Poster Abstracts

Poster Abstracts 

(Posters will be on display for the entire duration of the workshop.)

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Genetic Algorithms Based on the Principles of Grundgestalt and Developing Variation

Carlos Almada

This article integrates a broad research project, whose main objective is to study systematically the principles of developing variation and Grundgestalt under analytical and compositional perspectives. Both principles were elaborated by Arnold Schoenberg and are associated with the organic musical creation, or else, the gradual production of a large set of motive-forms and themes from a small group of basic and abstract elements (the Grundgestalt) by recursive use of transformational processes, which corresponds to the developing variation techniques. The present paper describes a set of three sequential and complementary computational programs, modeled as a genetic algorithm complex, forming a system created for organic production of variations from a basic structure, i.e., the system’s Grundgestalt, or, in the research’s terminology, its Axiom. An Axiom is defined as a brief two-dimensional musical fragment (i.e., it presents both rhythm and intervalar configurations), like a conventional motive. The first and principal program is subdivided into three phases: (1) From the Axiom—input as a monophonic MIDI file—are abstracted two one-dimensional structures (its intervalar and rhythmic contours), that are transformed into numeric vectors. (2) These structures (named, respectively, I- and R- chromosomes) become the basis for parallel production of rhythmic and intervalar abstracted variations (geno-theorems or gT’s) by sequential and/or recursive application of a number of algorithms (production rules) in several generations. This procedure corresponds to which is named developing variation of first order, in other words, the production of abstracted offspring. A special algorithm calculates the “parenthood” degree of each produced gT considering its referential form (one of the chromosomes in the first generation or its gT’s ancestor, in the following generations) which is expressed as a real number, the coefficient of similarity (Cs), ranging between 0 (absolute dissimilarity) and 1 (identity). (3) The crossing-over of pairs of gT’s (rhythmic + intervalar) results in a matrix of feno-theorems (fT’s, concrete melodic structures). The second program is employed to form more complex musical unities (axiom-groups or aG’s) by concatenating two or more selected fT’s according to several user’s decisions. The well-formed aG’s can be considered as “patriarchs” of their own lineages, and are stored in a special matrix (LEXICON), which comprises the basis for the operations in the third program, which is dedicated to the processes of developing variation of second order, or else, to yield concrete offspring, by similar application of production rules (including “mutational” ones, responsible to random, and almost imperceptible transformations) at the pre-produced aG’s. The resulting forms are labeled as theorem-groups (tG’s) and are also stored in the LEXICON matrix, “genealogically” organized by a special index, the GödelVector. The method has already been applied to the composition of several pieces, of distinct instrumentations, characters and styles, by providing a large number of variants that can be used by the composer melodically and harmonically.

Keywords: genetic algorithms, developing variation, Grundgestalt, organic musical construction, algorithmic composition

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History and Perspectives on Rhythmic Tiling Canons

Hélianthe Caure

Dan Tudor Vuza introduced the musical concept of rhythmic tiling canons in the nineties as the art of filling the time axis with some finite rhythmic patterns and their translations, without the superposition of onsets. This extensive work has been done without the awareness of Hajós's 1949 paper, which introduced the mathematical concept of group factorizations while solving the Minkowski conjecture. Both of these papers have been linked as two distinct approaches to solving the same problem: tiling the time line. Its beauty comes from the variety of its exploration methods. Besides the use of both musical and mathematical points of view to understand it, the age of the problem and its mathematical implications have made it a crossroads of many mathematical fields.

For sixty years, numerous mathematical tools have been used, with constant progress to the exhaustive understanding, though never attained, of how and when a given pattern would tile. The history and modern research on the subject are fascinating but extensive and intricate. Mathematical fields as diverse as analysis (for instance, the use of the Fourier transform is one of the main modern research areas), computational models (in the fifties, de Bruijn introduced the use of trees for the study of quasi periodic tiling), or even combinatory have been used. More recent works are trying new approaches through field theory and set theory. The clear and thorough display on the poster of every major step made on the subject, and what mathematical tools have been used, would be enriched with their chronological and logical links, and the perspectives they open. As rhythmic tiling canon is still a domain of rich enough interest to be one of the main thematic sessions of the Mathemusical Conversations, but is very dense, such a summary would be welcome for any interested mathematicians.

The benefit of a poster presentation, compared to a talk, is the geometrical and musical representations of rhythmic tiling canons. These make this difficult problem musician-friendly and really easy to perceive and understand for non-mathematicians. Composers have been developing these representations while using rhythmic canons as composition tools, turning them progressively into a powerful aid for mathematics education. A conference for a large audience on tiling and music was held in Montpellier (France) in September 2014, which was a formidable opportunity to bring mathematics to everyone with the help of music.

There is currently no reference guide for interested researchers and musicians on the complex field of rhythmic tiling canons. Making one in a visual way would allow using it also as a pedagogic tool, intriguing for the well-informed as well as the general public.

