Physics and Mathematics in Musical Composition

At the beginning the basic question is asked: “How is music born?”. Is it created spontaneously? We contemplate the birth of music and, in parallel, that of sculptures. The historical role of physics and mathematics is introduced. Furthermore, we summarize the basic knowledge of the tonal system which will be helpful for readers who are not familiar with the many rules in this field. The five-line notation system is explained. This comprises the meaning of clef, pitch, time signature, tempo, and other terms in music.

Here various tone intervals are described (e.g., unison, major third, octave, and other intervals). Subsequently, different scales such as the pentatonic, hexatonic, and heptatonic scales, follow. Among the heptanonic scales the major and minor scales are in focus. Since Pythagorean times tunings have become very complicated and problematic - even today, especially when instruments are played together. We will try to present them in as much detail as necessary to understand the tonal problems involved. Furthermore, the notion of harmonics and the color of sound (timbre) are briefly characterized. As special topics the horn fifth, the ancient musical instrument guqin, the Picardy third, the problem of the third and the devil’s interval are described.

This chapter is conceived for music friends, who are not familiar with physics, especially not with (musical) acoustics. We describe the differences between sound, tone and noise. Then, with respect to the physics of sound, several important quantities are introduced (sound pressure, particle velocity, wave speed, just to mention a few). These have to be used for comprehending, in particular, the formation of standing sound waves on strings and in tubes, and what happens, when both ends of a tube filled with air are open or one end is closed. After these preparative parts, the construction of musical instruments and the way how they generate sound are discussed in a compact way. The structure and physics of typical instruments are explained: a recorder and a clarinet as examples for aerophones, a violin and its family representing the chordophones, timpani and drums for membranophones, bells for idiophones and a theremin and synthesizers for electrophones.

As a new approach for analyzing musical scores, we present the phase-portrait method. Our purpose is to visualize objectively musical images. The phase plot approach is successfully applied to characterize the dynamical behavior of complex systems in physics. For the purposes of simplicity we illustrate this approach for the oscillation of a mass-spring system, both for a harmonic oscillation and a damped system. The displacement x and the velocity \(y=dx/dt\) of the mass are the two variables plotted in a plane defined with help of an Cartesian coordinate system (without the explicit time dependence). In the case of music the frequency of the tone is attributed to x and the change in frequency per unit time to y. Different from the physical system such a mass-spring example where the form of the trajectory is essential, in music it is necessary to visualize the details of the time dependence. For such cases introducing a third coordinate proves to be useful. Examples of such 3D portraits for pieces of music are presented.

This chapter is devoted to traditional European music which is analyzed on the basis of phase plots. One can clearly recognize in the phase plots the difference between monophony, polyphony, canons and concertos. Ornaments are usually written in musical scores with tiny letters or symbols and are sometimes misinterpreted by beginners. By contrast, phase plots of ornaments show substantial effects along the main line of melodies. Rhythms can also be presented clearly by this kind of plots, as underlined by examples of triplets and multiplets. The divergence and convergence in Bach’s music, the small joke of Mozart’s 180\(^\circ \) rotation of sheet music, symmetric scales of Messiaen and a (quite ambitious) discussion on the Fibonacci series in Bartók’s works are described.

We look at music from cultures outside Europe. While collecting the necessary original material we felt quite sad and sometimes even angry, because very often we were and still are too late: In many places native music had been forbidden and had disappeared. Another difficulty arises here, because many cultures have different notation systems or there exist none at all. In rare cases one finds very few transcriptions to our notation system or one can listen to luckily conserved early recordings. Before discussing music in different cultures, we report our small but effective research on lullabies collected from all over the world. In contrast to musical pieces written by certain composers, most lullabies are born and sung spontaneously and are passed on from generation to generation. We point out their interesting common features. We study music from northern people (from the Vikings, Sami, Ainu, and Inuit), Arabic music, Indian music, African music, music from East Asia, music from North American Indians, music from many countries of Central and South America and from some Pacific islands.

We discuss the characteristic features of several composers. Why do we know it is Mozart, when we hear a piece of Mozart? The music of Mozart is different from the music of Bach, Beethoven, or other composers, but how and why? In this context we will take a closer look at the music of Mozart, Beethoven and Schubert. Furthermore, we discuss why we can remember some music well and other music not, by using pieces from the music of the iconic film “2001: A Space Odyssey”. There is also some music of which we know immediately what it is, when we hear only a few bars at the beginning. “The Fate Symphony” of Beethoven is a perfect example of this kind. A regular rhythm generally helps us to recall the music’s origin. Finally, Pictures at an Exhibition of Mussorgsky and the music of the Beatles are viewed through a magnifying glass.

Visualization of music is the first theme here. Dukas was inspired by Goethe’s ballad “Der Zauberlehrling” and composed “The Sorcerer’s Apprentice”. Later Disney made a trick film to visualize this story, a remarkable achievement at the time. There are many pieces of music that express animal features: and behavior: “Flying of the Bumblebee” in a piece of Rimsky-Korsakov, a bird, a wolf, a duck and a cat in “Peter and the Wolf” by Prokofiev, and 12 animals in “The Carnival of the Animals” by Saint-Saëns. Among animals the birds are the most commonly used species for inspiring music, from the Middle Ages to the era of New Music. The second theme in this Chapter is the role of noise in music. Debussy expressed the sound of the sea (noise) and composed “La Mer”. Other composers recorded noises such as those stemming from trains and used this directly for their compositions. Some try to compose music from a sequences of symbols and numbers (serial music). We consider the music after 1910, and ponder the future of music: Has the resource of music expired? If not, where can we find one nowadays? How can we find them? Let us be optimistic and have a nice time when reading this volume!