The harmonic relationships between the frequency ratios composing a chord are analysed to provide an easy way to evaluate the chord's harmonicity properties. The harmonic space is used to describe a chord, where the harmonic distance and the harmonic length and co-length inform us about the degree of the chord's resonance and its deviation from harmonic symmetry. The distribution of the harmonics relative to the common fundamental is quantified by the harmonic dispersion and is visualized from the chord spheroid, which also accounts for the harmonic asymmetry. Its opposite, the harmonic likeness, is correlated with the perception of consonance.
Musical harmony, as the ancient problem of relating tunings and scales to the commensurability of sound waves, has been approached by Helmholtz in terms of a dissonance curve in the frequency domain. This model for musical intervals is here recast in an elementary number-theoretic and thermodynamic formalism, connecting the biophysical insight with Riemann's zeta function on the critical line and Minkowski's question mark measure. The former models rational relationships from a harmonic timbre in the Mellin-Fourier domain conjugate to log-frequencies, capturing the acoustic aspect. The latter minimally models the neurocognitive effort to assess commensurability in frequency pairs, emerging as the maximum entropy distribution on rational approximations generated by Euclid's algorithm. Motivated by just intonation, this ensemble extends the primon gas, a statistical mechanics model based on prime energy contributions. This framework gives Helmholtz's phenomenological dissonance curve a theoretical basis. The spectrum of the resulting fractal curve predicts the quasi-periods of widely used scales, from the pentatonic to microtonal divisions of the octave, providing a biophysical and mathemusical common ground for harmony across musical genres and cultures.
I stumbled by chance on a nice musical interpretation of an esoteric theorem due to Pierre Beauguitte that more clearly demonstrates its perceptual consequences. It is a translation of a statement on Fourier coefficients in terms of the $ \operatorname {IFunc} $ IFunc between two generated pc-sets, and it characterizes the non-periodic maximally even sets, those generalizing pentatonic or diatonic scales. As a simple example, a chromatic pentachord always has three notes in common with any diatonic scale, except for one case when there are only two; and the diatonic and pentatonic scales are the only pc-sets harbouring that relationship with these chromatic spans. A very slightly different and simpler condition characterizes periodic pc-sets.
Properties related to the discreteness of the phases of the Discrete Fourier Transform (DFT) of pitch-class (pc)-sets in the 12-tone universe were studied. We primarily investigated the conditions where the DFT phases are integer multiples of $ \pi /12 $ pi/12 for pc-sets of up to six tones (effectively covering all the pc-sets in the 12-tone universe). Pc-sets were classified into five classes according to whether or not each DFT component's phase was an integer multiple of $ \pi /12 $ pi/12. It was also found that phases of non-integer multiples of $ \pi /12 $ pi/12 tend to be associated with larger Fourier components. Additionally, discontinuities associated with pc-sets with indeterminate phases were also investigated by tracking the phase changes in certain chord progressions.
Between 2002 and 2003, the tiling process was one of the main inspirations for Tom Johnson, during which he composed several pieces dedicated to tiling the temporal axis: Tilework: 14 Pieces for 14 Solo Instruments, Tilework for String Quartet, and Tilework for Piano. The originality of his approach lies in the use of short cycle lengths, thus excluding Vuza canons, and in assigning pitch values to each voice rather than a single fixed pitch. This results in complex pitch sequences from several voices executing the same pitch values. However, these pitch sequences are periodic, i.e. they can be expressed as the repetition of a shorter sequence. In this article, we propose to adapt this compositional process to generate non-periodic pitch sequences. In particular, following Tom Johnson's tiling method, we generate pitch sequences from a tiling in which the voices use the same set of pitch values, but not from the same starting point. We explore the results of this approach and illustrate its application in the piece Rotation recorded in early 2025.
The Eulerian tonnetz, which associates three minor chords to each major chord and three major chords to each minor chord, can be represented by a bipartite graph with twelve white vertices denoting major chords and twelve black vertices denoting minor chords. This so-called Levi graph determines a configuration of twelve points and twelve lines in ℝ^2 with the property that three points lie on each line and three lines pass through each point. Interesting features of the tonnetz, such as the existence of the four hexatonic hexacycles and the three octatonic octacycles, crucial for the understanding of nineteenth-century harmony and voice leading, can be read off rather directly as properties of this {12_3} and its Levi graph. Analogous tone networks together with their associated Levi graphs and configurations can be constructed for pentatonic music and twelve-tone music, offering the promise of new methods of composition. When the constraints of the Eulerian tonnetz are relaxed so as to allow movements between major and minor triads with variations at exactly two tones, the resulting bipartite graph has two components, each of which generates a tessellation of the plane, of a type known to Kepler, based on hexagons, squares and dodecagons. When the same combinatorial idea is applied to tetrachords of the Tristan genus (dominant sevenths and minor sixths) the cycles of the resulting bipartite graph are sufficiently ample in girth to ensure the existence of a second geometrical configuration of type {12_3}, distinct from the Eulerian tonnetz as an incidence geometry, which can be used as the basis for a new approach to the analysis of the music of Chopin, Wagner, Tchaikovsky, Brahms and their contemporaries.
