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Quantum approach to jam session J. Math. Music (IF 1.1) Pub Date : 2024-03-04 Bogusław Fugiel
A quantum mechanical approach to the psychoacoustic effect of Shepard tones has been explored. Melodic intervals are represented by qubits. The phenomenon of bistable perception of intervals and th...
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An algebra of chords for a non-degenerate Tonnetz J. Math. Music (IF 1.1) Pub Date : 2024-02-29 Rafael Cubarsi
For an n-TET tuning system, we propose a formalism to study the transformations of k-chords over a generalized non-degenerate Tonnetz generated by a given interval structure. Root and mode are the ...
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Renaissance canons with asymmetric schemes J. Math. Music (IF 1.1) Pub Date : 2023-12-26 Evan M. O'Dorney
By a scheme of a musical canon, we mean the time and pitch displacement of each entering voice. When the time displacements are unequal, achieving consonant sonorities is especially challenging. Us...
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Correction J. Math. Music (IF 1.1) Pub Date : 2023-10-11
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Ahead of Print, 2023)
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Exploring Musical Spaces: A Synthesis of Mathematical Approaches J. Math. Music (IF 1.1) Pub Date : 2023-08-27 Jordan Lenchitz
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 17, No. 3, 2023)
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A different kind of meantone temperament and a novel mapping from the harmonic series to Pythagorean pitch classes J. Math. Music (IF 1.1) Pub Date : 2023-08-22 Konstantin L. Gurin
The logarithm mapping of natural numbers is a sum of products of coefficients. If these coefficients are arbitrary parameters, a new mapping of natural numbers to some subset of real numbers appears. This mapping preserves some crucial logarithm properties and constructs a new musical sound with a spectrum of inharmonic overtones. The simplest one-parameter mapping to a subset of polynomials with integer
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Parsimonious sequences of finite sets and their applications to chord progressions and music composition J. Math. Music (IF 1.1) Pub Date : 2023-08-17 Hugues Bedouelle
Parsimony is a broad concept with applications in music theory and composition. Two sets A and B of finite cardinality n (n-sets) are in parsimonious relation if there exists a (n–1)-set C that is included in both A and B. A sequence of n-sets is parsimonious if any two successive sets in the sequence are in parsimonious relation. This work demonstrates that for any set of finite cardinality p, there
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Musical stylistic analysis: a study of intervallic transition graphs via persistent homology J. Math. Music (IF 1.1) Pub Date : 2023-08-10 Martín Mijangos, Alessandro Bravetti, Pablo Padilla-Longoria
We develop a novel method to represent a weighted directed graph as a finite metric space and then use persistent homology to extract useful features. We apply this method to weighted directed grap...
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Algebraic tunings J. Math. Music (IF 1.1) Pub Date : 2023-07-24 Carl P. Dettmann, Liam Taylor-West
We propose an approach to tuning systems in which octave doubling ratio is replaced by a suitable algebraic unit τ, and note frequencies are proportional to a subset of the ring Z[τ]Z[τ] . Then it is possible for many difference tones between notes in the tuning to also appear in the tuning. After outlining more general principles, we consider in detail some natural examples based on the golden ratio
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Mathematical foundations of complex tonality J. Math. Music (IF 1.1) Pub Date : 2023-07-17 Jeffrey R. Boland, Lane P. Hughston
Equal temperament, in which semitones are tuned in the irrational ratio of 21/12:121/12:1 , is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of 2, 3, and 5, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system
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A framework for topological music analysis (TMA) J. Math. Music (IF 1.1) Pub Date : 2023-06-21 Alberto Alcalá-Alvarez, Pablo Padilla-Longoria
In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We...
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Iterative method of construction for almost smooth rhythms J. Math. Music (IF 1.1) Pub Date : 2023-06-04 Fumio Hazama
The present article introduces the notion of marked rhythm and its almost smoothness through a certain transformation Rep, called repulsion map. A marked rhythm consists of a rhythm together with a...
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Composing Cagean silence J. Math. Music (IF 1.1) Pub Date : 2023-05-31 Michael D. Fowler
In this article I model John Cage's pragmatics of silence using the mathematics of category theory with the framework of ontological logs. I use this approach in order to represent knowledge within...
