Twelve Theorems for Music and Engineering

Luis Nuño, a Professor and researcher at the ITACA Institute of the Universitat Politècnica de Valencia, has just published a new paper in the internationally renowned Journal of Mathematics and Music.

Mathematics has always been a solid foundation of music theory. Thus, for instance, we currently use the equal temperament with 12 notes per octave; and, during the “common practice” period (around 1650 to 1900), the compositions are based on “tonal harmony”, specifically chords with 3 or 4 notes (triads or tetrads), whose notes belong to a few types of 7-note or heptatonic scales, mainly the major scale and some types of minor scales. Additionally, given a set of notes, its “complement” consists of the rest of the twelve notes. Regarding “post-tonal” music, since the early 20th century, some composers began experimenting with specific techniques based on transpositions, inversions, and other mathematical relations among the 12 notes.

It is common among musicians to know that a major scale contains 3 triads of the major-chord type. However, what is not so common is to know that, reciprocally, the complement of a major chord (which is a set of nine notes) also contains 3 complements of major scales (which are of the major-pentatonic type). This “complementary reciprocity” is established in the first of twelve recently published theorems. All of them were obtained considering an arbitrary number n of notes per octave (mathematically, for Zn) and relate the contents of the different scale types (with c notes) with respect to the different chord types (with m notes). The results are expressed as vectors and matrices, with a purely mathematical formalism.

From the engineering point of view, the elements of those vectors and matrices are “higher-order autocorrelations”, also known as “k-deck”, which are present in many fields of knowledge. Particularly, the 2-deck is used in “phase retrieval” problems, which arise when measuring a complex quantity (normally, the Fourier transform of a real quantity) and, due to different factors, only the magnitude can be obtained, but not the phase. Thus, the problem is finding the phase from the magnitude, for which some authors have proposed using the 3-deck. On the other hand, a “coded” version of the k-deck, with different values of k, is used in the “reconstruction problem” of specific information from fragments of it, which also gives rise to complex combinatorial problems.

We can find examples of those kinds of problems in such diverse scientific and technological areas as microscopy, holography, crystallography, neutron radiography, optical coherence tomography, diffraction grating design, reconstruction of character sequences, reconstruction of graphs, radar signal processing or quantum mechanics, to name a few.

This paper focuses on the discrete periodic problem and obtaining the vector and matrix relations among the k-deck and other related values, considering from k=0 to k=n, which provides a global insight and a detailed knowledge of these parameters. Furthermore, it seems likely to apply these ideas to other problems. The paper can be downloaded here.