The most commonly used tonnetz in the mathematical analysis of musical relationships represents different-sized intervals as being equal in length. However, a shift in perspective toward a geometrically-constrained tonnetz allows the analyst to account for differences in tuning among 12-tone tuning systems that are not equal-tempered. In this work, we construct a metric version of the 12-tone tonnetz with edge lengths proportional to the musical intervals they represent. Our approach begins with frequency ratios to construct the metric tonnetz out of individual triangles, calculating side lengths and internal angles. This construction provides insight into the curvature of the tonnetz as a geometric object, leaving us with a set of discrete invariants characterizing our metric tonnetz’s curvature concentrated at individual points. From these, we derive two additional invariants using Fourier methods to obtain a continuous function and a scalar measure for a more efficient comparison between tunings. All three of these invariants remain unchanged under the operations of transposition and inversion of the starting metric tonnetz. In the future, we aim to use these three invariants for the quantitative comparison of unequal-tempered tuning systems across the world and throughout history.
Conventional audio quality metrics such as SNR, THD+N, PESQ, and SI-SDR are designed primarily from engineering perspectives and often fail to reflect perceptual and musical judgments of sound quality. In musical contexts, small spectral or temporal distortions may have a disproportionate perceptual impact depending on harmonic structure, timbral salience, and cognitive expectation. This paper proposes a mathematically grounded framework for perceptually aligned audio quality assessment in musical signals. We model musical audio using perceptually weighted time–frequency representations and construct spectral graphs that encode harmonic, temporal, and psychoacoustic relationships between spectral components. Audio degradation is formulated as an operator-theoretic error acting on this structured representation. A psychoacoustic kernel is introduced to model masking, loudness, and temporal integration, while eigenvalue-weighted operator norms are employed to emphasize perceptually salient distortions. Based on this framework, we derive a novel music-aware quality metric that integrates spectral topology, perceptual weighting, and cognitive salience. The proposed metric is evaluated on diverse musical datasets with controlled distortions and validated against subjective listening tests. The experimental results demonstrate robust correlation with human judgments, highlighting the value of structural modeling alongside standard objective measures. The proposed approach establishes a unified mathematical foundation for perceptually meaningful audio testing and provides a practical tool to evaluate musical audio systems, codecs, and playback devices. This work bridges signal processing, music theory, and perceptual modeling, contributing a new class of quality metrics tailored to musical listening.
Writing for the modern pedal harp presents significant challenges due in part to the instrument occupying a space between the purely diatonic and fully chromatic, mediated by the constraints of the seven pedals. Idiomatic writing often involves the use of enharmonic respellings to minimize the number of pedal changes, especially for more chromatic passages. We address this problem with a graph-theoretic approach that maps sequences of harp pedal settings to a trellis graph, weights edges according to a cost function modeling the physical constraints of harp pedaling, and uses a modified Dijkstra’s algorithm for finding shortest (minimum-weight) paths through the graph, yielding optimal enharmonic respellings of the passage. Practical application of this approach is demonstrated in several examples, including an extended passage from the author’s composition Dear Matafele Peinam.
This research explores a component of the Labyrinth of Time (LDT), a polyartistic work by composer-researcher Stéphane de Gérando. The LDT integrates collaborations into its theoretical and experimental development phases, including work with Franck Jedrzejewski and Christophe Mourougane (homometric sets), Jérôme Pétri (relativistic spacetime), Louis Bigo (STIOZ, STISMI), and Gilles Baroin (hypersphere of spectra). Several of these collaborations have led to world premieres. To broaden the perceptual and compositional horizon, de Gérando introduces the concept of the n-dimensional polysensory hyperobject (HOPS), defined both mathematically and perceptually through control parameters engaging the five senses within an interactive immersive space. These hyperobjects encompass sound and visual objects (SVO), the principal challenge being to compose n correlated or decorrelated dimensions. This article provides a overview of this research since 2014 (Homometric Attractors). Subsequently, de Gérando restates the definitions of HOPS and SVO, and Baroin presents perspectives on transposing SVO into virtual reality within a four-dimensional geometric space.
