A Multi-Level Tonal Interval Space for Modelling Pitch Relatedness and Musical Consonance
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Detecting Harmonic Change in Musical Audio
Detecting Harmonic Change In Musical Audio Christopher Harte and Mark Sandler Martin Gasser Centre for Digital Music Austrian Research Institute for Artificial Queen Mary, University of London Intelligence(OF¨ AI) London, UK Freyung 6/6, 1010 Vienna, Austria [email protected] [email protected] ABSTRACT We propose a novel method for detecting changes in the harmonic content of musical audio signals. Our method uses a new model for Equal Tempered Pitch Class Space. This model maps 12-bin chroma vectors to the interior space of a 6-D polytope; pitch classes are mapped onto the vertices of this polytope. Close harmonic relations such as fifths and thirds appear as small Euclidian distances. We calculate the Euclidian distance between analysis frames n + 1 and n − 1 to develop a harmonic change measure for frame n. A peak in the detection function denotes a transi- tion from one harmonically stable region to another. Initial experiments show that the algorithm can successfully detect harmonic changes such as chord boundaries in polyphonic audio recordings. Categories and Subject Descriptors J.0 [Computer Applications]: General General Terms Figure 1: Flow Diagram of HCDF System. Algorithms, Theory has many potential applications in the segmentation of au- dio signals, particularly as a preprocessing stage for further Keywords harmonic recognition and classification algorithms. The pri- Pitch Space, Harmonic, Segmentation, Music, Audio mary motivation for this work is for use in chord recogni- tion from audio. Used as a segmentation algorithm, this approach provides a good foundation for solving the general 1. INTRODUCTION chord recognition problem of needing to know the positions In this paper we introduce a novel process for detect- of chord boundaries in the audio data before being able to ing changes in the harmonic content of audio signals. -
Melodies in Space: Neural Processing of Musical Features A
Melodies in space: Neural processing of musical features A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Roger Edward Dumas IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Dr. Apostolos P. Georgopoulos, Adviser Dr. Scott D. Lipscomb, Co-adviser March 2013 © Roger Edward Dumas 2013 Table of Contents Table of Contents .......................................................................................................... i Abbreviations ........................................................................................................... xiv List of Tables ................................................................................................................. v List of Figures ............................................................................................................. vi List of Equations ...................................................................................................... xiii Chapter 1. Introduction ............................................................................................ 1 Melody & neuro-imaging .................................................................................................. 1 The MEG signal ................................................................................................................................. 3 Background ........................................................................................................................... 6 Melodic pitch -
Program and Abstracts for 2005 Meeting
Music Theory Society of New York State Annual Meeting Baruch College, CUNY New York, NY 9–10 April 2005 PROGRAM Saturday, 9 April 8:30–9:00 am Registration — Fifth floor, near Room 5150 9:00–10:30 am The Legacy of John Clough: A Panel Discussion 9:00–10:30 am TextMusic Relations in Wolf 10:30–12:00 pm The Legacy of John Clough: New Research Directions, Part I 10:30am–12:00 Scriabin pm 12:00–1:45 pm Lunch 1:45–3:15 pm Set Theory 1:45–3:15 pm The Legacy of John Clough: New Research Directions, Part II 3:20–;4:50 pm Revisiting Established Harmonic and Formal Models 3:20–4:50 pm Harmony and Sound Play 4:55 pm Business Meeting & Reception Sunday, 10 April 8:30–9:00 am Registration 9:00–12:00 pm Rhythm and Meter in Brahms 9:00–12:00 pm Music After 1950 12:00–1:30 pm Lunch 1:30–3:00 pm Chromaticism Program Committee: Steven Laitz (Eastman School of Music), chair; Poundie Burstein (ex officio), Martha Hyde (University of Buffalo, SUNY) Eric McKee (Pennsylvania State University); Rebecca Jemian (Ithaca College), and Alexandra Vojcic (Juilliard). MTSNYS Home Page | Conference Information Saturday, 9:00–10:30 am The Legacy of John Clough: A Panel Discussion Chair: Norman Carey (Eastman School of Music) Jack Douthett (University at Buffalo, SUNY) Nora Engebretsen (Bowling Green State University) Jonathan Kochavi (Swarthmore, PA) Norman Carey (Eastman School of Music) John Clough was a pioneer in the field of scale theory. -
Fourier Phase and Pitch-Class Sum
