifies the degree to which a phrase loudness peak location is characteristic among the performances polled. The ideas behind these descriptors extend to phrase information derived from tempo variation. We illustrate the LMPD method using preliminary results from its application to eleven commercially available audio recordings of a solo violin Bach Sonata.

Eric Cheng, Elaine Chew


Subgroup Relations among Pitch-Class Sets within Tetrachordal K-Families

In 1990 and 1991, Henry Klumpenhouwer and David Lewin introduced

Klumpenhouwer networks (K-nets)

as theoretical tools that display transformational interpretations of dyads contained within pitch-class multisets (

Lewin 1990


Klumpenhouwer 1991

). Informally, K-nets are directed graphs that employ pitch classes as nodes and elements of the T/I group as edges. In order for a K-net to be well defined, its edges must commute throughout the directed graph and its nodes must map to adjacent nodes according to the corresponding edge transformations. Several types of K-nets emerged by varying the cardinalities of the underlying pitch-class multisets, the number of constituent dyads subject to transformational interpretation, the number of transpositional and inversional operators employed, and the relative positions of these operators. We will work exclusively with two common types of K-nets:

trichordal K-nets


box-style tetrachordal K-nets

. See Examples 1a and 1b, respectively, for representatives of these two types.

Jerry G. Ianni, Lawrence B. Shuster

K-Net Recursion in Perlean Hierarchical Structure

Klumpenhouwer networks

, or


, are graphic representations of the intervallic relationships among elements of a set.


Theorist David Lewin has suggested that K-nets may be applicable to

Perle cycles

, entities referred to by George Perle in his theory of twelve-tone tonality as cyclic sets. These entities are created through the alternation of inversionally related interval cycles. The present study seeks to broaden the applicability of K-nets in Perle’s theory by exploring their recursive nature at varying levels of structure.

Gretchen C. Foley

Webern’s Twelve-Tone Rows through the Medium of Klumpenhouwer Networks

The theory of Klumpenhouwer networks (K-nets) in contemporary music theory continues to build on the foundational work of (


) and (


), and has tended to focus its attention on two principal issues: recursion between pitch-class and operator networks and modeling of transformational voice-leading patterns between pitch classes in pairs of sets belonging to the same or different T




I classes.


At the core of K-net theory lies the duality of objects (pitch classes) and transformations (T


and T


I operators and their hyper-T


and hyper-T


I counterparts). Understood in this general way, K-net theory suggests other avenues of investigation into aspects of precompositional design, such as connections between K-nets and Perle cycles, K-nets and Stravinskian rotational or four-part arrays, and between K-nets and row structure in the “classical” twelve-tone repertoire.

Catherine Nolan

Isographies of Pitch-Class Sets and Set Classes

One of the major differences between tonal and atonal music is the significantly larger number of available pitch-class sets in the latter. The categorization of pitch-class sets and the analysis and classification of their relations stand as a major strand in the 20th century music theory. The equivalence relation induced by the group of transpositions and inversions is the de facto standard classification of pitch-class sets. Music theorists, however, have recently suggested alternative approaches, for instance, relations based on similarity or voice leading.

Tuukka Ilomäki

Leonard Euler at the Crossroads of Music Theory

The Transmission of Pythagorean Arithmetic in the Context of the Ancient Musical Tradition from the Greek to the Latin Orbits During the Renaissance: A Computational Approach of Identifying and Analyzing the Formation of Scales in the De Harmonia Musicorum Instrumentorum Opus (Milan, 1518) of Franchino Gaffurio (1451–1522)

On the occasion of the three hundredth anniversary of Leonhard Euler (1707–1783) and the focus of his contribution to the mathematical explanation of scales, the quest arises to examine similar contributions of music theorists that lived


to Euler. In this paper we selected from the trilogy (

Theorica musicae

, 1492;

Practica musicae

, 1496;

De harmonia musicorum instrumentorum opus

, 1518) of the eminent Renaissance scholar, musician, theorist and composer Franchino Gaffurio (1451–1522) his third work of 1518.

In particular, using a modern elementary number theoretic approach, we discovered a number generating function in closed form, which we called a

Gaffurio number generating function

, that generates the tones of the Pythagorean scale. The number generating function is of the form

f (n, m) = p




, where




are primes, and




are elements of /



. Here, Gaffurio chooses the smallest primes


= 2 and


= 3, thereby with this ratio

p / q

denoting the interval of the


and he varies the integral exponents




from 0, 1, 2, 3, ... and in the process constructing a series of numbers signifying the precise placement of the tones within the gamut. His approach is in line with medieval mathematics, which focuses on elementary number theory and can be seen as a precursor to Euler’s


(grid of proportions). This paper embeds our results in Gaffurio’s oeuvre and his times and includes a concise of overview of his biography with an emphasis on his pedagogical and literary contributions.