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Minimum description length for selection of models of musical rhythm J. Math. Music (IF 1.1) Pub Date : 2023-05-29 Verónica Rumbo, Ernesto Mordecki, Martín Rocamora
We apply the minimum description length (MDL) methodology from information theory in order to select mathematical models for musical rhythm. We consider six models proposed by David Temperley and mathematically formalize them, which allows for an MDL analysis. As a consequence, we find that two of the models are not suitable to apply this methodology, as their codification strategy does not represent
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Machine composition of Korean music via topological data analysis and artificial neural network J. Math. Music (IF 1.1) Pub Date : 2023-05-03 Mai Lan Tran, Dongjin Lee, Jae-Hun Jung
Common AI music composition algorithms train a machine by feeding a set of music pieces. This approach is a blackbox optimization, i.e. the underlying composition algorithm is, in general, unknown ...
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On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation J. Math. Music (IF 1.1) Pub Date : 2023-04-30 Rafael Cubarsi
For well-formed generalized Pythagorean scales, it is explained how to fill in a bidimensional table, referred to as a scale keyboard, to represent the scale tones, arranged bidimensionally as iter...
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Topological data analysis of Korean music in Jeongganbo: a cycle structure J. Math. Music (IF 1.1) Pub Date : 2023-03-08 Mai Lan Tran, Changbom Park, Jae-Hun Jung
Jeongganbo is a unique music representation invented by Sejong the Great. Contrary to the Western music notation, the pitch of each note is encrypted and the length is visualized directly in a matrix form. We use topological data analysis (TDA) to analyze the Korean music written in Jeongganbo for Suyeonjang, Songuyeo, and Taryong, those well-known pieces played among noble community. We define the
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An integer linear programming model for tilings J. Math. Music (IF 1.1) Pub Date : 2023-03-01 Gennaro Auricchio, Luca Ferrarini, Greta Lanzarotto
In this paper, we propose an integer linear programming model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how it can be used to define an iterative algorithm that, given a period n, finds all the rhythms which tile with a given rhythm A and also to efficiently check the necessity of the Coven-Meyerowitz condition (T2). To conclude, we run several experiments to validate
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Fishing for complements with chord, scale, and rhythm nets J. Math. Music (IF 1.1) Pub Date : 2023-02-09 Roger Asensi Arranz, Daniel Harasim, Thomas Noll
The aim of this paper is to argue that complementation is an operation similarly fundamental to music theory as transposition and inversion. We focus on studying the chromatic complement mapping that translates diatonic seventh chords into 8-note scales which can also be interpreted as rhythmic beat patterns. Such complements of diatonic seventh chords are of particular importance since they correspond
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Letters on P-Relations, 1992–1997 J. Math. Music (IF 1.1) Pub Date : 2023-02-03 Jack Douthett, Richard Cohn, Dani Zanuttini-Frank
Jack Douthett wrote a number of letters to John Clough and Richard Cohn concerning Cohn's “P-Relations,” single-semitone voice-leading relationships. The ideas in these letters led to graph-theoretic and geometric models. The following selection has been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.
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Exhaustive chord progressions and their use in music composition J. Math. Music (IF 1.1) Pub Date : 2023-01-20 Hugues Bedouelle
A general approach for the design of mild chord progressions from the n-chords (unordered pitch-class sets of cardinal n) of a p-tonic scale was developed. Four relations between n-chords were cons...
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Genredynamics: a perceptual calculus of genre J. Math. Music (IF 1.1) Pub Date : 2023-01-09 Noah R. Fram
Prevailing theories of genre, derived primarily from literary and musical scholarship, differ in characteristics they ascribe to genre itself. Here, the temporally dynamic and culturally contingent nature of genre informs a computational framework that is reducible to extant theories of genre and connected to psychological theories of perceptual categorization. This framework, called genredynamics
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Letters on Hook's group, 2000 J. Math. Music (IF 1.1) Pub Date : 2023-01-05 Jack Douthett, Richard Cohn, Dani Zanuttini-Frank
After encountering Julian Hook's work on uniform triadic transformations, Jack Douthett wrote letters to John Clough suggesting further group-theoretic generalizations of Hook's idea. These letters have been edited and prepared for publication by Richard Cohn and Dani Zanuttini-Frank.
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Turning the volvelle: Exploring Jack Douthett's voice leading dynamics J. Math. Music (IF 1.1) Pub Date : 2023-01-04 Roger Asensi Arranz, Thomas Noll
This paper explores Jack Douthett's model of dynamical voice leading on the level of harmonic states. It investigates global contiguous and stroboscopic dynamical systems on the entire state space and introduces a measure of effectiveness for the trajectories under consideration. Special attention is paid to the harmonic state spaces behind second-order Clough-Myerson scales, such as diatonic triads
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Jack Douthett’s letters J. Math. Music (IF 1.1) Pub Date : 2022-12-29 Richard Cohn
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 16, No. 3, 2022)
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Two letters by Jack Douthett on uniform triadic transformations J. Math. Music (IF 1.1) Pub Date : 2022-12-19 Julian Hook
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 16, No. 3, 2022)
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Jack Douthett and mathematical music theory J. Math. Music (IF 1.1) Pub Date : 2022-11-29 Jason Yust
Jack Douthett's work over three decades was central to defining an era in mathematical theory. The present special issue attests to his abiding influence over the field, as well as the energy he brought to research in all areas of mathematical music theory through his collaborations, correspondence, and relationships.
