Blog on Applied Mathematics

Atlanta: Mathematics and Music

This is a short report about a conference on Mathematics and Music. In presence. In the United States. With a vibrant community of researchers. Because in a crazy world it is more important than ever to hold on to beautiful things. In the conference logo, you can see a peach, a symbol of Atlanta, with a torus of musical intervals and a bass clef.

Sunrise in Atlanta, from my hotel room. Pic  ⒸM. Mannone

The rhythm of jazz, the sparkle of the trumpet, the wind stirring  cotton plants' leaves. "Summertime" sung by a charming black voice. It could be the beginning of a movie. Or the context of a serious math conference. Or rather, of math and music. At Georgia State University, where there are black, or rather, blue panthers.

The mascot of Georgia State University is a blue panther. Pic ⒸM. Mannone

I am talking about the 2022 edition of the biennial Mathematics and Computation in Music (MCM 2022), which took place in Atlanta, Georgia, in the United States. It should have taken place in 2021, but Covid decided otherwise. (Covid is also present in 2022, but we are used to it by now).

MCM 2022 was a hybrid conference, with most participants in person, and some attending the event remotely. All of them were happy to be in touch again, to see each other again alive. Because, between pandemics, wars, droughts, energy crises, and so on, seeing each other alive and being able to meet in person is less trivial than ever.

Fountain near the building where the conference took place. Pic ⒸM. Mannone

According to G. H. Hardy, "When the world around you goes insane, mathematics provides an incomparable anodyne." Taking refuge in the Castalia of mathematics and music, we talked about geometry, category theory, algebra, computer science, and combinatorics. And all this, up and down the musical scales, through historical treatises, immersed in multidimensional representations of musical symmetries.

Detail of a fountain in Centennial Olympic Park. Pic ⒸM. Mannone


The conference took place from 20 to 24 June 2022, and was organized by some gurus of the mate-musical world: Mariana MontielOctavio A. Agustín-Aquino, Francisco Gómez, Jeremy Kastine, Emilio Lluis-Puebla, Brent Milam. The proceedings of the conference were published by Springer.

A succession of presentations and concerts took stock of the situation on the main current research topics. An afternoon of outreach at the Museum of Design Atlanta saw, for the first time, a heterogeneous audience in terms of age and preparation confronted with interactive topics and mate-musical exhibits. One of the demonstrations was my (Hyper) CubeHarmonic, which I will talk about later, and the first virtual museum of mathematics and music, conceived and developed by Gilles Baroin. The reference community is the Society for Mathematics and Computation in Music, chaired by Moreno Andreatta and having as its official venue the Journal of Mathematics and Music.

The presentations were articulated into four sessions: (1) rhythm and musical scale theory analyzed from a combinatorial point of view; (2) categorical and algebraic approaches to music; (3) mathematics for musical analysis; (4) algorithms for modeling musical phenomena. Let's see in detail the key ideas presented in the various sessions. I present below a quick review of the topics of each session. (On Math is in the Air, we had previously talked about category theory).

Let's start with the first session, with rhythm, scales and combinatorial models.

Richard Cohn (New Haven, USA) presents a model of metric relations based on set theory, focusing on “metric dissonances.” In his video from France, Franck Jedrzejewski, interested in microtonal scales since years, proposes a new definition of “microdiatonic” scales. He defines the degree of "majorness" of a musical scale, through limited transposition sets. We recall that the definition of major / minor pertains to traditional diatonic scales. In music, in the "modes with limited transposition", used by Messiaen, after a certain number of transpositions up or down, the notes return to their starting pitches.

Representation of circular scales.

In the research by Luis Nuño (Spain),  eight heptatonic-scale types are selected, together with their corresponding pentatonic complements. These scales are represented by novel "parsimonious graphs", called 7- and 5-Cyclops. It has also been shown an example of application to musical analysis, to engage both theorists and composers.


Colored cube dance

Moreno Andreatta (Italy-France) presents the study developed in collaboration with Alexandre Popoff and Corentin Guichaouha. The authors investigate the "Cube Dance," a concatenation of musical tonal transformations through "cubic" diagrams, proposed by Douthett e Steinbach, extending it to a monoid of binary relations, defined upon a set of major, minor, and augmented triads. After a discussion on the automorphisms' group of the considered transformations, it follows the presentation of an application web to try and listen. For detailed math-musical information, check out the blog by Alexandre.

Robert Peck (Louisiana, USA) focuses on combinatorial sets, and he considers the classical  transposition (P), inversion (I), retrogradation (R), and inverse retrogradation (RI). His research reminds us of Bach's music and more.