Our approach fills a gap in research, because although Latin music theorists from the era of musical humanism frequently include mathematical approaches to the Greek

systema teleion

and Guidonian system, as projected onto the monochord, they do


disclose the nature and specific details of the mathematical calculations which inform the natural numbers included in both the manuscript and early printed editions.

Herbert Kreyszig, Walter Kreyszig

Combinatorial and Transformational Aspects of Euler’s Speculum Musicum

The paper examines the structural and conceptual differences between the

Speculum musicum

from Euler’s 1774

De Harmoniae

, and the nineteenth-century


, but also examines how Euler’s conception of intervals as paths within the


anticipates a combinatorial conception of interval that underlies contemporary transformational theories.

Edward Gollin


Structures Ia Pour Deux Pianos by Boulez: Towards Creative Analysis Using OpenMusic and Rubato

Pierre Boulez introduced the concept of

creative analysis


Boissière 2002

) in the late 1980s suggesting that the aim of analysis should be the production of new pieces. Marcel Mesnage and André Riotte followed this path in their work on computer-aided analysis and composition (

Mesnage and Riotte 1993

). Our current study focuses on Ligeti’s analysis of

Structures pour deux pianos Ia

(first book in 1951–1952) by Boulez, where the compositional process is described in detail and further set a model. In


, Boulez uses series consisting of 12 pitches, 12 durations, 12 attacks and 12 dynamics borrowed from Messiaen’s

Mode de valeurs et d’intensité

(1949). Following Boulez’s analytic concept, our task is to start from his compositional model and consider how it might be used to produce another piece.

Our study uses the software OpenMusic, which was developed at the IRCAM and is built on a functional paradigm. This graphical environment can be exploited to imitate the model used in


, stressing a functional point of view.

Parallel to this, a second approach will be implemented using Rubato, an universal music software environment that has been developed at the University of Zurich. The keystone of this application is a categorical point of view, theorized by Guerino Mazzola. Since category theory and functional languages are strongly linked, the two software applications are complementary. However, Rubato brings a different level of abstraction and therefore offers new possibilities that have yet to be developed, for instance creating metapieces that could give different



Yun-Kang Ahn, Carlos Agon, Moreno Andreatta

The Sieves of Iannis Xenakis

Xenakis’s article’ sieves’ was published in 1990, but the first extended reference to Sieve Theory is found in the final section of ‘Towards a Metamusic’ published 1967. These two writings mark two periods, where there is a progression from the decomposed formula to the simplified one. This progression reflects Xenakis’s aesthetic of sieves as timbres and is here explored under the light of the idea of inner periodicities.

Dimitris Exarchos

Tonal, Atonal and Microtonal Pitch-Class Categories

This paper reviews and generalizes Pitch-Class Set Theory using Group Theory (groups acting on pc-sets) and Category Theory, which provides methods for mapping the structure of a


-tone system onto another


-tone system. This paper also suggests a new implementation approach that represents pitch-class sets as bit-sequences, which are equivalent to integer values. Forte’s

best normal order

is shown to be equivalent to the smallest integer, among the cyclic permuted pc-sets. Further, transposition of pc-sets is shown to be equivalent to bit-shifts; and their inversion, to bit-reversal. The tonal (diatonic) pc-category is presented as a subset of the atonal (12-tone) pc-category, which, similarly, can also be contained in a microtonal pc-category. Functors between those categories present properties that preserve relationships while still using the same operations: tonal relationships are preserved even though atonal music operations, such as transposition and inversion are applied, allowing motives to be mapped into different modes, scales, or even microtonal scales. The appendix offers an implementation of this new approach to calculate and represent pc-sets with an arbitrary number of pitch-classes.

Fernando Gualda

Using Mathematica to Compose Music and Analyze Music with Information Theory

In this paper we present two case studies for the application of the technical computing software


in the domain of music creation and music research. The first section describes an experimental interface for the usage of random points, parametric curves and other mathematical objects in the role of three-dimensional musical scores. Similar to the technology of the old-fashioned player piano roles which encode any arbitrary piece for mechanical player piano in three basic dimensions (onset, pitch, duration) we provide an interface where a 3-dimensional score is created, visualized and played. With this software the scores can be created with the assistance of a rich arsenal of mathematical functions and also the sound of each single note can be controlled in terms of mathematical functions. The aim of this software is the creation of experimental musical pieces which explore the musical potential of certain mathematical functions. In this paper we restrict ourselves to sketch the interface. The more interesting aspects, namely the ‘musicality’ of concrete sonifications of certain mathematical objects, are subject to our live demonstrations in Berlin.