Fourier Phase and Pitch-Class Sum Dmitri Tymoczko1 and Jason Yust2(B) Author Proof 1 Princeton University, Princeton, NJ 08544, USA [email protected] 2 Boston University, Boston, MA 02215, USA [email protected] AQ1 Abstract. Music theorists have proposed two very different geometric models of musical objects, one based on voice leading and the other based on the Fourier transform. On the surface these models are completely different, but they converge in special cases, including many geometries that are of particular analytical interest. Keywords: Voice leading Fourier transform Tonal harmony Musical scales Chord geometry· · · · 1Introduction Early twenty-first century music theory explored a two-pronged generalization of traditional set theory. One prong situated sets and set-classes in continuous, non-Euclidean spaces whose paths represented voice leadings, or ways of mov- ing notes from one chord to another [4,13,16]. This endowed set theory with a contrapuntal aspect it had previously lacked, embedding its discrete entities in arobustlygeometricalcontext.AnotherpronginvolvedtheFouriertransform as applied to pitch-class distributions: this provided alternative coordinates for describing chords and set classes, coordinates that made manifest their harmonic content [1,3,8,10,19–21]. Harmonies could now be described in terms of their resemblance to various equal divisions of the octave, paradigmatic objects such as the augmented triad or diminished seventh chord. These coordinates also had a geometrical aspect, similar to yet distinct from voice-leading geometry. In this paper, we describe a new convergence between these two approaches. Specifically, we show that there exists a class of simple circular voice-leading spaces corresponding, in the case of n-note nearly even chords, to the nth Fourier “phase spaces.” An isomorphism of points exists for all chords regardless of struc- ture; when chords divide the octave evenly, we can extend the isomorphism to paths, which can then be interpreted as voice leadings. -
Generalized Tonnetze and Zeitnetz, and the Topology of Music Concepts
June 25, 2019 Journal of Mathematics and Music tonnetzTopologyRev Submitted exclusively to the Journal of Mathematics and Music Last compiled on June 25, 2019 Generalized Tonnetze and Zeitnetz, and the Topology of Music Concepts Jason Yust∗ School of Music, Boston University () The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geomet- rical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces, and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Ton- netze in that their objects are pitch-class distributions (real-valued weightings of the twelve pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. The present article examines how these du- plications can relate to important musical concepts such as key or pitch-height, and details a method of removing such redundancies and the resulting changes to the homology the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth-chords, scalar tetrachords, scales, etc., as well as Zeitnetze on a common types of cyclic rhythms or time- lines. -
Music and Operations Research - the Perfect Match?
This article grew out of an invited lecture at the University of Massachusetts, Amherst, Operations Research / Management Science Seminar Series organized by Anna Nagurney, 2005-2006 fellow at the Radcliffe Institute for Advanced Study, and the UMass - Amherst INFORMS Student Chapter. Music and Operations Research - The Perfect Match? By Elaine Chew Operations research (OR) is a field that prides itself in its be a musical Google, whereby one could retrieve music files by versatility in the mathematical modeling of complex problems to humming a melody or by providing an audio sample. find optimal or feasible solutions. It should come as no surprise Beyond such industrial interests in mathematical and that OR has much to offer in terms of solving problems in music computational modeling of music, such approaches to solving problems composition, analysis, and performance. Operations researchers in music analysis and in music making have their own scientific and have tackled problems in areas ranging from airline yield intellectual merit as a natural progression in the evolution of the management and computational finance, to computational disciplines of musicology, music theory, performance, composition, and biology and radiation oncology, to computational linguistics, and improvisation. Formal models of human capabilities in creating, in eclectic applications such as diet formulation, sports strategies analyzing, and reproducing music serve to further knowledge in human and chicken dominance behavior, so why not music? perception and cognition, and advance the state of the art in psychology I am an operations researcher and a musician. Over the past ten and neuroscience. By modeling music making and analysis, we gain a years, I have devoted much time to exploring, and building a career at, deeper understanding of the levels and kinds of human creativity the interface between music and operations research. -
Messiaen's Regard IV: Automatic Segmentation Using the Spiral Array
