The Musical-Mathematical Mind

A counterpoint theory for the whole continuum of the octave is obtained from Mazzola’s model via extended counterpoint symmetries, and some of its properties are discussed.

The present paper integrates a broad research project, based on the principles of developing variation and Grundgestalt (both formulated by the Austrian composer Arnold Schoenberg), which aims at a systematical production of musical variants through employment of a group of genetic algorithms. The article examines a specific aspect of the process for production of these variants, namely, the creation of an adequate and efficient method for their genealogical organizing and labeling. This led to the elaboration of a couple of complementary concepts, the Gödel-vector and the Gödel-address, inspired by a function created by the Austrian mathematician Kurt Gödel. The results obtained by the application of both elements in the process of variant production allowed a decisive improvement of the procedures employed for classiying and retriving the derivative musical forms.

Discrete Fourier Transform may well be the most promising track in recent music theory. Though it dates back to David Lewin’s first paper (Lewin, J. Music Theory (3), 1959) [33], it was but recently revived by Quinn in his PhD dissertation in 2005 (Quinn, Perspectives of New Music 44(2)–45(1), 2006–2007) [35], with a previous mention in (Vuza, Persp. of New Music, nos. 29(2) pp. 22–49; 30(1), pp. 184–207; 30(2), pp. 102–125; 31(1), pp. 270–305, 1991–1992) [40], and numerous further developments by (Andreatta, Agon, (guest eds), JMM 2009, vol. 3(2). Taylor and Francis, Milton Park) [5], (Amiot, Music Theory Online, 2, 2009) [8], (Amiot, Rahn, (eds.), Perspectives of New Music, special issue 49 (2) on Tiling Rhythmic Canons) [9], (Amiot, Proceedings of SMCM, Montreal. Springer, Berlin, 2013) [10], (Amiot, Sethares, JMM 5, vol. 3. Taylor and Francis, Milton Park (2011) [16], (Callender, J. Music Theory 51(2), 2007) [17], (Hoffman, JMT 52(2), 2008) [29] (Tymoczko, JMT 52(2), 251–272, 2008) [38], (Tymoczko, Proceedings of SMCM, Yale, pp. 258–272. Springer, Berlin, 2009) [39], (Yust, J. Music Theory 59(1) (2015) [42]. I chose to broach this subject because I have had a finger in most, or all, of the pies involved (even using Discrete Fourier Transform without consciously knowing it, in the study of rhythmic tilings).

We present a motivation and a proposal for the definition of gestures and hypergestures on locales and localic topoi. Further possible generalizations, including gestures on Grothendieck topoi are discussed.

From 1959 to 1969 I composed music as most others do and have done—by direct transference from the imagination to a musical instrument (in my case the piano) and from there to a written score. During this period, I found myself relying increasingly on traditionally structured techniques such as canon, fugue, dodecaphony, serialism and electronics. In 1970 I was struck for the first time by a mathematical rule-based idea for an ensemble piece, which necessitated my learning to program a computer. Since then I have composed over fifty works (half my total output) with computer help—works for piano, organ, chamber ensemble, orchestra and electronics. Of these fifty-odd pieces, about half are partially and sometimes wholly based on abstract mathematical principles. This paper describes eight of these pieces or relevant sections of them in varying detail.

Braille Musicography is the most used system by blind people for reading and writing music in the world. It is a transcription from the conventional music notation to Braille system, which symbols are generated by a matrix of raised dots of 2 columns and 3 rows. It shows two main difficulties that make it a hard tool for the blind musicians: (1) The number of music symbols exceeds by far the number of possible Braille dots combinations and (2) it is a linear system representing a bidimensional system. These two problems result in the need of using combinations of up to 4 Braille boxes to represent one musical symbol, and the repetition of Braille symbols that change meaning depending of the context. In order to give more clarity or simplicity in Braille scores, abbreviations and contractions are used, thus a fragment of music can sometimes be written in several ways. Because of all this, automatic transcription to Braille is complicated, sometimes not possible at all, and as a consequence blind people do not have full access to Braille scores. Besides the many efforts of people around the world, music scores transcription to Braille musicography is still a problem. In this work some of the Braille musicography problems are identified, and the need of a more efficient musicography is established. In order to create a new set of symbols for the blind and a useful system, which is an objective of a later stage of this work, it is important to notice that our fingertips have a delimited zone in which the density of receptors is high and allows a clear reading of a symbol. Outside this zone, the produced mental image is unclear and makes the reading tiresome and difficult. This and some other physiological and cognitive considerations have to be taken into account. Experience and ideas from the blind must be always regarded.