In the second section we show how information theory may be used in the analysis of musical scores and how specialized packages of the software Mathematica may assist such investigations. As a particularly interesting topic we describe the calculation of the transfer entropy between selected instrumental parts in Beethoven symphonies.

Christopher W. Kulp, Dirk Schlingmann

Mathematical Approaches to Music Theory

Invited Lectures

A Diatonic Chord with Unusual Voice-Leading Capabilities

Diatonic set theory, as established by John Clough and others (see

Clough 1979




Carey and Clampitt 1989


Carey 1998

), applies the tools of standard set theory of 12-tone ET to the heptatonic set of seven tones of the diatonic scale. The two universes differ from each other in a number of ways other than simple cardinality. Although 12 is nearly twice as big as 7, the fact that 7 is prime and 12 composite contributes to a number of subtle differences between them. Every positive integer less than 7 is a unit mod 7, thus every diatonic interval generates the entire set. In the set of 12 tones, there are only four units, 1, 5, 7, and 11. Further, because of tuning, the geographies, if you will, of the sets also differ. In the equal-tempered 12-tone landscape, every place looks like everyplace else. In the diatonic scale, each generic span is inhabited by several different specific intervals. Because of this, the terrain is everywhere distinct, contributing to the phenomenon of gravitational asymmetry and of tonality.

Norman Carey

Mathematical and Musical Properties of Pairwise Well-Formed Scales

The short paper below presents the definition of the

pairwise well-formed scale

concept, and a few of the significant mathematical and musical features entailed by that definition. The verifications that are easily available are supplied here; for the more difficult proofs which are here omitted, the reader is directed to my dissertation (

Clampitt 1997

). While the definition itself is quite abstract, the body of implications and equivalences that constitute the theory include several musically attractive properties. With the significant exception of one structural subcategory, all other pairwise well-formed scales participate in

modulating cycles

that generalize the

maximally smooth cycles

defined in Cohn 1996 and intersect with the

Cohn functions

defined in

Lewin 1996


David Clampitt

Eine Kleine Fourier Musik

The discrete Fourier transform, or DFT, of a pc-set was first introduced, for musical purposes, by David Lewin in his very first paper (

Lewin 1959

). His aim was the characterization of a pc-set by its intervallic relationship with another. This approach fails for some specific cases which exhibit interesting symmetries. It was by and large forgotten when Ian Quinn exhumated it in his dissertation (

Quinn 2004

), this time as an efficient tool for pinpointing in the landscape of all chords the most salient ones (‘prototypes’). One of several fascinating by-products of Quinn’s study is that such prototypes appear as solutions of an optimization problem of the DFT. In this conference I showed that the DFT is not only a good way to introduce and indeed define some of these prototypes (the famous Maximally Even Sets) but also essential or handy for many other musical notions, such as interval content and Z-relation, tilings and rhythmic canons, and the very recent Flat Interval Distribution of pc-sets first introduced by Jon Wild on the same day.

Emmanuel Amiot

Towards New Music-Theoretical Concepts

WF Scales, ME Sets, and Christoffel Words

With a few exceptions (

Chemillier and Truchet 2003

), (

Chemillier 2004

), musical scale theory and combinatorial word theory have remained unaware of each other, despite having an intersection in methods and results that by now is considerable. The theory of words has a long history, with many developments coming in the last few decades; see Lothaire 2002 for an account. The authors thank Franck Jedrzejewski for an initial reference in word theory. The purpose of this paper is to translate between the language of two closely related scale theories and that of the theory of words.

Manuel Domínguez, David Clampitt, Thomas Noll

Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study

Interval preservation—wherein intervals remain unchanged among varying musical objects—is among the most basic means of manifesting coherence in musical structures. Music theorists since (


) seminal publication of “Twelve-Tone Invariants as Compositional Determinants” have examined and generalized situations in which interval preservation obtains. In the course of this investigation, two theoretical contexts have developed: the group-theoretical, as in (


) Generalized Interval Systems; and the graph-theoretical, as in (


) K-net theory. Whereas the two approaches are integrally related— the latter’s being particularly indebted to the former—they have also essential differences, particularly in regard to the way in which they describe interval preservation. Nevertheless, this point has escaped significant attention in the literature. The present study completes the comparison of these two methods, and, in doing so, reveals further-reaching implications of the theory of interval preservation to recent models of voice-leading and chord spaces (

Cohn 2003


Straus 2005


Tymoczko 2005

, among others), specifically where the incorporated chords have differing cardinalities and/or symmetrical properties.