REGARDS ON TWO REGARDS BY MESSIAEN: AUTOMATIC SEGMENTATION USING THE SPIRAL ARRAY Elaine Chew University of Southern California Viterbi School of Engineering Epstein Department of Industrial and Systems Engineering Integrated Media Systems Center [email protected] ABSTRACT the same space and uses spatial points in the model’s interior to summarize and represent segments of music. Segmentation by pitch context is a fundamental process The array of pitch representations in the Spiral Array is in music cognition, and applies to both tonal and atonal akin to Longuet-Higgins’ Harmonic Network [7] and the music. This paper introduces a real-time, O(n), tonnetz of neo-Riemannian music theory [4]. The key algorithm for segmenting music automatically by pitch spirals, generated by mathematical aggregation, collection using the Spiral Array model. The correspond in structure to Krumhansl’s network of key segmentation algorithm is applied to Olivier Messiaen’s relations [5] and key representations in Lerdahl’s tonal Regard de la Vierge (Regard IV) and Regard des pitch space [6], each modeled using entirely different prophètes, des bergers et des Mages (Regard XVI) from approaches. his Vingt Regards sur l’Enfant Jésus. The algorithm The present segmentation algorithm, named Argus, is uses backward and forward windows at each point in the latest in a series of computational analysis techniques time to capture local pitch context in the recent past and utilizing the Spiral Array. The Spiral Array model has not-too-distant future. The content of each window is been used in the design of algorithms for key-finding [2] mapped to a spatial point, called the center of effect and off-line determining of key boundaries [3], among (c.e.), in the interior of the Spiral Array. -
The Tonal Diffusion Model.Transactions of the International Society for Music Information Retrieval,3(1), Pp
Lieck, R., et al. (2020). The Tonal Diffusion Model.Transactions of the International Society for Music Information Retrieval,3(1), pp. 7,60,5 153–164. DOI: https://doi.org/10.5334/tismir.46 RESEARCH The Tonal Diffusion Model Robert Lieck, Fabian C. Moss and Martin Rohrmeier Pitch-class distributions are of central relevance in music information retrieval, computational musicology and various other fields, such as music perception and cognition. However, despite their structure being closely related to the cognitively and musically relevant properties of a piece, many existing approaches treat pitch-class distributions as fixed templates. In this paper, we introduce the Tonal Diffusion Model, which provides a more structured and interpretable statistical model of pitch-class distributions by incorporating geometric and algebraic structures known from music theory as well as insights from music cognition. Our model explains the pitch-class distributions of musical pieces by assuming tones to be generated through a latent cognitive process on the Tonnetz, a well-established representation for harmonic relations. Specifically, we assume that all tones in a piece are generated by taking a sequence of interval steps on the Tonnetz starting from a unique tonal origin. We provide a description in terms of a Bayesian generative model and show how the latent variables and parameters can be efficiently inferred. The model is quantitatively evaluated on a corpus of 248 pieces from the Baroque, Classical, and Romantic era and describes the empirical pitch-class distributions more accurately than conventional template-based models. On three concrete musical examples, we demonstrate that our model captures relevant harmonic characteristics of the pieces in a compact and interpretable way, also reflecting stylistic aspects of the respective epoch. -
Visualization of Low Dimensional Structure in Tonal Pitch Space
VISUALIZATION OF LOW DIMENSIONAL STRUCTURE IN TONAL PITCH SPACE J. Ashley Burgoyne and Lawrence K. Saul Department of Computer and Information Science University of Pennsylvania {burgoyne,lsaul}@cis.upenn.edu ABSTRACT d] F] f] A a C c g] B b D d F f In his 2001 monograph Tonal Pitch Space, Fred Ler- dahl defined an distance function over tonal and post-tonal c] E e G g B[ b[ harmonies distilled from years of research on music cog- nition. Although this work references the toroidal struc- f] A a C c E[ e[ ture commonly associated with harmonic space, it stops b D d F f A[ a[ short of presenting an explicit embedding of this torus. It is possible to use statistical techniques to recreate such an e G g B[ b[ D[ d[ embedding from the distance function, yielding a more a C c E[ e[ G[ g[ complex structure than the standard toroidal model has heretofore assumed. Nonlinear techniques can reduce the Figure 1. Weber’s diagram of tonal space dimensionality of this structure and be tuned to emphasize global or local structure. The resulting manifolds high- light the relationships inherent in the tonal system and of- over harmonies. No other research to date has attempted fer a basis for future work in machine-assisted analysis to embed it explicitly. David Temperley used a MIDI- and music theory. based approach to implement many components of the theory [9] but has not yet treated the topology of the space. The Mathematical Music Theory Group at the Techni- 1. -