In the first two sections of this contribution we construct the groups $$(S_n,+)$$ and $$(L(S_n),\circ )$$ in order to have an intuitive way to represent musical phrases by their melodic contour. Later we derive an algorithm for composing music using a given number and the group $$(L(S_n),\circ )$$. Finally we offer a variation of the same algorithm to be able to translate a piece of music in a finite digit number, with analytic and deconstructive aims.

In this note we give a full proof of Mazzola’s Escher Theorem (Mazzola, J Math Music, 3(1):31–58, 2009, [4]). This theorem is needed for the development of the theory that Mazzola seeks to realize, and it helps us to understand better the concept of hypergesture as used in his work (Mazzola, J Math Music 3(1):31–58, 2009, [4], Mazzola, Musical performance-A comprehensive approach: theory, analytical tools, and case studies, 2011, [5], Mazzola and Andreata, J. Math. Music, 1(1):23–4, 2007, [6], Mazzola et al., Musical creativity-strategies and tools in composition and improvisation, 2011, [7]). A gesture is a morphism from a digraph into a topological space, and is one of the fundamental blocks in the Mathematical Theory of Performance. A hypergesture is a gesture built upon another gesture, describing, in a way, the variation of the latter. The non-trivial fact that the variation of the former gesture, as described by the latter, is given by the same hypergesture is essentially the content of the Escher Theorem.

Tipping points are an observable and experienced natural phenomenon that has been invoked metaphorically across various domains external to physics. This article introduces the tipping point analogy for musical timing, and presents three case studies illustrating the concept. Quantitative data from recorded performances presented in score-time graphs support the illustrations. The examples show how musicians employ tipping points in performance, and demonstrate how tipping points play on the listener’s expectations to elicit emotion. Tipping points form principal tools for the performer’s choreography of expectation; the pervasiveness of tipping points in human experience make them an important strategy also for ensemble coordination.

The context of this paper is the theory of modes of non-degenerate well-formed scales (generalized diatonic or pentatonic scales), within the framework of algebraic combinatorics of words, specifically musical modes encoded as members of the monoid of words in $$A^{*}$$ over a two-letter alphabet A, and the monoid of Sturmian morphisms that act on $$A^{*}$$. The paper relates lexicographic orderings of words modes of (non-degenerate) well-formed scales (especially the canonical examples, the diatonic modes) and lexicographic orderings of the special Sturmian morphisms associated with the modes, to the musical scale and circle-of-fifths orderings. These lexicographic orderings are related to Zarlino’s 1571 re-ordering of Glarean’s 1547 listing of six authentic diatonic modes.

We illustrate the basic ideas and principles of quantum geometry, by considering mutually complementary quantum realizations of circles. It is fascinating that such a simple geometrical object as circle, provides a rich illustrative playground for an entire array of purely quantum phenomena. On the other hand, the ancient Pythagorean musical scale, naturally leads to a simple quantum circle. We explore different musical scales, their mathematical generalizations and formalizations, and their possible quantum-geometric foundations. In this conceptual framework, we outline a diagramatical-categorical formulation for a quantum theory of symmetry, and further explore interesting musical and geometrical interconnections.

