*Saba Goodarzi, William Sethares*

**Inharmonic Vibrations of Three-String Networks**

Connecting three strings together and setting them into motion produces a sound that is not a periodic (harmonic) vibration. We study the resonant frequencies of three-string networks by examining the roots of the spectral equation (as given in the work of Gaudet, Gauthier, and colleagues). A collection of scaling laws is 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. Numerical optimization is used to solve the inverse problem (where a desired set of frequencies is specified and the parameters of the system are adjusted to best realize the specification). The intrinsic dissonance of the overtones provides an approximate way to measure the inherent inharmonicity of the sound.

*Gennaro Auricchio, Luca Ferrarini, Greta Lanzarotto*

**An Integer Linear Programming Model for Tiling Problems**

In this talk, we will present an Integer Linear Programming model for computing all the aperiodic tiling rhythmic canons given a generic rhythmic motif and a period. We characterize those rhythms as the solutions of an integer linear problem. To conclude, we will show the results of several experiments done to validate the efficiency of our model.

*Maria Mannone*

**Color and Timbre Gestures**

Visual forms can be thought of as the result of drawing gestures; similarly, musical articulations, phrases, contours can be related to performing gestures. Thus, the mathematical description of gestures becomes a tool to compare the world of visual forms with the world of musical shapes. Such a comparison can be moved forward, defining gestures

in the space of timbres and gestures in the space of (visual) colors. This challenging development may help face the complexity of similarity between color and timbre variations. Along with formalism, the results of a recent experiment are mentioned as well.

*Julio Estrada*

**The interval classes d1 theory in the chromatic continuum of scales. **

**Their representation through the identity and the permutahedron: the case of the 12 tone scale.**

The division of the musical matter into discontinuous – any scale of rhythm, sound or space – and continuous -imperceptible high resolution of the previous elements – asks for the exploration of scales from smaller to those whose numerical dimensions are at the limits of our cognitive capacities. In the d1 theory – distance 1, minimal distance between any of its operations – the range of the scalar domain is arbitrarily fixed by its dimension, D, from D2 to D64 terms. Prime numbers serve to classify the scales according to their numerical affinities and combinations. The cognitive necessity of simplifying and generalizing the treatment of pulses, pitches and spatial points, asks the method to operate with interval classes. Any scale is classified through its interval accumulations, from 1 to n intervals; those are ordered from the smaller to the bigger to obtain a series of identifiers – freely denominated “identities” in d1 theory. The permutation of intervals of any identity – an expansion of the “inversions” in harmony – contribute to synthetize scale combinatorics, reduced again through the identity combinatorial content – a, ab, aaabc… – represented by the “permutahedron”. An example: D12 generates 77 identities, whose combinatory content is reduced to 4 groups and 10 subgroups producing a total of 36 permutahedron. Those last, identical in scales of smaller or bigger in dimension, contribute through d1 to interrelate the dense discontinuous space. Fragments of musical works illustrate the ideas.

*Jason Yust*

**Applying Fourier Transform of Pitch Classes to Theories of Tuning and Tone Systems**

There is a growing body of research applying the discrete Fourier transform on pitch classes to theory and analysis of tonality and harmony. One simplifying assumption made in most of this work is that pitch classes occupy can only twelve equally spaced locations per octave. While this is a reasonable assumption for most Western art music, it does not necessarily apply more generally. Even where it does apply, neglecting the contingency of this limitation can be misleading. For instance, the fifth DFT coefficient is often used as a measure of diatonicity, which works well in the twelve-tone context. But if the octave is differently discretized, the fifth and seventh coefficients will typically measure different things, which might be described as “pentatonicity” and “heptatonicity,” neither of them equivalent intuitively to diatonicity. In this paper I consider Fourier coefficients abstracted from divisions of the octave and apply them to questions about tuning and scales in Persian and Indonesian contexts. I also address some more abstract scale-theoretic questions: the connection of Carey and Clampitt’s generalized Pythagorean system to non-linear distortion effects in Fourier spectra, interactions of fifth-periodicity of scales with octave periodicity, and a new way of explaining the twelve-tone system as overdetermined, a solution not only to a well-known acoustic problem, but also special in its ability to support a multi-dimensional harmonic system at a relatively low cardinality.

*Jordan Lenchitz, Anthony Coniglio*

**Continuous Chromagrams, Generalized Octave Reduction, and Pseudometric Spaces of Sound Spectra**

In this talk, we extend the ubiquitous MIR technology of the chromagram (or chroma feature) to propose a continuous chromagram, an octave-reduced spectrogram that collapses the frequencies of a sound spectrum into the chroma octave [f, 2f). We prove that any chroma octave bounded by both the upper and lower frequencies of the spectrum in question yields the same continuous chromagram as any other chroma octave up to logarithmic scalings and

rotation as a corollary of a proof that this principle generalizes for collapsing spectral frequencies into [f, sf) for any positive real number s. We then propose families of pseudometrics on the set of all sound spectra based on generalized octave reduction and discuss their potential applications to analysis and composition.