Keywords: rhythmic tiling canons, mathematical models, representations, history

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Complexity of Musical Patterns

Yuping Ren and Charo I. Del Genio

In this project we analyse music using pitch and time information by employing a network projection. Each network represents the music played by a single instrument; multi-layer networks are constructed for ensemble music pieces. The nodes in the networks correspond to pitch and duration pairs, and the edges are added when the nodes are connected in the music as a time sequence. The statistics of such a network, such as adjacency matrices, degree distributions, clustering coefficients and centralities are calculated and plotted. Through the application to a variety of music pieces, we characterize network observations in which exponential, power-law distributions are found, implying the robustness of musical structures. The number of components and the loop structures of the networks are studied by employing the topological data analysis (TDA) method. Using TDA, a modern data-driven idea of persistent homology, we calculate the Betti-number barcode intervals and plot the persistence diagrams of individual musical networks, using weights for filtration. From the annotated barcode intervals, we can retrieve musical patterns frequently used by the composer. The difference between the persistences are summarised by plotting the distribution of interval groupings by varying music and varying instruments. Compared with results from random, complete, scale-free and small world networks, we show salient differences between the structure of those networks and the music networks.

Keywords: music, network, TDA (Topological data analysis)

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Generating Two-Part Inventions Using a Machine Learning Approach

Somnuk Phon-Amnuaisuk

A two-part invention is a style of musical composition that has two simultaneous independent melody lines. Both melody lines are written as a counterpoint where each line expresses independent melodic and rhythmic characteristics. However, logical harmonic progression emerges from both independent lines. Traditionally, music students learn rules for writing counterpoints. These rules are patterns extracted by humans and form an essential theoretical music knowledge.

In this poster, we discuss a computational model that learns the prominent characteristics of two-part inventions from real world examples and then use the learned model to generate new two-part inventions. The crucial points in learning knowledge of structural patterns in music concerns the following important questions: (i) what kind of knowledge is extracted? and (ii) how this knowledge is exploited?

Algorithmic composition has been investigated by AI-music researchers and various computation techniques have been employed to model this creative process. Music knowledge has been abstracted at a symbolic level in which music theorists converse and a computational model such as a rule-based system can exploit this knowledge. Music knowledge has also been exploited using other soft-computing approaches.

Here, data mining techniques are employed to extract useful patterns from examples, at the abstraction level of melody, harmony and rhythmic patterns. These patterns are summarised into constraints which will be used by SARSA, a reinforcement learning model. SARSA learns an optimal policy by sampling the state space to estimate the utility of state-action pairs Q(s,a):

Q(s,a) ← Q(s,a) + a[r+γQ(s',a' )-Q(s,a)],

where a is the learning rate and γ is the discount rate. By carefully selecting the representation of states, actions, rules and contexts, a complex problem such as algorithmic composition could be dealt with and reasonable output is obtained with comparatively less effort.

Keywords: algorithmic composition, data mining, reinforcement learning, SARSA

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A Study of Sound Micro-Organization Random Processes for Perception Purposes

Jocelyne Kiss and Serge Lacasse

Organizing micro events for sound composition in order to consider the user’s perception is the question we would like to discuss here in the context of building real-time automatons. More specifically, we wish to propose the application of a random function that could provide a large range of data (Haramoto, 2008) and at the same time participate in the process of organizing sound in such a way that it provides an impression of polarization (Forbes, 2012). Random functions are by default not efficient and do not display real aleatory function (Goldreich, 1986). (Also these randomisations do not provide the impression of organization.) Rather, it only produces a display of data. We are looking for a function that could both provide a huge bank of data and constrain this data within a given pool that would give the impression to the user that an organisation is at play (Loy, 2006). Organizing the sound in order to produce an impression that an orientation is given is an important factor in artistic applications, even, and especially at a micro level. Indeed, the random effect gives most of the time the opposite impression of an organisation. So if we could introduce, in the same process, both randomization and an impression of direction, this type of function would be closer to a musical way of thinking (Kugel, 1990)1. In previous studies we used strange attractors (Kiss, 2010, 2003) in order to build a similar impression. However, depending on the kind of data used, this system fails to come back to “the same point” quickly enough to get noticeable (and thus to give a sense of organization). In this paper, we propose instead to have recourse to a Brownian motion construction in dimension 2 (Legall, 1986). In dimension 3, the system would be great for randomization only. In dimension 2 (Revuz & Yor, 1991), the movement will eventually have to come back to the starting zone. Admittedly, the time variable is a challenge; but it can be resolved by considering it as a variable that could be scaled independently. Controlling this factor allows us to control the system so it gives an impression of polarization within the confine of a given effect or process. We will present Pure Data building blocks using few patches as examples in automatic music composition.