An example of tonnetz

About classics. Also a treatise of past centuries can tell us something on math and music. Sonia Cannas, in collaboration with Maria Polo, analyzes the Tentamen novae theoriae musicae, where Euler (yes, that Euler) elaborates a new musical theory using math. Sonia illustrates Eulerian theoretical system devised to justify pleasure in music listening, considering differences and similarities with other consonance theories. The side image is one of the first representations of  "tonnetz," a lattice with notes and musical intervals.

The second session is about algebraic and categorical models.

Second-species counterpoint and its mathematical formalization

Octavio Agustín-Aquino (Mexico) analyzes and further develops the mathematical model of counterpoint (first species, note against note) proposed by Guerino Mazzola, based on similar symmetries and strong dichotomies. Agustín-Aquino's research extends Mazzola's ideas to the microtonal sphere. He shows how to handle dissonances and extend the formalization to second-species counterpoint species (two notes against one). The mathematical analysis of counterpoint was also addressed by Dmitri Tymoczko. In the style of the mathematical duels of the past centuries, we don't miss different opinions and mutual criticism!

Poster of the virtual museum by G. Baroin on math-musical objects. On the background: the museum and the Pyrenees

I mentioned above the virtual museum on mate-musical objects. With a view of the Pyrenees from the windows to look out on with the VR viewer, the museum devised by Gilles Baroin (scholar with a 20 years of experience in 3D animated CGI movies) contains an interactive collection, including intertwined hyperspheres, the Möbius strip, and the CubeHarmonic. These models allow us to visualize Traditional Harmony, Atonal, Spectral, Non-Equal Temperaments, Fourier Phases, and Microtonal Music, projected on different geometric objects from circles to hyperspheres.

David Orvek (Indiana, USA) e David Clampitt (Ohio, USA) focuses on "SUM" classes. They are sets of pitch classes  whose elements sum to a given value. The two Davids  develop algebraic properties of SUM-class systems and define quotient generalized interval systems, extending the group-theoretic concept.

Remotely connected from Italy, Greta Lanzarotto and Ludovico Pernazza present their research on rhythmic canons. They focus on aperiodic tilings, i.e., tiling canons where inner and outer voices are aperiodic. Until today, we do not know a recipe to build all aperiodic tiling rhythmic canons (also called “Vuza canons”). Greta and Ludovico propose new algebraic constructs for extending the Vuza canons.

Gesture (on the left) and band (right) in the space of colors

During the second session, I presented research developed in collaboration with Juan Sebastián Arias Valero (Colombia), concerning the use of category theory to compare variations of orchestral timbres with (visual) color variations. The subjectivity of single associations is absorbed into classes of equivalence of perceptive similarity and similarity of transformational processes. We extend the notion of "musical gesture" to paths in the space of colors (as a Euclidean space R3 of RGB, or, in general, a variety) and in the space of timbres, including simplicial complexes and infinity-groupoids. A complementary discussion is based on bigroupoids.

The session is ended by Thomas Noll (Germania-Spagna) e David Clampitt (Ohio, USA). In their study, the pairwise well-formed (PWWF) modes, represented as words over a 3-letter alphabet, are studied transformationally. The authors prove that all PWWF modes may be generated by certain transformations, which are, however, not closed under composition. Thomas and David also present a new construction for the generation of PWWF modes, with transformations of words over a 4-letter alphabet, conjecturing that these transformations form a monoid.


Conference proceedings & breakfast

In the third session, we consider methods for musical analysis.

Gonzalo Romero-García (Spain-France) presents the research developed in collaboration with Isabelle Bloch (France) and Carlos Agon (Colombia-France) on "musical morphology." The authors define musical operators, that are responsible for transformations in a musical score, through time-frequency groups. Two examples: erosion and dilation, to remove and add information, respectively, from/to fundamental structures of a musical piece.

Paul Lascabettes (France) and co-authors (Carlos Agon, Moreno Andreatta, and Isabelle Bloch) define matrices of similarity in musical structures, and they pursue research with filters and musical operators. Matrices of similarity are frequently used also for signal processing. A particular use of matrices to characterize symbolic musical sequences is proposed in an article on quantum mechanics and music.

Fourier coefficients for musical analysis

Emmanuel Amiot (Francia) e Jason Yust (Massachusetts, USA) investigate the meaning of Fourier coefficients for musical objects such as scales and rhythms. Whereas in usual musical spaces, coordinates independently testify to the absence or presence of some note, Fourier coefficients appear to carry musical characters (such as diatonicity, chromaticism, major or minor…) with unerring precision and could possibly mirror some part of human perception of music. For instance, the high value of the fifth coefficient in the side picture indicates the prevalence of fifth intervals in any music composed with the notes involved in this interval. Emmanuel and Jason analyze the musical meaning of different products of Fourier coefficients, characterizing major/minor modes and diatonic/pentatonic scales.

