On the mathematical structure of Tonal Harmony

In this article we solve this ancient problem of perfect tuning in all keys and present a system were all harmonies are conserved at once. It will become clear, when we expose our solution, why this solution could not be found in the way in which earlier on musicians and scientist have been approaching the problem. We follow indeed a different approach. We first construct a mathematical representation of the complete harmony by means of a vector space, where the different ton...

Mathematical music theory has assumed without proof that musical notes can be associated with the equivalence classes of $\mathbb{Z}_n$. We contest the triviality of this assertion, which we call the Pitch-class Integer Theorem (PCIT). Since the existing literature assumes the PCIT without proof, the mathematical language to rigorously treat the PCIT does not yet exist. We axiomatically construct an abstract study of harmony to generalize its fundamental tenets and to show th...

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. Without confining the study to the twelve-tone equal-temperament (12-TET) system, we propose a set of basic axioms based on musical observations. The axioms may lead to scales that are reasonable both mathematically and musically in any equa...

This paper provides a new mathematical axiom system for classical harmony, which is a prescriptive rule system for composing music, introduced in the second half of the 18th century. The clearest model of classical harmony is given by the homophonic four-part pieces of music. The form of these pieces is based on the earlier four-part chorale adaptations of J. S. Bach. Our paper logically structures the musical phenomena belonging to the research area of classical harmony. Its...

In this paper we present a mathematical way of defining musical modes and we define the musicality of a mode as a product of three different factors. We conclude by classifying the modes which are most musical according to our definition.

We consider the space of all musical scales with the ambition to systematize it. To do this, we pursue the idea to view certain scales as basic constituents and to "mix" all remaining scales from these. The German version of this article appeared in Mitteilungen der DMV, volume 25, issue 1.

This paper attempts to look for a mathematical method of composing music by incorporating Schonbergs idea of tone rows and matrix theory from linear algebra. The elements of a note set S are considered as the integer values for the natural notes based on the C Major Scale and rational numbers for semitones. The elements of S are effectively mapped by a polynomial function to another note set T. To accomplish this, S is treated as a column vector, applied to the matrix equatio...

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 whole positive numbers. The structure of the chord can be analysed mathematically by finding functions which are unchanged upon chord transposition. These functions are are denoted invariant, and are important for understanding the structur...

Understanding the structural characteristics of harmony is essential for an effective use of music as a communication medium. Of the three expressive axes of music (melody, rhythm, harmony), harmony is the foundation on which the emotional content is built, and its understanding is important in areas such as multimedia and affective computing. The common tool for studying this kind of structure in computing science is the formal grammar but, in the case of music, grammars run...

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 t...