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Colleague, collaborator, friend Jack Douthett (1942–2021) J. Math. Music (IF 1.1) Pub Date : 2022-11-18 Richard Krantz
Remembrances of Jack Douthett from his students and colleagues.
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Musicological, computational, and conceptual aspects of first-species counterpoint theory J. Math. Music (IF 1.1) Pub Date : 2022-11-16 Juan Sebastián Arias-Valero, Octavio Alberto Agustín-Aquino, Emilio Lluis-Puebla
We re-create the essential results of a 1989 unpublished article by Mazzola and Muzzulini that contains the musicological aspects of a first-species counterpoint model. We include a summary of the mathematical counterpoint theory and several variations of the model that offer different perspectives on Mazzola's original principles.
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Teponazcuauhtla, or “Forest of Resonances” Mesoamerican Plot of Harmony J. Math. Music (IF 1.1) Pub Date : 2022-11-10
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Ahead of Print, 2022)
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Structural properties of multi-octave scales J. Math. Music (IF 1.1) Pub Date : 2022-11-08 Emmet Crowley, Francisco Gómez-Martín
Whilst not widely extended, non-octave-repeating scales are present in a variety of musical settings, yet have received scarce attention in the existing literature. This paper provides a brief general historical contextualization before focusing on a specific group of two-octave scales based on properties in common with the most widely used scales in Western music. After characterizing them in mathematical
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Three-string inharmonic networks J. Math. Music (IF 1.1) Pub Date : 2022-11-07 Saba Goodarzi, William A. Sethares
This paper studies the resonant frequencies of three-string networks by examining the roots of the relevant spectral equation. A collection of scaling laws are established which relate the frequencies to structured changes in the lengths, densities, and tensions of the strings. Asymptotic properties of the system are derived, and several situations where transcritical bifurcations occur are detailed
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Type and class vectors and matrices in ℤn. Application to ℤ6, ℤ7, and ℤ12 J. Math. Music (IF 1.1) Pub Date : 2022-10-18 Luis Nuño
In post-tonal theory, set classes are normally elements of Z12Z12 and are characterized by their interval-class vector. Those being non-inversionally-symmetrical can be split into two set types related by inversion, which can be characterized by their trichord-type vector. In this paper, I consider the general case of set classes and types in Zn and their m-class and m-type vectors, m ranging from
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Partitions, their classes, and multicolour evenness J. Math. Music (IF 1.1) Pub Date : 2022-10-11 Jack Douthett, Peter Steinbach, Robert Peck, Richard Krantz
We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe Uc. Interactions measure the average evenness of colour sets (partitioning sets) of Uc. For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximally even partitions, it is necessary to use computational
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Explicit presentations of topological categories of gestures J. Math. Music (IF 1.1) Pub Date : 2022-08-31 Juan Sebastián Arias-Valero, Emilio Lluis-Puebla
Thanks to Mazzola's notion of gestures on topological categories we can appreciate how the notion of gesture transcends its manifestation as the movement of the body's limbs and takes a more abstract form that blends diagrammatic (discrete gesturality) and bodily aspects (continuous gesturality). These two aspects are strongly related to two main branches of mathematical music theory, namely, a discrete
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Correction J. Math. Music (IF 1.1) Pub Date : 2022-08-31
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 17, No. 1, 2023)
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The scale of the Old Hispanic chant J. Math. Music (IF 1.1) Pub Date : 2022-08-11 Elsa De Luca
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 17, No. 2, 2023)
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In Memoriam J. Math. Music (IF 1.1) Pub Date : 2022-07-15 Darrell Conklin
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 17, No. 1, 2023)
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Rhythm, mathematics, and Godfried Toussaint J. Math. Music (IF 1.1) Pub Date : 2022-07-13 Jason Yust, Christopher William White, Leigh VanHandel
Godfried Toussaint occupied a unique place in music theory. The contributions in this special issue honour his legacy and continue the work that he started in his uniquely creative approach to introducing mathematical and computational tools for the analysis of cyclic rhythms.