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1We choose the proposition of Revuz and Yor (1991) in order to build a Brownian motion β. This model is based on Fourier series such as for the time interval t ∈ [0,1] and the variables Nk, Nk", in which all the coefficients are independent Gaussian, we have: β_t = tN_0 + ∑_(k=1)^(+∞) ((√2/2πk)(Nk cos(2πkt-1) + Nk' sin(2πkt)).

Keywords: Brownian motion, sound micro-organization, random processes, perception, composition

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Coarsening Maps as Mathematical Foundations of Reductive Analysis

Immanuel Albecht, Martin Rohrmeier, and Stefan Schmidt

Reductive forms of analysis have been central in music research and constitute a topic that bridges music theory, music cognition, computational and mathematical approaches to music. Originating from Schenkerian analysis (Schenker, 1935), a number of approaches of reductive analysis have been proposed based on harmonic, voice-leading or contrapuntal representations (e.g. Lerdahl & Jackendoff, 1984; Keiler, 1978, 1983; Narmour, 1992). While many of the music theoretical approaches remained largely formally underspecified, several formalizations were proposed recently that used formal languages and related computational approaches (e.g. Steedman, 1996; Rohrmeier, 2007, 2011; De Haas et al, 2010; De Haas, 2012; Marsden, 2011; Katz & Pesetsky, submitted; Hamanaka et al, 2006; Granroth-Wilding & Steedman, 2013).

Our contribution analyses the concise algebraic foundations underlying reductive analyses. We may assume that events in musical time cannot take place prior to themselves and that if event A takes place prior to B, which takes place prior to C, then A also takes place prior to C (see also: Marsden, 2000). We propose the notion of coarsening maps between irreflexive and transitive musical time relations as a device that allows one to structure pieces of music hierarchically in order to analyze their deep structure (abstracting from harmonic or other building block choices) in a way arising/derived naturally from the properties of the perception of time and the elapsing of non-synchronous events. In this context, coarsening maps may further be regarded as the pure mathematical concept underlying the common process of combining contiguous musical elements into bundles with a common surface or function as it is done for harmonic analysis on different levels of granularity. Therefore, a complete analysis ranging from the smallest musical elements to the most rough reduction of large parts of a musical piece gives rise to/specifies an underlying classification tree that coincides with a chain of musical time relations that are linked by coarsening maps, and an annotation of that underlying coarsened time relation.

Further employing an appropriate scoring function that scores the grammaticality, coherence or preferability according to style-specific or cognitive constraints for each piece of information that is annotated to the coarsened time relation, it is possible to compute an order of preferability of different given reductive analyses of a piece of music.

Finally, all such chains of coarsened time relations with respect to any fixed piece of music have the mathematical structure of a complete lattice, which is due to the relative independence of the finer grained analyses of smaller musical elements that are too far apart of each other in order to be necessarily correlated. In a computational context, this structural feature of locality may be exploited in order to find the best possible analysis, i.e., the chain of coarsened time relations together with an annotation, with regard to some given scoring function; and it also may be exploited in order to find an estimate of a scoring function that maximizes the score of a set of given analyses of musical pieces.

Keywords: reductive analysis, coarsening maps, musical time relations

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A Creative Patterning of Pitch-Class Sets: A Mathemusical Approach

Stace Constantinou

Composers such as Arnold Schoenberg, Milton Babbitt, and Elliott Carter have created a diverse range of musical material using combinatorial methods for the organising of pitch material, and the resulting harmonic content. Indeed, Elliott Carter has published a book detailing his pre-compositional work called Harmony Book.

The Pitch-Class sets, as provided by Allen Forte in his book The Structure of Atonal Music, offer a comprehensive collection of pitch materials for composers to work with, in realising harmonic content. But they offer little by way of guidance, in organising combinations of patterns within the sets. This is perhaps because the number of possible combinations is so large that such a work would be incomprehensible. Yet pitch order may be an important factor in the realisation of novel musical compositions. The number of possible combinations for any given hexachord is 720, for example. And there are fifty unique hexachords in Forte’s collection. Writing out each permutation would mean ordering each of the 3600 individual patterns, then deciphering which to use. Selecting the most compelling set of patterns is a choice, perhaps, best left to the individual composer, who may find that certain permutations, whilst mathematically discrete, sound too similar, or insignificantly (musically) varied, to be of use.

Using ideas developed during my recently completed doctorate study, I reveal an approach for filtering the large number of possible permutations of Pitch-Class sets. By finding patterns within the Pitch-Class sets, then applying these within the music, it is possible to filter unwanted, or unnecessary, permutations, thus reducing the number of possibilities to a usable sample. Such an approach can also be extended to include microtones, by distorting the existing patterns by microtonal steps.

The actual patterns used to filter and distort the Pitch-Class sets may vary; indeed, part of the rationale for this paper is to encourage experimentation with different types of pattern. Nevertheless, as a simple example, consider any four-note Pitch-Class set and the pattern of prime numbers: 2 to 23. Given that there are 24 (four-note) permutations, these may be filtered down, using the prime numbers 2, 3, 5, 7, 11, 13, 17, 19 and 23, to nine. In addition, each of the nine selected patterns may be distorted, also using prime numbers, e.g. 2, 3, 5, and 7. Whereby the prime number represents the number of microtonal steps of distortion applied in, say, one-eighth tones.