This paper integrates a broad research about the pragmatic modelling of compositional process, and some mathematical abstractions that arises from the relations between textural configurations. As the available choices for textural organization are limited, it is possible to provide a global map of all possible configurations for a given number of sources (exhaustive taxonomy) and assess all the kinship and metrics between them (topology). The graphic called Partitiogram, in fact, constitutes this phase space, where three basic nets of parsimonious relations are drawn: mnet, vnet and tnet. Each net deals with a different kind of textural transformation. This framework is part of Partitional Analysis (PA) — an original proposal of mediation between mathematical abstractions derived from the Theory of Integer Partitions and compositional theories and practices. The main goal of the theory is the study of compositional games. It has been used in the pedagogy of composition and in the creation of new pieces.

In the mid-1970s, Alain Louvier worked out microtonal scales called modes of progressive transposition and used them in many musical works. These modes have similar properties to major modes and are related to diatonicism. Some of them were known by Ivan Wyschnegradsky and Georgy Rimsky–Korsakov, the grand-son of Nikolai. Deep scales are well known in diatonic theory, and are special cases of these modes. However, their algebraic structure is not known. Although the diatonic theories have been developed by many musicologists, such as Agmon, Balzano, Carey, Clampitt, Noll, Zweifel and others, many questions remain open. In this paper, we describe some studies on microtonality, published over the last century, and we review what is known and what remains to understand in this field, in both theoretical and compositional aspects. In the first section, we study the modes called by Alain Louvier “imperfect modes”, a special case of modes of progressive transposition. He used them in several important works as Le Clavecin non tempéré (1973), Canto di Natale (1976) and Anneaux de lumière (1983), written in the 24 tone equal temperament. The mathematical study would be to find a criterion to easily determine all the modes of progressive transposition in any equal temperament, and in particular to determine all deep scales. In the second section, we study the enumeration of the modes of limited transposition, also known as Messiaen modes in any equal temperament. In the last section, we present another classification of the modes related to the plactic monoid. Along this paper, we question what, if they exist, the microtonal diatonicism and the microtonal modality could be.

This investigation is a departure point for understanding what Pythagoreanism can mean today, how can harmony be conceived at several time scales and what might a hierarchical model of form together with an algebra of perception entail for music composition. The study of qualitative aspects of music through mathematics is made by taking James Tenney’s theory of musical form together with Alain Badiou’s ‘objective phenomenology’ in order to imagine new ways of composing music.

Based on the spatial composition method proposed by the author and its application in the piece, Materia Oscura (This work was premiered at MediaLab-Prado, Madrid, Spain on October 11, 2013.), some geometric representations that allow the description and documentation of the relation between cognition, sound and space are proposed. The purpose of developing this analysis is to establish a formal precedent for future studies related to different perceptual skills with which we abstract three-dimensional space information through our ears. The applications of these studies range from artistic creation to the development of educational tools for music and mathematics.

Generative art emphasizes processes. On the other hand, mathematical models are used to understand underlying biological, economic, physical or social phenomena (among others). The outcome of such models can be considered as processes in their own right. For instance, physiological processes give rise to a wide variety of signals which can, in turn, be detected by changes in pressure, temperature, electrical potential and so on. When measured and converted with an appropriate transducer, they constitute the raw material which algorithms and models may translate into sound. In this paper we explore a mathematical model of the human circulatory system based on differential equations. We then use this model as a generator of melodic and rhythmic structures in a compositional multimedia context.

What makes a musical work successful? In Darwinian terms, music is successful if listeners attend to it, repeatedly, for then it can live on. However, attention is fleeting: successful music holds listeners’ interest by manipulating their expectations using deception and confirmation. The ratio of the rate at which listeners follow music to the rate at which music unfolds is a predictor for musical success. This paper informally presents a theory of musical interest, based on some ideas from music theory, cognitive psychology, and information theory.