Towards Automatic Extraction of Harmony Information from Music Signals
Towards Automatic Extraction of Harmony Information from Music Signals Thesis submitted in partial fulfilment of the requirements of the University of London for the Degree of Doctor of Philosophy Christopher Harte August 2010 Department of Electronic Engineering, Queen Mary, University of London I certify that this thesis, and the research to which it refers, are the product of my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline. I acknowledge the helpful guidance and support of my supervisor, Professor Mark Sandler. 2 Abstract In this thesis we address the subject of automatic extraction of harmony information from audio recordings. We focus on chord symbol recognition and methods for evaluating algorithms designed to perform that task. We present a novel six-dimensional model for equal tempered pitch space based on concepts from neo-Riemannian music theory. This model is employed as the basis of a harmonic change detection function which we use to improve the performance of a chord recognition algorithm. We develop a machine readable text syntax for chord symbols and present a hand labelled chord transcription collection of 180 Beatles songs annotated using this syntax. This collection has been made publicly avail- able and is already widely used for evaluation purposes in the research community. We also introduce methods for comparing chord symbols which we subsequently use for analysing the statistics of the transcription collection. To ensure that researchers are able to use our transcriptions with confidence, we demonstrate a novel alignment algorithm based on simple audio fingerprints that allows local copies of the Beatles audio files to be accurately aligned to our transcriptions automatically. -
Interval Vectors
Ilhan M. Izmirli George Mason University [email protected] The exploration of the profound and intrinsic cohesion between mathematics and music is certainly nothing new – it actually dates all the way back to Pythagoras (c. 570 BCE – c. 495 BCE). However, the introduction of the dodecaphonic (twelve-tone) system developed by Arnold Schoenberg (1874 – 1951) has taken this study to entirely new levels, and has instituted such concepts as set theory, ordered sets, vectors, and various types of spaces as useful tools in music theory. In this paper we will look into one of these tools, namely the notion of interval vectors. Around 1908, the Viennese composer Arnold Schoenberg developed a system of pitch organization in which all twelve unique pitches were to be arranged into an ordered row. This row and the rows obtained from it by various basic operations were then used to generate entire pitch contents, giving rise to a method of composition now usually referred to as the dodecaphonic (twelve-tone) system or serialism. This new system not only bolstered the existing ties between mathematics and music, but helped introduce some new ones as well. In fact, the field of musical set theory was developed by Hanson (1960) and Forte (1973) in an effort to categorize musical objects and describe their relationships in this new setting. For more information see Schuijer (2008) and Morris (1987). Let us first review the basic terminology, starting with a notational convention. We will call the octave from middle 퐶 to the following 퐵 the standard octave. If 퐶 denotes the middle 퐶, we will use the convention 퐶 = 0. -
A Unified Helical Structure in 3D Pitch Space for Mapping Flat Musical Isomorphism
HEXAGONAL CYLINDRICAL LATTICES: A UNIFIED HELICAL STRUCTURE IN 3D PITCH SPACE FOR MAPPING FLAT MUSICAL ISOMORPHISM A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Science University of Regina By Hanlin Hu Regina, Saskatchewan December 2015 Copytright c 2016: Hanlin Hu UNIVERSITY OF REGINA FACULTY OF GRADUATE STUDIES AND RESEARCH SUPERVISORY AND EXAMINING COMMITTEE Hanlin Hu, candidate for the degree of Master of Science in Computer Science, has presented a thesis titled, Hexagonal Cylindrical Lattices: A Unified Helical Structure in 3D Pitch Space for Mapping Flat Musical Isomorphism, in an oral examination held on December 8, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Dominic Gregorio, Department of Music Supervisor: Dr. David Gerhard, Department of Computer Science Committee Member: Dr. Yiyu Yao, Department of Computer Science Committee Member: Dr. Xue-Dong Yang, Department of Computer Science Chair of Defense: Dr. Allen Herman, Department of Mathematics & Statistics Abstract An isomorphic keyboard layout is an arrangement of notes of a scale such that any musical construct has the same shape regardless of the root note. The mathematics of some specific isomorphisms have been explored since the 1700s, however, only recently has a general theory of isomorphisms been developed such that any set of musical intervals can be used to generate a valid layout. These layouts have been implemented in the design of electronic musical instruments and software applications.