The purpose of this paper is to share my findings with a view to encourage critical feedback, and offer the possibility of collaborating with other composers, theorists, mathematicians, and programmers. For example, the processes I outline may perhaps be computerised.

Keywords: patterning, creative, Pitch-Class sets, filtering, distortion, prime numbers, permutations, combinations

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An Automatic Performance Analysis System Focused on Vibrato using Filter Diagonalization Method and Statistics

Luwei Yang, Khalid Z. Rajab, and Elaine Chew

Different performers display different interpretations of the same piece of music. For scripted music, these differences lie mainly in the expressive dimensions, such as tempo, dynamics, vibrato and portamento. The mathematics underlying these expressive nuances can further performer-style and cross-cultural analyses. Our work focuses on automatic detection and analysis of vibratos, a fundamental expressive device, and systematic analysis of their mathematical properties.

We propose a probabilistic automatic vibrato performance analysis system that can greatly speed systematic research and advance performance pedagogy. This system features the end-to-end process from audio signal vibrato extraction to vibrato analysis statistics output. The system consists of two modules: vibrato detection, and vibrato analysis. The vibrato detection module features a novel application of the Filter Diagonalization Method (FDM) to frame-wise vibrato detection. The FDM’s advantage over the traditional Fourier transform is that its output remains robust over short time frames, allowing frame sizes to be small enough to accurately identify local vibrato characteristics and pinpoint vibrato boundaries. For each frame, the FDM returns the frequency and amplitude of a given number of harmonics. We apply and compare two decision-making methods—Decision Tree and Bayes’ Rule—to determine vibrato existence. The Decision Tree requires the frequency and amplitude range thresholds to be set manually. These two threshold ranges can be set according to the known vibrato statistics. For example, previous studies showed that violin playing results in higher vibrato rates and significantly smaller vibrato extents than voice and erhu. Bayes’ Rule takes advantage of the estimated probability density distribution of the frequency and amplitude of the FDM output. Once vibrato existence has been confirmed, the vibrato analysis module computes and returns the statistics of rate, extent and sinusoid similarity.

The vibrato detection results show that the FDM (F-measure around 0.88 for Bayes’ Rule) significantly outperforms the FFT for short time frames <0.25s, the FFT having insufficient frequency resolution to provide an answer in many cases. The FDM is further compared to a recent cross-correlation-based vibrato detection method and is shown to consistently perform better for both short (<12s) and long (50-60s) audio samples. The results also show that Bayes’ Rule usually has better performance than the Decision Tree. For vibrato analysis, an evaluation on 1800 synthesized vibratos show high F-measure values for vibrato detection (0.9873), and high correct rates for vibrato parameter extraction (99.53% for the vibrato rate, 98.43% for the extent, and 96.85% for the sinusoid similarity.) The system is also applied to a real dataset consisting of six recorded solo erhu and two unaccompanied violin performances to demonstrate the differences in vibrato characteristics between erhu and violin. The system’s outputs are compared to, and shown to be consistent with, those produced using manual segmentation in our previous research, thus demonstrating the feasibility of this approach to vibrato performance analysis.

Keywords: vibrato detection, performance analysis, filter diagonalization method

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Partitional Analysis and Melodic Texture

Pauxy Gentil-Nunes

Partitional Analysis (PA; Gentil-Nunes & Carvalho, 2003) is an original proposal of mediation between mathematical abstractions derived from the Theory of Integer Partitions (Euler, 1748; Andrews 1984; Andrews & Ericksson, 2004) and compositional theories and practices. Its main goal is the study of compositional games and has been used in the pedagogy of composition and creation of new pieces. The abstractions involved allow the biunivocal homology between heterogeneous fields, like texture, orchestration, timbre, melodic structures and spatialization. It also provides an exhaustive taxonomy for the fields as well as its topology and metrics. The point of departure of Partitional Analysis, inspired by the work of Wallace Berry (1976), is the consideration of binary relations between concurrent agents of a musical plot. The relations are categorized in collaboration and contraposition types, according to given criteria (congruence between time points and duration, belonging to a line inside a melody, proximity of timbre or orchestral group, spatial location in the stage, and so on). This categorization underlies the constitution of the partitions and at the same time leads to the establishment of the agglomeration and dispersion indices (a, d). Two graphs are constructed from the indices. The partitiogram, where the indices are plotted one against the other, constitutes a phase space and generates a graph called the Partitional Young Lattice (PYL), where all the adjacency relations are categorized and assessed. The indexogram arises from plotting both indices against a time axis and presents the temporal progression of partitions, in a wave-like representation. Linear Partitioning is the application of PA to melodic texture, based on fundamental Schenkerian concepts of melodic conjunction and disjunction and the concept of line as an internal compounding element of melodic structures. Although this concept has been central to the work of several authors, like Hindemith (1937), Costère (1954), Meyer (1973), Lester (1982) and Narmour (1992), among others, there has not been so far a quantitative and relational evaluation of lines under the framework of melodic texture. The categorization of linear types (line, arpeggiation and compound melody, for instance) in fact can be represented by partitions, following the criteria of independence, promoted by the development of adjacent conjuctions, and interdependence, created by consecutive adjacent disjunction. Through Linear Partitioning, six basic operations are found in linear structures: portamento, opening, activation, prolongation, convergence and closure. The analysis of the interactions between these procedures leads to the formulation of linear partitioning, the exhaustive taxonomy of melodic behaviors and the possibility of its handling as a formal construct. Some functions were developed inside the MATLAB environment and are integrated in the author’s software Parsemat® for Windows. The module Partlin® is presented as a button in the interface and can generate graphs (partitiogram and indexogram) and tables. The research also covers some sub-projects including the reassessment of melodic analysis of other authors, mainly Meyer and Narmour; analysis of solo pieces from Darmstadt period; and expansion of the proposal to analysis of contour in general.

