相关论文: Musical intervals under 12-note equal temperament:…

Why are white and black piano keys in an octave arranged as they are today? This article examines the relations between abstract algebra and key signature, scales, degrees, and keyboard configurations in general equal-temperament systems.…

This paper deals with the algebraic structure of the sequence of harmonics when combined with equal temperaments. Fractals and the golden ratio appear surprisingly on the way. The sequence of physical harmonics is an increasingly enumerable…

The mathematics of musical intervals and scales has been extensively studied. Vastly simplified, our ears seem to prefer intervals whose frequency ratios have small numerator and denominator, such as 2:1 (octave), 3:2 (perfect fifth), 4:3…

A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…

In this article, for positive integers $n\geq m\geq 1$, the parameter spaces for the isomorphism classes of the generic point arrangements of cardinality $n$, and the antipodal point arrangements of cardinality $2n$ in the Eulidean space…

While motivated by structural problems in mathematical music theory, this article introduces a novel combinatorial framework that advances the classification of cyclic cubic bipartite graphs. We extend the classical study of Levi graphs by…

Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of…

Western music is predominantly based on the equal temperament with a constant semitone frequency ratio of $2^{1/12}$. Although this temperament has been in use since the 19th century and in spite of its high degree of symmetry, various…

Many music theoretical constructs (such as scale types, modes, cadences, and chord types) are defined in terms of pitch intervals---relative distances between pitches. Therefore, when computer models are employed in music tasks, it can be…

We study classes of musical scales obtained from shift spaces in symbolic dynamics through the first symbol rule, which yields scales in any $n$-TET tuning system. The modes are thought as elements of orbit equivalence classes of cyclic…

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…

Monads in category theory are algebraic structures that can be used to model computational effects in programming languages. We show how the notion of "centre", and more generally "centrality", i.e. the property for an effect to commute…

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…

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a…

In classical music and in any genre of contemporary music, the tonal elements or notes used for playing are the same. The numerous possibilities of chords for a given instance in a piece make the playing, in general, very intricate, and…

We propose a geometric hypothesis for neutrino mixing: twice the sum of the three mixing angles equals $180^\circ$, forming a Euclidean triangle. This condition leads to a predictive relation among the mixing angles and, through…

We consider a notion of "numerosity" for sets of tuples of natural numbers, that satisfies the five common notions of Euclid's Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show…

Two subsets of $\mathbb{Z}_n$ are said to be homometric if they have the same multiset of pairwise cyclic (i.e., Lee) distances. Homometric subsets necessarily have the same cardinality, say $k$. In this paper, for all positive integers…

Musical chords, harmonies or melodies in Just Intonation have note frequencies which are described by a base frequency multiplied by rational numbers. For any local section, these notes can be converted to some base frequency multiplied by…

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the…