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A comparative analysis of melodic rhythm in two corpora of American popular music J. Math. Music (IF 1.1) Pub Date : 2022-06-06 Christopher William White, Joe Pater, Mara Breen
This paper compares two corpora of melodies drawn from premillennial and postmillennial American popular music, and identifies several notable differences in their use of rhythm. The premillennial corpus contains melodies written between 1957 and 1997 [deClercq and Temperley (2011. “A Corpus Analysis of Rock Harmony.” Popular Music 30 (1): 47–70)], while the postmillennial corpus (compiled for this
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Ostinatos in Black-Atlantic traditions: generic-specific similarity and proximity J. Math. Music (IF 1.1) Pub Date : 2022-05-23 Jay Rahn
Ostinatos of sub-Saharan Africa, South America, and the Caribbean are continually repeated rhythms also known as “timelines.” Taking as a starting point the ostinato termed the “Standard Pattern,” generic and specific features of Black-Atlantic ostinatos are analyzed: in European-derived theory, these features correspond to the quantity and quality of musical intervals. Like the standard pattern, other
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The inconstancy of music J. Math. Music (IF 1.1) Pub Date : 2022-05-16 Florence Levé, Gianluca Micchi, Jean-Paul Allouche
A melody is often described as a line of music that evolves through time and, therefore, it is possible to draw its 2D pitch-time representation as a series of points implicitly defining a curve. We introduce to computational musicology a descriptor of this music curve: the inconstancy, a function that gives information on the curve's smoothness as well as some of its topological properties. A mathematical
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A three-dimensional timbre model via Peano curves J. Math. Music (IF 1.1) Pub Date : 2022-04-13 Daniele Ghisi, Carmine-Emanuele Cella
Creating a formal model for timbre is one of the most compelling open questions in music research. In contrast to more traditional perceptually-oriented approaches, often aimed at sound analysis, we introduce a three-dimensional geometric model of timbre, specifically designed for sound synthesis. The proposed model relies on the properties of space-filling curves for multidimensional scaling, and
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Hadamard transforms of pure-duple rhythms J. Math. Music (IF 1.1) Pub Date : 2022-04-07 Jason Yust
Recent research has demonstrated a number of applications of the discrete Fourier transform to cyclic rhythms, addressing amongst other issues questions about conceptualizations of metre. One can apply a similar operation, the Hadamard transform, when the rhythmic cycle is a power of two. This paper explores some analytical applications of the Hadamard transform to repertoires where pure-duple metrical
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Quantum-like melody perception J. Math. Music (IF 1.1) Pub Date : 2022-04-05 Bogusław Fugiel
I propose a quantum-like approach to the description of melody perception where classic intervals that constitute a melody are replaced by acoustical qubits, i.e. two-level acoustic systems, using Shepard tones for this purpose. Each of such qubits is considered to be a superposition of two intervals, ascending and descending, that form an octave when put together. Any melody perception can thus be
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The line of fifths and the co-evolution of tonal pitch-classes J. Math. Music (IF 1.1) Pub Date : 2022-03-17 Fabian C. Moss, Markus Neuwirth, Martin Rohrmeier
In this study, we determine the fundamental role of the line of fifths for the organization of tonal material by applying dimensionality reduction to a large historical corpus of pitch-class counts (ca. 1360–1940). We observe a historically growing trend in the exploitation of the fifths range, i.e. the size of segments that pitch-class distributions cover on the line of fifths. Moreover, we introduce
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A review of Godfried Toussaint's The Geometry of Musical Rhythm J. Math. Music (IF 1.1) Pub Date : 2022-02-19 Francisco Gómez-Martín
This is both a personal and academic review of Godfried Toussaint's The Geometry of Musical Rhythm.
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Hierarchical set theory J. Math. Music (IF 1.1) Pub Date : 2022-01-10 Dmitri Tymoczko
Musicians often operate with a hierarchy of scale-like collections, each embedded within the next, and with transposition and inversion available at every level. A particularly common technique is to counteract a transformation at one level with an analogous transformation in the intrinsic scale consisting of a chord’s own notes.