Keywords: Partitional Analysis, melodic texture, Theory of Integer Partitions, musical composition, musical analysis

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Textural Contour: Relations Between Textural Progressions and Melodic Contour in Claude Debussy’s Prélude à l'après-midi d'un faune

Daniel Moreira de Sousa and Pauxy Gentil-Nunes

This paper proposes the extension of basic principles of the Musical Contour Theory (MCT), developed mainly by Robert Morris (1987), to the textural domain. MCT is constructed from abstractions of levels, ordering them from 0 (lowest level) up to n–1 (where n is the number of different levels in the structure), and assessing the relations between the levels. The transposal of such abstraction to the field of texture can reveal the movements of textural complexity, toward the measuring and comparing between different textural configurations. This present proposal is called Textural Contour. The methodological tools and concepts provided by the Partitional Analysis (PA; Gentil-Nunes & Carvalho, 2003) can handle textural organization using numeric representations (partitions) that express the relations of compounding musical elements in the texture, allowing for its measurement. The PA is a new and important analytical tool that makes possible the examining of musical texture using Wallace Berry’s proposal (1987) and the Theory of Integer Partitions (Euler, 1748; Andrews, 1984), constituting a partially ordered organization that relates partitions by its transformational process in a taxonomy that encompasses all partitions from 1 to a determined number using mainly the inclusion relation (Andrews & Ericksson, 2004). This organization is presented using a Hasse Diagram based on the Partitional Young Lattice (PYL). Such organization allows the creating of a contour based on the adjacency relations between partitions and its complexity levels, revealed by the internal configuration, therefore establishing textural progression curves. The Textural Contour provides an abstraction of textural progressions enabling a broader view of the progressions and relations of texture. This contour can be manipulated through canonic operations from Musical Contour Theory (inversion, retrogradation, and retrograded inversion) and can also be analyzed by Michael Friedmann’s (1987) descriptive tools for analysis of contour. Two computational tools (Partitional Operators and Contour Analyzer) were developed to facilitate the systematization and application of this proposal both as an analytical tool and as a compositional possibility. This paper concludes with an application of methodological use of the Textural Contour in the analysis of Claude Debussy’s Prélude à l'après-midi d'un faune, comparing possible relations between the melodic contour of the flute introduction and the overall textural progression.

Keywords: Musical Contour Theory, Partitional Analysis, textural contour, melodic contour, Claude Debussy’s Prélude à l'après-midi d'un faune

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Validating an Optimisation Technique for Estimating the Focus of a Listener

Jordan B. L. Smith and Elaine Chew

Many musicologists and music psychologists are interested in the phenomenon of attention: what do listeners pay attention to while listening to music, and how does this affect their perception of musical structure? Studies of disagreements between listeners’ analyses have found that such differences can usually be attributed to (among other factors) listeners paying attention to different musical attributes than one another, but repeating such findings at a large scale remains a challenge. Alas, while those studying visual attention have the benefit of eye-tracking devices, there is no comparable ear-tracking technology, so auditory attention cannot be directly observed.

With this in mind, we proposed in previous work an algorithm for estimating what a listener has focused on in the music by examining how the listener’s analysis, represented as a set of structural boundary annotations and segment labels, relates to features extracted from a recording. Self-similarity matrices (SSMs) consist of the computed similarity of all pairs of points in a recording and are widely used to visualise and estimate structural analyses. Our algorithm uses quadratic programming to minimise the distance between the listener’s analysis and a set of SSMs based on different musical features. By decomposing the matrices into section-wide blocks based on the listener’s analysis, the optimal solution estimates the relative attention paid to each feature in each section.