We prove a modulation theorem for diatonic scales that is based on the theory of hypergestures and vector fields derived from inner symmetries of diatonic scales and Lie bracket fields. It yields the same modulation degrees as the classical model (Mazzola, Gruppen und Kategorien in der Musik, 1985, [1], Geometrie der Töne, 1990, [2], Mazzola et al., The Topos of Music-Geometric Logic of Concepts, Theory, and Performance, 2002, [3]), which confirmed Schoenberg’s modulation theory (Schoenberg, Harmonielehre 1911, Universal Edition, Wien 1966, [4]). In this hypergestural model, integration of differential forms is considered. In this context, we can model and prove Stokes’ theorem for hypergestures, generalizing the classical case. Stokes’ theorem is a central result in differential geometry, relating the integral of the derivative of a form to the boundary of its domain of integration. It has important application in physics, such as in mechanics (integral invariants, see (Abraham, Foundations of Mechanics, 1967, [5])) or in electrodynamics (relating differential and integral forms of Maxwell’s equations (Jackson, Classical Electrodynamics, 1998, [6])). The basic form of this theorem deals with integration on singular hypercubes. In (Mazzola, J Math Music 6(1):49–60, 2012, [7]) we have extended singular homology on hypercubes to singular homology on hypergestures. It was therefore straightforward to try to extend Stokes’ theorem to hypergestures.

In the present work an analysis is made of several passages from Manuel M. Ponce’s Sonata No. 2 for piano (Ponce, Sonata No. 2 for Piano, 1916/1968, [1]), employing Julian Hook’s theoretical development of signature transformations and proper spelled heptachords. A signature transformation reinterprets a diatonic object in the context of a different key signature. The signature transformations form a cyclic group of order 84; indeed, the chromatic transpositions (Tn) and the diatonic transpositions ($$t_n$$) form subgroups of this cyclic group, hence contributing with yet another way of unifying diatonic and chromatic structures. After giving an introduction to the theory behind the signature transformations, we proceed to an analysis of illustrative passages of the Sonata, using units of varying size called diatonic fragments. During this analysis we realized that the classes of proper spelled heptachords, a generalization of the signature transformations, could explain the constant transition between 7-note nearly diatonic scales. These classes also have a clear mathematical structure, with a transposition operator $$\tau $$ (they are also called $$\tau $$-classes), and possess some of the symmetries as well as the seven modes of the diatonic class. This analysis made us look for both intra-class transformations, similar to the ones we find in the diatonic class, and inter-class transformations that can explain the fluid movement between classes found not only in this sonata, but in other pieces that are classified as “chromatic” without more detail.

This paper proposes an organisation of textural progressions through the ranking of partitions lexical set (lexset). Departing from the Musical Contour Theory, mainly developed by Michael Friedmann and Robert Morris, and extending its principles to the music textural domain, it is possible to generate the Textural Contour. The Partitional Analysis, which emerges from the approximation between Wallace Berry’s approach and the Theory of Integer Partitions, is applied as a methodological approach for the textural parameter. The Textural Contour provides some tools for textural analysis based on the relative variation of textural complexity. The paper concludes with an example of methodological application of the Textural Contour and ranking of partitions.

Rameau [15] redundantly defines the subdominant (1) as the fifth under the tonic and (2) as the scale degree immediately below the dominant. In the context of scale theory this motivates the interpretation of this definition as an equation. It states that the diazeuxis (the difference between the generator and its octave complement) is a step interval of the scale. The appropriate scale-theoretic concept for the formulation of this equation is that of a Carey–Clampitt Scale (a non-degenerate well-formed scale). The Rameau equation then imposes a constraint on the associated Regener transformation which converts note intervals from generator/co-generator coordinates into step/co-step coordinates. The equation takes two forms depending on the sign of the diazeuxis (positive or negative). The solutions then come either with two (flatward directed) co-steps or two (sharpward directed) steps, accordingly. In addition to this generic characterization the paper closes with a corollary on the specific scale properties of reduced Clough–Myerson scales. These scales solve Rameau’s equation if and only if they are Agmon scales.

