Twelve theorems for music and engineering

Mathematics are one of the fundamental pillars of music; they are the basis, for example, of different musical theories; or algorithmic composition, a technique based on the use of algorithms for the creation of new scores.

And both disciplines have also been the passion, for years, of Luis Nuño, a researcher at the Polytechnic University of Valencia (UPV) in Spain.

In 2020, Luis Nuño created an innovative musical periodic table and a few years before he presented different musical abacuses. Now, he has publicly presented the results of the fusion of his two great passions: twelve new theorems for music and engineering. His work makes it possible to make musical composition even more mathematical and will also facilitate the analysis of musical pieces, “because many of the modern works cannot be analyzed with classical harmony”, Nuño justifies.

As Nuño adds, “with this work we have the complete information that relates each set of notes with any other and with its complement. This makes it easy to compose using all this information. Now you have a set of notes with a certain loudness, and you can establish relationships from their subsets. With this proposal it is relatively easy to generate this type of coherence in the musical composition. It comes to add more theory to the analysis of contemporary composition”, highlights Luis Nuño, who carries out his research activity at the University Institute of Information and Communication Technologies (ITACA) of the UPV.

Luis Nuño. (Photo: UPV)

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

“This work goes one step further. Aimed at mathematician musicians and mathematician musicians, it proposes a new way of composing, new theorems, being able to use n notes per octave and relating the contents of the different types of scales with respect to the different types of chords. The results are expressed as vectors and matrices, in a purely mathematical way, and facilitate composition and musical analysis”, adds Luis Nuño.

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

The study is titled “Type and class vectors and matrices in ℤn. Application to ℤ6, ℤ7, and ℤ12. And it has been published in the academic journal Journal of Mathematics and Music. (Source: UPV)