*Jeremy Kastine*

**Can You Canon?**

I will introduce a video game based on rhythmic canons and discuss its use in promoting interest in and appreciation of the study of rhythmic canons (and mathematical approaches to music in general). In particular, I will discuss how I recently used the game in a general audience course on math and music, and how I intend to use it at a public outreach event during the upcoming International Conference on Mathematics and Computation in Music.

*Robert W. Peck*

**Enumeration and Musical Motives**

Aspects of equivalence classes function significantly in several mathematical musical theories: for instance, it is common to say that two pitch-class sets are equivalent under transposition or inversion, or that two beat-class sets are equivalent under metric shift or retrogradation. Enumerating these equivalence classes is tantamount to counting the

numbers of orbits of a group of transformations on these spaces This task can be accomplished with Burnside’s Lemma, or, in finer granular detail, by using Pólya’s Enumeration Theorem. Musical motives, however, involve aspects of both pitch and rhythm. Counting their equivalence classes is a significantly more complex endeavor. In this

case, two groups are acting on the relevant space. Those two groups need to be combined into a single action on the space: a structure called a “Power Group” (following Harary and Palmer 1966). We use the Power Group Enumeration Theorem (PGET) to count orbits of motives under the action of the transposition-and-inversion and metric-shift-and-retrogradation group. The PGET in its constant form determines the total number of these equivalence classes, whereas the polynomial form of the theorem determines the numbers of equivalence classes for motives of varying cardinalities.

*Mariana Montiel*

** A generative model of jazz solo improvisations using LSTM neural networks framed in the language of applied category theory.**

This talk will introduce an algebraic graphical language of neural networks, represented by string diagrams in a hypergraph category equipped with a monoidal endofunctor that endows the category with a diagrammatic operation of concatenation. This language is used to construct a neural network architecture to model sequences conditioned to time-structured context such as a music solo given a chord progression. To test our architecture, a generative model of jazz solo improvisation is implemented. This is a work in progress and further implementations are proposed before the improvisations attain a virtuosic or even acceptable level of performance.

*Alexandre Popoff*

**Musical Networks: a Categorical Journey**

Musical networks, since their introduction by Lewin and Klumpenhouwer in 1990, have been used to formalize not only the relationships between musical elements but also their evolution, through the notion of network transformations usually called ‘isographies’. In their original formulation, these so-called Klumpenhouwer networks (K-nets) are usually applied to the study of pitch classes and their interrelations via the action of the T/I group, or of major/minor chords via the action of either the T/I group or the neo-Riemannian PLR group. While Klumpenhouwer networks seem rather intuitive in their use, the lack of a proper formalization of their structure has led to discrepancies in their definition. Emphasizing their algebraic properties, in particular the compositional action of groups on sets of musical elements, general definitions relying on category theory have emerged, first by Mazzola and Andreatta in 2006 using the framework of denotators, then by Popoff, Andreatta, and Ehresmann in 2015 using diagrammatic functors. The purpose of this talk is to review the work developed on this latest approach since 2015. We will start with the definition of Poly-

Klumpenhouwer networks (PK-nets) as a categorical generalization of Klumpenhouwer networks, showing that the use of category theory allows us to readily define network transformations, among which the usual K-net isographies. This framework can then be extended to relational networks (Rel-PK-nets), using the category Rel instead of Sets. Relational presheaves, on which relational networks are built, establish bridges with graph-theoretic music ideas and open new possibilities for transformational music theory, which will be demonstrated through applications to harmony and meter analysis. Emerging perspectives for PK-nets will be also be presented, for example, their possible connection to DFT analysis.

*Dmitri Tymoczko*

**Repetition, Sequence, Transformation**

In my talk I will define a new theoretical tool, the “repeated contrapuntal pattern,” which can be found throughout music from the Renaissance to the present day. Repeated contrapuntal patterns unify a host of common devices from sequences to rounds to atonal “wedge” voice leadings to Renaissance idioms having no common name. Musically, they combine a voice leading with a permutation determining how that voice leading is to be reapplied. Geometrically, they are closely related to the parallel transport of vectors. I will also show how my approach can be generalized into a new and more concrete version of transformational theory, incorporating K-nets, neo-Riemannian transformations, motivic development, and a host of other devices.

*James Hughes*

**Generalizing Chord Space Orbifolds to Incorporate Loudness**

The utility of modeling chords using orbifolds and voice leading as paths in those orbifolds is well established. We extend the orbifold approach to include a loudness parameter for each voice, in particular allowing for the possibility of resting voices. Along the way, we take the opportunity to clarify some mathematical background on orbifolds, orbifold paths, and groupoids, and explore visualization of the models using braids.

*Giovanni Albini, Marco Paolo Bernardi*

**Limit and wonder**

We will present some research that the authors have conducted in the context of the application of mathematics in compositional practice and music theory. The dual potential for cognitive and creative purposes of mathematical instruments, capable of describing, revealing and manipulating elements of music in formal terms, will be emphasized. The apparent limit of abstraction is contrasted by the wonder of discovery, of the beauty on which it can shed light. Specifically, we will introduce some geometric and algebraic research that can be framed in neo-Riemannian theories and we will propose examples of compositional techniques and scores born in the wake of such research.