Robert Peck

Pseudo-diatonic Scales

The generalization of diatonic scales in a given tone system has been investigated by Eytan Agmon (see

Agmon 1989


Agmon 1991

), John Clough (see

Clough 1979


Clough and Myerson 1985

), and in relation with microtonality by Gerald Balzano and Mark Gould (see

Gould 2000

). Recently, (


) gave a new synthetic approach of pseudo-diatonic scales (Model


). From our first essay (

Jedrzejewski 2002

) until the last article (

Jedrzejewski 2008

), we developed a new model of generalized diatonic scales based on a new arrangement of the Stern-Brocot tree (Model


). With our new definition of diatonicism, we recover Wyschnegradsky’ diatonic scales in the quarter tone universe, a concept that he called

diatonicized chromatism


Wyschnegradsky 1979

), studied in (

Jedrzejewski 1996

) and (

Jedrzejewski 2003

). In the present article, we point out the differences of the two models (





Franck Jedrzejewski

Dasian, Diatonic and Dodecaphonic Set Theory

Affinity Spaces and Their Host Set Classes

This paper proposes the organization of pitch-class space according to the notion of affinities discussed in medieval scale theory and shows that the resultant arrangement of intervallic affinities establishes a privileged correspondence with certain symmetrical set classes. The paper is divided in three sections. The first section proposes a pitch-class cycle, the Dasian space, which generalizes the periodic pattern of the dasian scale discussed in the ninthcentury Enchiriadis treatises (

Palisca 1995

). The structure of this cycle is primarily derived from pitch relations that correspond to the medieval concepts of transpositio and transformatio.1 Further examination of the space’s properties shows that the diatonic collection holds a privileged status (host set class) among the embedded segments in the cycle. The second section proposes a generalized construct (affinity spaces) by lifting some of the intervallic constraints to the structure of the Dasian space, while retaining the relations of transpositio and transformatio, and the privileged status of host set classes.2 The final section examines some of the properties of host set classes, and in turn proposes “rules” for constructing affinity spaces from their host sets. The study of affinity spaces will give us insights regarding scalar patterning, inter-scale continuity, the combination of interval cycles, voice leading, and harmonic distance.

José Oliveira Martins

The Step-Class Automorphism Group in Tonal Analysis

Until recently, researchers who have dealt formally with tonal hierarchy (prolongation) have considered only models in which the objects of the hierarchy are musical events (where a musical event might be a chord or a note in a particular voice).


In contrast, (


) proposes a concept called “dynamic prolongation” in which the objects of tonal hierarchy are

motions between

events rather than events themselves. The events in the model of (


) are chords made up of harmonically related pitches from several voices. In the present study I develop a different approach to dynamic prolongation. Rather than expressing harmonic relations between notes by grouping them into chords, we can treat harmonic relations as intervals and mix them in a hierarchy with melodic motions. This creates a model that can posit long-range harmonic relationships and blur the boundaries between intervals of harmonic and melodic significance.

Jason Yust

A Linear Algebraic Approach to Pitch-Class Set Genera

The concept of interval-class vector (ICV) plays an important role in musical set theory. ICV can be seen as a six-dimensional representative of the intervallic content of a set class (SC), which often forms the basic tool in harmonic analysis of twentieth century music. Interrelations between SCs have been evaluated by means of similarity functions and clustering techniques in many contexts. SCs have also been classified into ’families’ called genera. Among others, six-, sevenand twelve-part systems have been outlined. Using the linear algebraic concept of spanning, the six-part genera is unambiguously justified. A group of interval-class vectors, which actually represent points, restrict a finite area in a six-dimensional space. Using the determinant of a matrix, the volume of the area formed by a six-member set class combination1 can be calculated. All possible set groups among TnI-type trichords or hexachords, which produce maximum volumes, are detected. Both the extreme points on the edges of SC space as well as the most neutral set classes in the middle of SC space are recognized using three different methods derived from linear algebra, namely the determinant of a matrix, cosine distance and principal component analysis. A short final demonstration concerns volumes of hexachord combinations used by Finnish composer Magnus Lindberg in his chaconne chains.

Atte Tenkanen


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