As an extension to this work, we validate our algorithm by applying it to a large set of artificial stimuli. The stimuli have been systematically composed in order to have different features (harmony, melody, rhythm and timbre) change at different times, creating musical passages with ambiguous forms: for example, a passage may have form AAB with respect to rhythm, and form ABB with respect to melody. We demonstrate that our algorithm is able to correctly estimate the feature that best explains a given analysis.

These stimuli were previously used in an experiment showing that paying attention to one feature led listeners to choose the analysis matching that feature more often. The present results establish that our algorithm could have accurately predicted the attentional focus of the listeners based on their responses, and hence demonstrates that our algorithm is a valid means of estimating a listener’s focus based on their analysis.

Keywords: formal analysis, structure, segmentation, quadratic programming, attention

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Linking Dynamic Score Markings with Statistical Properties of Perceived Loudness Changes in Audio Recordings

Katerina Kosta, Oscar F. Bandtlow and Elaine Chew

Expressivity is an essential part of musical performance. Although many computational models and probabilistic and statistical functions have been established for modelling tempo and intensity, and changes in these properties, few models attempt to map these parameters to an ontology for musical expression. Our study aims to discover the connections between performed loudness in audio recordings and dynamic representations in the score by analysing the mathematical essence of their connections. Dynamic markings prescribe loudness levels in performance. In a musical piece, the same dynamic markings may not always correspond to the same loudness levels. Analysing changes in audio dynamics forms an important part of understanding the meaning of dynamic markings as employed in the score such as p (piano) and f (forte) and their practical manifestations in the audio signal. We posit that reasons for significant differences in dynamic values between instances of the same dynamic markings can be formalized as rules that take into account the local sequential context for the dynamic marking.

Large-scale analysis of the practical meanings of dynamic markings is hampered by a lack of accurate and efficient methods for automatic beat annotation and segmentation by loudness. Our study uses as data the collection of recorded performances in the Mazurka project (www.mazurka.org.uk), which offers a large collection of interpretations of the same Chopin piano pieces. First, we propose and test a multiple-file alignment method that allows score-beat positions to be propagated from one annotated file to multiple recordings so as to align beats in the score to loudness values (in sones). Next, we present two complementary studies on dynamic markings and loudness levels.

The first study considers the loudness levels in the music at score positions corresponding to dynamic markings. Statistical analysis of the data reveal how the absolute meanings of dynamic markings change, depending on the intended (score defined) and projected (actual) loudness levels of the surrounding context in the particular music performed. Next, we propose a method for abstracting the shaping of the dynamics through score time within the performances, by presenting the results as dynamic landscapes. The analysis shows that while a considerable number of performances are unique in their dynamic landscapes, nevertheless specific patterns can be discerned from groups of recordings. These patterns help us create an intended dynamic marking map, represented by a loudness envelope, for each musical piece. The resulting map represents the interpretation of dynamic markings as the piece progresses; conversely, the map is crucial for understanding dynamic changes in the audio as they correspond to intended loudness.

The second study seeks to establish the mappings from performed loudness to score features. We introduce a novel use of the Pruned Exact Linear Time (PELT) change-point detection statistical algorithm to segment loudness information into different dynamic levels. Change-point algorithms have been applied to domains such as climatology, bioinformatics, finance, oceanography, medical imaging, and birdsong detection. To our knowledge, changes in audio dynamics have not been considered as data to be analyzed in such a fashion. We show that significant dynamic score markings do indeed correspond to change points. Also, change points in score positions without dynamic marking serve to enhance the expressive representation of musical structure, such as the ending of a musical phrase.

The findings of the two studies provide key steps towards dynamic marking transcription, dynamic level synthesis and shaping, and performance studies on large corpora of recordings.

Keywords: performance, analysis, statistics

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Tritonet - A New Approach to Music Harmony

Tolga Zafer Özdemir

Tritonet is a new concept in music theory. Based on the circle of fifths, the geometric simplicity of its mechanism reveals a clear observation of harmonic relationships in the music. It opens a new window for a composer to think about harmony in terms of tendency and resistance.

Harmony, in a broader sense, is a science which deals with the relationships between frequencies. Keeping the term in mind with this manner, music theory lessons can be expected to explain the harmonicity and inharmonicity of sounds, which is aesthetics-free. It should deal with the laws of physics and explore the duality between tendency and resistance in sound motions.

The circle of fifths is one of the best and probably the oldest visual helper for identifying the relationship in between the sounds. Having twelve notes placed at a perfect fifth apart from each other, the circle of fifths looks like a watch. Adding two new parts, anchor and the nightline, schema gains symmetry and ensures the correct note-naming.

