On the mathematical structure of Tonal Harmony

In this paper we present a mathematical way of defining musical modes, we derive a formula for the total number of modes and define the musicality of a mode as the total number of harmonic chords whithin the mode. We also give an algorithm for the construction of a duet of melodic lines given a sequence of numbers and a mode. We attach the .mus files of the counterpoints obtained by using the sequence of primes and several musical modes.

It is common for most people to think of science and art as disparate, or at most only vaguely related fields. In physics, one of the biggest successes of thermodynamics is its explanation of order arising from disordered phases of matter through the minimization of free energy; In 2019, Berezovsky showed that the mechanism describing emergent order from disorder in matter can be used to explain how ordered sets of pitches can arise out of disordered sound, thus bridging the ...

An optimal auditory tunable well (circular) temperament is determined. A temperament that is applicable in practice is derived from this optimum. No other historical temperament fits as well, with this optimum. A brief comparison of temperaments is worked out.

This article revisits the generation, classification and categorization of all-intervals 12-tone series (AIS). Inspired by the seminal work of Morris and Starr in 1974 (Morris and Starr, The Structure of All-Interval Series 1974), it expands their analysis using complex network theory and provides composers and theorists with the re-ordering scheme that links all AISs together by chains of relations.

We apply geometric group theory to study and interpret known concepts from Western music. We show that chords, the circle of fifths, scales and certain aspects of the first species of counterpoint are encoded in the Cayley graph of the group $\mathbb{Z}_{12}$, generated by $3$ and $4$. Using $\mathbb{Z}_{12}$ as a model, we extend the above music concepts to a particular class of groups $\mathbb{Z}_{n}$, which displays geometric and algebraic features similar to $\mathbb{Z}_{...

Motivated by the music-theoretical work of Richard Cohn and David Clampitt on late-nineteenth century harmony, we mathematically prove that the PL-group of a hexatonic cycle is dual (in the sense of Lewin) to its T/I-stabilizer. Our point of departure is Cohn's notions of maximal smoothness and hexatonic cycle, and the symmetry group of the 12-gon; we do not make use of the duality between the T/I-group and PLR-group. We also discuss how some ideas in the present paper could ...

We re-create the essential results of a 1989 unpublished article by Mazzola and Muzzulini that contains musicological aspects of a first-species counterpoint model. We include a summary of the mathematical counterpoint theory and several variations of the model that offer different perspectives on Mazzola's original principles.

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their d...

Sound consonance is the reason why it is possible to exist music in our life. However, rules of consonance between sounds had been found quite subjectively, just by hearing. To care for, the proposal is to establish a sound consonance law on the basis of mathematical and physical foundations. Nevertheless, the sensibility of the human auditory system to the audible range of frequencies is individual and depends on a several factors such as the age or the health in a such way ...

We propose a new way of analyzing musical pieces by using the exceptional musical icosahedra where all the major/minor triads are represented by golden triangles or golden gnomons. First, we introduce a concept of the golden neighborhood that characterizes golden triangles/gnomons that neighbor a given golden triangle or gnomon. Then, we investigate a relation between the exceptional musical icosahedra and the neo-Riemannian theory, and find that the golden neighborhoods and ...