In this paper, I argue that stored auditory and motor patterns are inserted into ongoing musical improvisations. This position aligns with the theoretical framework suggested by [11]. In support of this theory, I cite research in which artist-level improvisers describe their own thinking and mention learned patterns as one of the central mechanism underlying improvisation. I further outline how known solos by tonal jazz artists contain a large number of repeated patterns and that a computer algorithm using patterns is capable of producing new solos in similar style. Finally, an experimental study shows that improvisers use more patterns when their attention is diverted during improvisation. According to interviews with advanced improvisers in solo settings, their attention is focused on larger architectural structures during improvisation. Other research with advanced improvisers in group settings point out that interaction with other ensemble members also may be at the front of the improvisers mind in that setting [1, 5]. Therefore, I conclude that it may be the partially automatic process of inserting learned patterns into ongoing improvisations that allows the artist-level improviser to focus on planning and interaction.

This contribution introduces the concept of musical harmony as a geometric, physical mirror of human biologic proportionality. Although this idea is rather ubiquitous in many aspects and epochs of music theory, mathematical direct modelling is relatively a novelty within the field of dynamical systems. Furthermore, a hypothesis of atomic-molecular harmonicity is provided in order to explain how biologic proportionality is physically biased to perform harmonic patterns eventually codified by culture. This hypothesis is grounded on the topological properties of carbon, and its mapping and embedding within the characteristic geometry of music; from the graphene-Tonnetz analogy, to the map of musical harmony using the Arnold tongues analogy. The topological features of carbon are, then, conceived as crucially influential for the rising of human language and music, and for the development of an associated Euclidean intuition.

Idiophone instruments are classified through different methods, ranging from the Hornbostel–Sachs system of musical instrument classification, to time series organized according to features of frequency spectra and time span. We propose an alternative method for analyzing and classifying idiophones according to their timbral complexity, measuring timbral-body continuum and phase-synchronization degree. In order to simplify our exposition, we choose the teponaztli as a model of wooden idiophone, because of its structural unicity and its potential complexity through extended musical performance (e.g. through the instrument’s individual or group timbral experimentation). We start exposing organological and cultural topics on the teponaztli, and then we discuss its harmonic and musical implications. Finally we explain our experimental development, and discuss the implications for musicological research and eventual new musical output.

Algorithmic composition systems are now well-understood. However, when they are used for specific tasks like creating material for a part of a piece, it is common to prefer, from all of its possible outputs, those exhibiting specific properties. Even though the number of valid outputs is huge, many times the selection is performed manually, either using expertise in the algorithmic model, by means of sampling techniques, or some times even by chance. Automations of this process have been done traditionally by using machine learning techniques. However, whether or not these techniques are really capable of capturing the human rationality, through which the selection is done, to a great degree remains as an open question. The present work discusses a possible approach, that combines expert’s opinion and a fuzzy methodology for rule extraction, to model high level features. An early implementation able to explore the universe of outputs of a particular algorithm by means of the extracted rules is discussed. The rules search for objects similar to those having a desired and pre-identified feature. In this sense, the model can be seen as a finder of objects with specific properties.

Music and mathematics have a lot of common grounds. They both involve processes of thought, but where mathematics is concerned basically with symbols without any physical connection to the world, music has sound as its major category. Music, in this view, is characterized by a sonorous articulation over time, which can be described in physical terms. Yet, it is possible to conceive of music also at a virtual level of imagery and to carry out symbolic computations on mental replicas of the sounds. The major aim of this contribution, therefore, is to explore some basic insights from algebra, geometry and topology, which might be helpful for an operational description of the sounding music. Starting from a conception of music as a formal system, it argues for a broadening and redefinition of the concept of computation, in order to go beyond a mere syntactic conception of musical sense-making and a mere symbol-processing point of view.