With the help of system:

Live performance with software based on the Tritonet concept can be watched in demos:

vimeo.com/103606124

vimeo.com/95174714

vimeo.com/tolgaozdemir/tritonetmindmusic

Keywords: harmony, geometry, Tritonet

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Anthropomorphic Visualisation of Melodic Contour Based on Lissajous Figures

Tzu En Ngiao

Scientific evidence supports the existence of psycho-cognitive mechanisms for perceiving equivalence in spatio-temporality between the musical and visual modalities. We also have a deeply ingrained tendency to anticipate and apprehend dramatic and causal visual narratives involving anthropomorphic subjects and imageries. Composers have long seized upon the isomorphism between visual contours and musical contours, and attempted to derive melodic lines by tracing the visual profiles of thematically relevant subjects, underlying the creative potential of interrogating the difference between abstract pitch contours and anthropomorphic forms. Hence it would be interesting to develop an approach to pitch-spatial music visualisation where anthropomorphic visual feedback could be worked to the advantage of encouraging and delineating the processual creation of melodies. In addressing anthropomorphic forms, it is inevitable that we should further narrow in on the human facial form. Without the face—distinct facial profiles and character identity—figures of the human body and limbs are hardly more anthropomorphic than abstract representations that trace formal outlines quite similar to them. An anthropomorphic music visualisation approach could be adapted to the effort of setting up the right phenomenological conditions in comprovisation environments for coaxing a directional creative process of improvising a melody in parallel to sketching a face. The potential of seeing a human face could be the carrot (or stick) for improvisers to create melodies. Unlike most music visualisation approaches which seek to map individual musical pitches to visual elements or parameters in a syntagmatic fashion—whereby each musical pitch is parsed for visual mapping without regard to the larger context such as its relation to preceding pitches—the visualisation approach for processual melodic creation would need to adopt a more paradigmatic strategy as the formation of a melodic contour rests on the succession of a collection of music pitches. In this paradigmatic audiovisual mapping approach, melodic lines could be broken down into contour components approximating that of a sinusoidal waveform. To construct a visual homologue of a human face, there needs to be a way of aggregating these abstracted sinusoidal components into graphical entities that approximate the spatial ratios characterising the face, much like the polygon mesh figures used in computer graphics. Lissajous figures may provide the answer to this end. They can be generated from the fusion of two sinusoidal components, and this fusion could be performed continuously on consecutive sinusoidal components abstracted from melodic contours. We could capitalise on the different shapes and sizes of Lissajous figures to construct spatial ratios for the depiction of a variety of facial identities. The more asymmetrical the alignment between two sinusoidal components, the more complex the Lissajous mesh figure would be. This property becomes highly relevant to our intended mapping of highly melodic entities to the human face. Highly symmetrical and repetitive sinusoidal components or flat linear contours abstracted from less melodic (or non- melodic) figures—such as arpeggiated figure, ascending or descending gestures or long sustained tones—would be more inclined to produce simpler, or less mesh-like, hence less anthropomorphic Lissajous figures.

Keywords: music visualisation, pitch-spatial visualisation, Lissajous figures, sinusoidal waveforms, anthropomorphic, comprovisation environments

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The Opera Experience. Applying a Multimodal Theory of Music Performances

Aishwarya Nair and Patrick Becker

The traditional music theory is a text-based affair: to gain insights in the structure of a musical work it solely analyses the notated elements of music. Yet, experiencing music in everyday life or even in a concert performance has little to do with the score. It is rather a multimodal experience to visit an opera, go for a rock concert or to listen to a jazz ensemble for example. To think about music theory as a theory which describes the phenomenon of performed music at a whole, one has to leave the text-based view and add the visual, the auditive plus the lyrical dimension of a music performance.

This approach refers to music as a Gesamtkunstwerk, a fundamental form of art. At this point, it meets the theory of conceptual blending, since the different sensual stimuli merge together to form a unique band. For a new music theory, based on musical performances, this concept is a key for analyzing and explaining the multimodal perspective on music.

It was Richard Wagner who basically coined the term of Gesamtkunstwerk and put his main works under this aesthetic maxim. Therefore we are taking an example of his opera Parsifal to show how music theory can benefit from taking a new perspective on musical experiences. From a DVD-production by Daniel Barenboim and the Staatskapelle Berlin in 1992 of Parsifal, we took the aria of Kundry “Ich sah das Kind an seiner Mutter Brust” in which she reveals Parsifal’s origin and the fate of his parents. Parsifal resists her following seduction and becomes “welthellsichtig” instead; he turns from being a fool to a wise man, being capable of fulfilling his destiny. Therefore, Kundry’s lamentation is a key scene for the whole opera, its turning point, dramatically and musically highly interesting because of its connection of Kundry’s seduction, the reveal of Parsifal’s origin and the subsequent destruction of Klingsor and his magic garden.

In our analysis we are broadening our horizon and implementing the visual, auditive and lyrical dimension of the excerpt, showing how to achieve a higher understanding of the scene through a multimodal approach. We will show how this approach is capable of theorizing Wagner’s complex music which is strongly interconnected to the lyrical and dramatical dimension.

