arXiv subjects

Jeffrey R. Boland

Publications and source records attributed to Jeffrey R. Boland.

3 recordsLinked to original sources

arXiv2026

Tonnetz Theory, Classical Harmony, and the Combinatorial Geometry of Abstract Musical Resources

In a previous submission, we established a fundamental relation between tone networks and configurations. It was shown that the Eulerian tonnetz can be represented by a $\{12_3\}$ of Daublebsky von Sterneck type D222. We also constructed a tonnetz for Tristan-genus chords (dominant sevenths and half-diminished sevenths) and showed that this tonnetz can be represented by a $\{12_3\}$ of type D228. In both constructions the associated Levi graphs play an important role. Here we look at the tonnetze associated with some other musical systems, thereby offering concrete examples of an abstract view of music as combinatorial geometry. First, we look at the tonal harmonies of the classical period. In the case of diatonic triads, we show the existence of a bipartite graph of type $\{7_3\}$ and girth four that represents the relations between the seven diatonic degrees and their pitch classes. In the case of diatonic seventh chords, we obtain a Fano configuration $\{7_3\}$, which gives a characterization of the voice-leading relations that hold between such chords. Next, we construct a tonnetz for pentatonic music based on the Desargues configuration $\{10_3\}$ and we construct a tonnetz for the 12-tone system based on the Cremona-Richmond configuration $\{15_3\}$. Both can be used as resources for compositions. Finally, we show that the relation between the chromatic pitch class set and the major triad set is represented by a D222. The minor triads are in one-to-one correspondence with the members of a certain class of hexacycles in the Levi graph of this configuration. In this way, the characteristic duality between major and minor triads in the tonnetz can be broken.

math.CO↗

arXiv2026

Configurations, Tessellations and Tone Networks

The tonnetz, which is commonly represented as a tessellation of the plane by a triangular network of tones, can also be represented as a bipartite graph of degree three with twelve vertices denoting major triads and twelve vertices denoting minor triads. We show that this Levi graph can be realized geometrically as a system of twelve points and twelve lines in $\mathbb R^2$ with the property that three points lie on each line and three lines pass through each point, in a configuration $\{12_3\}$ of Daublebsky von Sterneck type D222. This tonnetz configuration, alongside various generalizations thereof, can be used as a new basis for the composition and analysis of music.

math.CO↗

arXiv2023

Mathematical Foundations of Complex Tonality

Equal temperament, in which semitones are tuned in the irrational ratio of $2^{1/12} : 1$, is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of $2$, $3$, and $5$, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system. The tritone, so problematic in just temperament, given ambiguously by the ratios $\tfrac{45}{32}$, $\tfrac{64}{45}$, $\tfrac{36}{25}$, $\tfrac{25}{18}$, none satisfactory, is in our scheme represented by the complex ratio $1 + \rm{i} : 1$. The major and minor whole tones, given by intervals of $\tfrac{9}{8}$ and $\tfrac{10}{9}$, can each be factorized into products of complex semitones, giving us a major complex semitone $\tfrac{3}{4}(1 + \rm{i})$ and a minor complex semitone $\tfrac{1}{3}(3 + \rm{i})$. The perfect third, given by the interval $\tfrac{5}{4}$, factorizes into the product of a complex whole tone $\tfrac{1}{2}(1 + 2\rm{i})$ and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of low powers of Gaussian primes leads to the construction of a complete system of major and minor scales in all keys.

cs.SD↗