Motivated by an investigation of the historical roots of set theory in analysis, this paper proposes a generalisation of existing spectral synthesis methods, complemented by the idea of an experimental algorithmic composition. The background is the following argument: already since 19th century sound research, the idea of a frequency spectrum has been constitutive for the ontology of sound. Despite many alternatives, the cosine function thus still serves as a preferred basis of analysis and synthesis. This possibility has shaped what is taken as the most immediate and self-evident attributes of sound, be it in the form of sense-data and their temporal synthesis or the aesthetic compositional possibilities of algorithmic sound synthesis. Against this background, our article considers the early phase of the foundational crisis in mathematics (Krise der Anschauung), where the concept of continuity began to lose its self-evidence. This permits us to reread the historical link between the Fourier decomposition of an arbitrary function and Cantor’s early work on set theory as a possibility to open up the limiting dichotomy between time and frequency attributes. With reference to Alain Badiou’s ontological understanding of the praxis of axiomatics and genericity, we propose to take the search for a specific sonic situation as an experimental search for its conditions or inner logic, here in the form of a decompositional basis function without postulated properties. In particular, this search cannot be reduced to the task of finding the right parameters of a given formal frame. Instead, the formalisation process itself becomes a necessary part of its dialectics that unfolds at the interstices between conceptual and perceptual, synthetic and analytic moments, a praxis that we call musique axiomatique. Generalising the simple schema of additive synthesis, we contribute an algorithmic method for experimentally opening up the question of what an attribute of sound might be, in a way that hopefully is inspiring to mathematicians, composers, and philosophers alike.

This work will present a method for algorithmic music composition based on concepts from graph theory and non deterministic finite state automaton. The core formulation lies on the construction of a basic mathematical formal set structure over music rhythmic elements with two arithmetic operations: sum and multiplication. This structure allows to generate a whole compositional structure where mathematical functions can be directly related, or interpreted as musical rhythmic generators. We present then a brief scheme of a proposed algorithmic music composition system which we call Automaplex and its implementation in programming language Supercollider.

In polyphonic music, melodic patterns (motifs) are frequently imitated or repeated, and transformed versions of motifs such as inversion, retrograde, augmentations, diminutions often appear. Assuming that economical efficiency of reusing motifs is a fundamental principle of polyphonic music, we propose a new method of analyzing a polyphonic piece that economically divides it into a small number of types of motif. To realize this, we take an integer programming-based approach and formalize this problem as a set partitioning problem, a well-known optimization problem. This analysis is helpful for understanding the roles of motifs and the global structure of a polyphonic piece.

How to analyze open form scores? Generally, analysis will not be able to proceed by describing the architecture of a sequence of events, which renders most traditional analytical tools of music theory (structural voice leading, harmonic progression, thematic development, etc.) inoperative. But even scores that do not prescribe anything about event order contain an idea about time, and the treatment, or architecture, of time that is implicit in them is part of their musical subject matter. The temporal architecture in such compositions will be a non-linear field, a network of possible performance developments, and this structure, which we will refer to as a “time field”, can be studied for its formal properties. What scores in open form express, then, is the character of a time field.

In this paper, we examine a number of minimal change musical morphologies. Each morphology has an analogous representation in mathematics. Our mathematical objects of study are Gray codes, de Bruijn sequences, aperiodic necklaces, disjoint subset pairs, and multiset permutations with musically motivated constraints that result in several open problems.

One of the reasons for the widely felt influence of Schenker’s theory is his idea of long-range voice-leading structure. However, an implicit premise, that voice leading is necessarily a relationship between chords, leads Schenker to a reductive method that undermines the structural status of keys. This leads to analytical mistakes as demonstrated by Schenker’s analysis of Brahms’s Second Cello Sonata. Using a spatial concept of harmony based on DFT phase space, this paper shows that Schenker’s implicit premise is in fact incorrect: it is possible to model long-range voice-leading relationships between objects other than chords. The concept of voice leading derived from DFT phases is explained by means of triadic orbits. Triadic orbits are then applied in an analysis of Beethoven’s Heiliger Dankgesang, giving a way to understand the ostensibly “Lydian” tonality and the tonal relationship between the chorale sections and “Neue Kraft” sections.

This contribution connects Mathematical Music Theory (MaMuTh) with Peircean semiotics, identifying general grounds for Gesture Theory (in Mazzola’s sense). In order to make clear this connection, some of the contributions included in this volume are refered, unveiling a common framework for semiotics in MaMuTh.