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Mathesis as the Philosophy of the Beauty of Music

Rima Povilioniene

The idea of the art of sounds based on mathematics that is referred to by Gioseffo Zarlino as numerus sonorus or numerus in sono has been developed for centuries. It is called perhaps the longest-lasting interdisciplinary dialogue. The logic of the structure of music is identified with the logic of a mathematical equation, and a musical composition is defined as “a dramatically, passionately told mathematical story” (Hammel, 1997). In the sphere of musical research the conception of musica mathematica is in essence related to one of the trends of music – attributing the theory of music to science and classifying as musica theorica / theoretica / contemplativa / speculative / arithmetica.

Most likely one of the answers to the question of where vitality of the idea of interaction between music and mathematics lies is the perception that mathematics is the principal cause and source of the all-embracing beauty. The idea of a mathematically substantiated world was developed as far back as antiquity and was in line with the outlook of the later epochs too: Galileo Galilei called mathematics the ABC by means of which God described the world, the Hungarian mathematician Paul Erdős spoke about the imaginary divine book which contained the most beautiful mathematical proofs, and Herbert E. Huntley who investigated the beauty of the Golden Section stated that the feeling of a scientist who is proving a theorem is identical to admiring a masterpiece of art because this is also creative work (Huntley, 1970). Alexei Losev related the definition of beauty to the number while stating that beauty was something “impersonal”—neither spirit nor personality but a non-qualitative structure—i.e., the number (“the primary kernel”, “the clasp of the entire construction”; see Losev, 1963, and Losev, 1999).

The attitude to mathematics as the art of the beauty of numbers had an effect on the environment too: operations with numbers, regularities of symmetry and proportions have become beauty formulas in different spheres of art. The aesthetised conception of mathematics made the original imprints of mathematicised composition of music in every epoch especially concentrating in the practice of composing contemporary music. The 20th-21st century music composing process is investigated as a compendium of experiences of the earlier epochs and traditions of musical numerology (e.g. the models of additivity, progressions, symmetry and proportions, the application of combinatorics of music, kabbalistic numbers, Christian numerology, the creation of musical messages encoded in numbers, etc.) as well as the integration of innovative ideas that are based on very formalised idioms of mathematical nature (such as complicated mathematical formulas and models, mathematical theories, fractals, chaos, groups, probabilities and others). The use of computer possibilities transforming the algorithm processes into the sphere of composing contemporary music extended considerably the spectrum of generating ideas and opened wider the immense layers of the implementation of the musica mathematica phenomenon. On the other hand, the practice of computerised composing of music opened the way to innovative analytical approaches as well as rising of the metacomposition term, which defines what is more than a traditional musical composition.

Keywords: mathesis as the background of music, numerus sonorous, beauty of numbers

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An Empirical Evaluation of Musical Distance using a Computational Model

Goktug T. Cinar, Jose C. Principe

Concepts of musical distance are top-down concepts, i.e. proposed by experts based on music theoretic concepts. It would be very interesting to link these concepts to the representations that resemble the ones found in the auditory cortex. To achieve this we should use a bottom-up approach to get the representations from acoustic signals, otherwise we will never know if the suggested organization is contained in the musical structure. This requires a self-organizing system that will extract such a structure from the acoustic signals, with as little number of free parameters as possible. Our aim in this work is to utilize the representation space created by the Hierarchical Linear Dynamical System to obtain distances between notes/triads.

The Hierarchical Linear Dynamical System (HLDS) was introduced in Cinar and Principe (2014) as a self-organizing architecture that can cluster time series. The idea of a nested HLDS is having a model that would consist of one measurement equation and multiple state transition equations. The system is nested in the sense that each state transition equation creates the causes/states that would drive the lower layer. This introduces a top-down flow of information. In the nested HLDS we drive the system bottom-up by the observations, and top-down by the states with an apriori constraint that provides a local stationary context for the dynamics since a sound lasts mostly unchanged for a short period of time. The design of the top layer stabilizes the input dynamics and provides a representation space. An important characteristic of the methodology is that it is adaptive and self-organizing, i.e. previous exposure to the acoustic input is the only requirement for learning and recognition.

In recent experiments we have shown that the representation space that the algorithm learns preserves the organization found in the musical scales for monophonic notes, and exhibits the properties suggested in the theory of efficient chromatic voice leading and Neo-Riemannian theory for triads. The models trained on isolated pitches place the pitches that are direct neighbors in the Riemannian Tonnetz close to each other. This suggests that the self-organizing model also puts emphasis on the harmonic relationship between the pitches. The models trained on isolated triads place triads with smaller voice leading distances close to each other. This is in line with the findings obtained in the monophonic case as shared harmonics between the triads increase as the voice leading distance decreases. We demonstrate these behaviors both in a case-by-case manner and by averaged results obtained by studying multiple independently trained models. These findings show that the self-organization of the HLDS, which is learned from acoustic signals in an unsupervised manner, resembles and supports music theoretic concepts of distance.

Keywords: musical distance, self-organizing models, linear dynamical systems