Jordan Lenchitz (Florida, USA) and Anthony Coniglio (New York, USA) synthesize the information contained in the spectrum of a musical audio file, through a suitable logarithmic "chromogram." Richard Leinecker and William R. Ayers (Florida, USA) work in the field of microtonal music, proposing a new synthesizer.

In the fourth and last session, we discuss techniques of computational modeling of musical phenomena.

Detail of timbral modeling

The research by Matthew Klassen (Washington, USA) focuses on audio signal modeling through cubic splines. With splines, the author represents cycles and short segments of audio, to produce a small model of an instrument sound. This process allows the mixing, or blending, of instrument sounds efficiently with very little data. Matthew also considers timbre and timbric variation as paths from a point to another one, made possible via computer science. In this sense, my research with Juan and the study by Matthew have elements in common.

The N2D3P9 function evaluated on a musical dataset

Dave Keenan (Australia) and Douglas Blumeyer (California, USA) develop a function to improve sagittal musical notation. Given a rational number n/d to indicate a pitch (relative to some tonic note), N2D3P9 estimates its "rank in popularity" among all rational pitches in musical use. A low value of N2D3P9 indicates that the ratio is used often, and so should have a simple accidental symbol, while a high value indicates that the ratio is used rarely and so can have a more complex symbol if necessary. The function N2D3P9 may also be useful in designing rational scales or tunings.

Kjell Lemström (Norway) presents two studies in collaboration with Antti Laaksonen and Otso Björklund on the automatic retrieval of musical patterns through time-frequency representations.

Tonal functions and degrees of the scale as a graph

Francisco Gómez and Isaac del Pozo (Spagna) discuss the tonal function classification. Whereas some definitions of tonal function are based on consonance and dissonance, they do not work for kinds of music where the dissonance is present in the basic chords such as jazz music or music from the "extended common practice" in the sense of Tymoczko, from Middle Ages to today). The work by Francisco (Paco) and Isaac introduces a model of tonal function based on optimal voice-leading, allowing one to define it when chords have different cardinalities.

The last talk of the conference has been my presentation on HyperCubeHarmonic, the 4-dimensional version of CubeHarmonic, made in collaboration with Japanese professors Takashi Yoshino and Yoshifumi Kitamura, and the Chinese-French researcher Pascal Chiu. We talked about CubeHarmonic in a post sulla NIME (in Italian).

Effect of a permutation on the eight cubes of the CubeHarmonic

Amongst the interesting presented posters, I mention in particular a study on quantum mechanics applied to music by Peter beim Graben (Germany) and Thomas Noll, and a study on homology applied to music analysis by Victoria Callet (France). On "Quantum Music," an emerging research field, there's a blog post (in Italian) on the homonymous conference. From this conference, there are available videos in English as well.

In Mathematics and Computation in Music 2022, there were no concurrent sessions. The plenary sessions in the sense of lectio magistralis were in the form of collective or individual panels, concerning the future of mate-musical research, the different ways of indicating the musical interval "Do-Mi" depending on the theoretical approach considered (Julian Hook), the background necessary to approach the studies between mathematics and music.

Flowers on campus. Pic @M. Mannone

Concerts have seen the realization of some of the presented ideas, as the musical composition based on a Hamiltonian chord-path (Moreno Andreatta), the visualization of complex musical geometries (Gilles Baroin), but also the graphic-rhythmic structure of tango. Classic pieces such as “Faust” Sonata op. 28 by Rachmaninoff and “Les Funérailles” by Liszt have been presented and interpreted on the piano by...  established mathematicians, respectively Emilio Lluis-Puebla ed Emmanuel Amiot. One of the panels and one of the concerts (with Agustín-Aquino) have been a homage to Jack Douthett, influencing math-musician, prematurely passed away.

Before ending, let me give some reading suggestions, besides the classic "preprints" posted on ArXiV and ResearchGate. To deepen understanding and be guided step-by-step, here's a collection of essays on pedagogy of mathematics and music, edited by Mariana Montiel and Francisco Gómez, and a collection of approaches to math-musical research, edited by Mariana Montiel and Robert Peck. (The drawing and the graphic respectively present on the cover of these books are my works). An in-depth study on the application of the Fourier series and Fourier transforms to musical analysis is present in the book by Emmanuel Amiot. An overview of mathematical spaces for music  is provided in the forthcoming book by Julien Hook. A monography on rhythm is the book by Jason Yust.

Collaboration between scientists and artists to develop new ideas and connect art and science is in itself an instrument of peace. Perhaps, not just an anesthetic, like Hardy's anodyne, but a real cure.

Greetings from me and the panther, and see you in 2024!

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