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The melodic beat: exploring asymmetry in polska performance J. Math. Music (IF 1.1) Pub Date : 2021-12-09 Olof Misgeld, Andre Holzapfel, Petter Kallioinen, Sven Ahlbäck
Some triple-beat forms in Scandinavian Folk Music are characterized by non-isochronous beat durations: asymmetric beats. Theorists of folk music have suggested that the variability of rhythmic figures and asymmetric metre are fundamental to these forms. The aim of this study is to obtain a deeper understanding of the relationship between melodic structure and asymmetric metre by analysing semi-automatically
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Barberpole tempo illusions J. Math. Music (IF 1.1) Pub Date : 2021-12-07 Daniele Ghisi
“Barberpole” tempo illusions are a family of auditory illusions based on the synchronization of faded rhythmic streams playing at different rates, often manufacturing experiences of seemingly eternal acceleration or deceleration. The forefather of all such illusions, based on layers whose rates are powers of two apart (“octaves”), was studied by Jean-Claude Risset in the late seventies and is now known
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Grammar-based compression and its use in symbolic music analysis J. Math. Music (IF 1.1) Pub Date : 2021-12-06 Tiasa Mondol, Daniel G. Brown
We apply Context-free Grammars (CFG) to measure the structural information content of a symbolic music string. CFGs are appropriate to this domain because they highlight hierarchical patterns, and their dictionary of rules can be used for compression. We adapt this approach to estimate the conditional Kolmogorov complexity of a string with a concise CFG of another string. Thus, a related string may
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A geometric framework for pitch estimation on acoustic musical signals J. Math. Music (IF 1.1) Pub Date : 2021-10-06 Tom Goodman, Karoline van Gemst, Peter Tiňo
This paper presents a geometric approach to pitch estimation (PE) – an important problem in music information retrieval (MIR), and a precursor to a variety of other problems in the field. Though there exist a number of highly accurate methods, both mono-pitch estimation and multi-pitch estimation (particularly with unspecified polyphonic timbre) prove computationally and conceptually challenging. A
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Tonal harmony and the topology of dynamical score networks J. Math. Music (IF 1.1) Pub Date : 2021-09-18 Marco Buongiorno Nardelli
I introduce the concept of dynamical score networks for the representation and analysis of tonal compositions: a score is interpreted as a dynamical network where every chord is a node and each progression links successive chords. This network can be viewed as a time series of a non-stationary signal, and as such, it can be partitioned for the automatic identification of tonal regions using time series
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Pattern in music J. Math. Music (IF 1.1) Pub Date : 2021-08-17
(2021). Pattern in music. Journal of Mathematics and Music: Vol. 15, Pattern in Music; Guest Editor: Darrell Conklin, pp. 95-98.
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Triadic patterns across classical and popular music corpora: stylistic conventions, or characteristic idioms? J. Math. Music (IF 1.1) Pub Date : 2021-06-17 David R. W. Sears, David Forrest
Many musical traditions – from Western art, to popular and commercial – organize pitch phenomena around a referential pitch class (or tonic) and feature triads and seventh chords. As a result, triadic progressions associated with one tradition sometimes resurface in others. How, then, are we to distinguish between the conventional harmonic patterns that span several time periods, and the characteristic
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Some observations on autocorrelated patterns within computational meter identification J. Math. Music (IF 1.1) Pub Date : 2021-06-28 Christopher Wm. White
The computational approach of autocorrelation relies on recurrent patterns within a musical signal to identify and analyze the meter of musical passages. This paper suggests that the autocorrelation process can act as a computational proxy for the act of period extraction, a crucial aspect of the cognition of musical meter, by identifying periodicities with which similar events tend to occur within
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Exploring annotations for musical pattern discovery gathered with digital annotation tools J. Math. Music (IF 1.1) Pub Date : 2021-07-21
The study of inter-annotator agreement in musical pattern annotations has gained increased attention over the past few years. While expert annotations are often taken as the reference for evaluating pattern discovery algorithms, relying on just one reference is not usually sufficient to capture the complex musical relations between patterns. In this paper, we address the potential of digital annotation
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Symbolic dynamical scales: modes, orbitals, and transversals J. Math. Music (IF 1.1) Pub Date : 2021-08-09 Ricardo Gómez Aíza
We study classes of musical scales obtained from shift spaces in symbolic dynamics through the first symbol rule, which yields scales in any n-TET tuning system. The modes are thought as elements of orbit equivalence classes of cyclic shift actions on languages, and we study their orbitals and transversals. We present explicit formulations of the generating functions that allow us to deduce the orbital
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José Manuel López López’s Chart for managing tempi J. Math. Music (IF 1.1) Pub Date : 2021-07-05 José L. Besada
Published in Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance (Vol. 16, No. 2, 2022)
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Minimally non-diatonic pc-sets J. Math. Music (IF 1.1) Pub Date : 2021-06-17 Jay Schweig, Aurian Kutner
We discuss and enumerate pc-sets that are both not contained in any diatonic collection and are minimal with respect to this property, and we generalize this idea to other collections. We also consider related simplicial complexes and examine how some of their geometric properties reflect qualities of the associated pc-sets.