Twelve Theorems for Music and Engineering

Mathematics is one of the fundamental pillars of music; it is the basis, for example, of different musical theories; or of algorithmic composition, a technique based on the use of algorithms to create new scores. And both disciplines have also been the passion, for years, of Luis Nuño, a researcher at the Universitat Politècnica de València.

In 2020, Luis Nuño created an innovative musical periodic table, and a few years earlier, he presented different musical abacuses. Now, the Journal of Mathematics and Music publishes the results of the fusion of his two great passions: twelve new theorems for music and engineering (under the title “vectors and matrices of types and classes”). His work makes musical composition even more mathematical. It facilitates musical piece analysis “because many modern works cannot be analysed with classical harmony”.

According to Nuño, “With this work, we have complete information that relates each set of notes to any other and to its complementary. This makes it easier to compose using all this information. Now you have a set of notes with a certain sonority and can establish relationships from their subsets. This approach makes generating this coherence in musical composition relatively easy.” Luis Nuño, who conducts his research activity at the University Institute of Information and Communication Technologies (ITACA) of the UPV, emphasises that “It adds more theory to the analysis of contemporary composition”.

As Nuño points out, mathematics has always firmly supported music theory. Thus, for example, twelve notes per octave are commonly used; and, during the period of what is known as “common practice” (approximately from 1650 to 1900), musical compositions were based on “tonal harmony”, namely 3- or 4-note chords belonging to a few types of 7-note scales, mainly the major scale and various types of minor scales. Concerning “post-tonal” music, since the beginning of the 20th century, compositional techniques based on transpositions, inversions and other mathematical relationships between the twelve notes have been used.

“This work goes one step further. Aimed at mathematical musicians and musical mathematicians, it proposes a new way of composing, new theorems, using n notes per octave and relating the contents of the different types of scales for the different types of chords. The results are expressed as vectors and matrices, in a purely mathematical form, facilitating musical composition and analysis,” adds Luis Nuño.

The UPV professor explains that the elements of these vectors and matrices are known as “k-decks”, which are also used in scientific and technological areas as diverse as microscopy, holography, crystallography, radar signal processing and quantum mechanics. “This work transfers these concepts to the field of music theory and composition, reinforcing the relationship that has always existed between mathematics and music”, concludes Luis Nuño.

The paper can be downloaded here.

Source: UPV’s Information Office