This article presents an application of mathematical morphology to the discovery of repeated patterns in a multidimensional representation of music. In particular, we demonstrate the possibility of discovering patterns from their onsets, defined as the translations of the pattern to obtain its occurrences in musical data. We first prove this result for fully compact patterns, meaning that they contain all dataset points in their convex hull. We then prove that patterns can be discovered from their onsets if and only if they are maximal with respect to their translational equivalence classes (MTEC), meaning that no points can be added to it without decreasing the number of occurrences. Moreover, we prove that applying twice the morphological erosion completes any pattern by adding the points required to make it MTEC, and yields an MTEC pattern and their onsets linked by erosion. Finally, we apply these results to musical examples, illustrating how morphological operators can be used as analytical tools for describing musical data.
The "selfRep melodies" as defined and explored by Tom Johnson in his compositions and books are a stunning example of the sheer mathematical power of Tom's questioning. Most of the results that he established empirically could be proved (using affine maps and their actions on cyclic groups modelling the indexes of periodic melodies), and a wealth of original results followed from his ideas. It shows the strength and depth of the link between mathematical and musical thinking, and very specially how Tom Johnson influenced and worked with the mathematical community in a way no one else could.
It has been wisely said that Tom Johnson was "quietly influential." In this article, I will illustrate how his work shaped my own by recounting the creative process behind my orchestral piece On and on and on and on and on and finally, which is entirely based on a continuously accelerated version of the theme from his La vie est si courte. Along the way, I will explore the mathematical challenges this process entailed, with a particular focus on the algebraic intricacies of Johnson's self-replicating melodies and the calculus-driven inner workings of in-phase eternal accelerandos.
Homometric pitch-class sets (pc-sets) share the same interval content, as captured by the interval vector or the magnitudes of their Fourier coefficients, yet they can sound noticeably different. This paper proposes homometric paths as a way to bridge the gap between structural characteristics and perceptual experience of homometric pc-sets and, more broadly, of real-valued pitch-class distributions (pc-distributions). These paths lie in the space of pc-distributions that are homometric to the given pair - that is, pc-distributions whose Fourier coefficients share the same magnitude profile - and are characterized by changes in Fourier phases. To support this idea, we introduce two interactive tools. The first is ZPath, a visualizer with synchronized audio for exploring homometric paths between pc-distributions. The second is HexaComp, an algorithmic composition platform that incorporates homometric paths into musical contexts. Together, these tools aim to render homometric relationships not only analyzable but also visible and audible.
In 2008, Tom Johnson composed a piano piece called Twelve. Twelve years later, he revisited the work he had done on the $ (12,4,3) $ (12,4,3) block design for that piece and composed Twelve Years Later, also for piano. By this time, we had exchanged many ideas that led to the creation of this work. I present some of them in this article, including the discussion of non-serial 12-note composition, and show that Tom, in a very intuitive way, reminiscent of Ramanujan's approach, had given a diagrammatic 'proof' of a theorem established by Luis Morales and Carlos Velarde stating that there are only 5 non-isomorphic resolvable block designs of $ (12,4,3) $ (12,4,3).
We propose a modest attempt to describe briefly some of the mathematical ideas behind some pieces of Tom Johnson, without claiming any sort of exhaustivity. Also we will give several references (papers, books, websites) that can help to understand some of the ideas that Tom used in his music.
This paper investigates the superhexatonic (hexatonic-plus-one) spelled heptachords. It aims to situate the superhexatonics (set class 7-21) among families of symmetrical and asymmetrical scales. The article is divided into four parts. Part one presents the structure of the superhexatonics and compares them with symmetrical scales. This comparison involves a new concept that I call "heptatonic symmetrical hybridization", by which a heptatonic asymmetrical hybrid is generated from two symmetrical scales. The second part begins by proposing a way of normalising the relationship between spelled heptachords and modes, using solf & egrave;ge syllables and the line of fifths. In the third part, I examine the superhexatonics in light of set theory and show how they can be generated by the same interval cycle as the double harmonic (or verbunkos) minor scale. I illustrate this by addressing two musical excerpts, one by Bart & oacute;k and the other by Liszt. I finish the article with an extensive analysis of a piece by Rosemary Brown, which, in my view, shows a highly rational and sophisticated use of a superhexatonic scale.
