Mathemusics

What is the acoustic difference between Bach’s sonatas and an engine roar in the emotionally blind eye of science? Where is the border that separates so-called music from other, different sounds? To what kind of audible waves do we choose to listen? What are the characteristics of sound waves that give us chills? The answers to such questions remain mysteries of the human mind. Only through the scholarly intersection of physics, musicology, cognitive science, biology and psychology can the answers to these questions begin to be explored.
In this article and following ones, I will attempt to scratch the surface in providing insight into the complex mystery of the human emotional perception of sound. This article requires a basic background in physics and a minimum familiarity with music.  To get started, let me talk about the main elements of music from a music theory point of view and then provide a basic physical interpretation of that.
Pitch (melody and harmony) and rhythm are widely considered to be the two main elements of music. It is common to model rhythm in a (semi-) mathematical form. Rhythm is the frame that encompasses the pitch and moves it along as music. Therefore, it can be mathematically represented as the series of sounds and their distribution over time, which usually repeats, independent of the pitch. On the other side, pitch is a sequence of sound waves with different frequencies that ride over the rhythm. Is there any mathematical relation among different wave lengths in a music piece? It is well known that we don’t hear arbitrary frequencies in a song; rather, the frequencies seem to take on certain values which form a so-called scale.  Two most common scales are major and minor scales. C major scale is perhaps the most famous one: Do Re Mi Fa Sol La Si (Ti), or C D E F G A B.  Once the scale finishes, a new scale starts where each note is within the octave distance of the corresponding note in the last scale.
There are several ways to assign frequencies to the notes in scales. However, there has been a consensus in both east and west that the pitch or frequency should double within an octave distance. For example, if you fix a note, for example A, with 440 hertz, you can receive different other As in an octave lower or higher on 220 hertz or 880 hertz. (Note: This is a famous frequency and it corresponds to the middle A on a piano, or the sound that you hear when the orchestra tunes right before the beginning of the concert).
Different intonation systems, however assign different frequencies to notes inside a scale. Interestingly enough, the Iranian scientist and philosopher, Abu Nasr Farabi was among the individuals that helped establish the equal temperament system. In short, in the proposed Farabi-Bach system, the notes within a scale are distributed uniformly according to logarithmic distance of the octaves. This is the way that modern western instruments are made and tuned nowadays.  The second column of the following table is the frequency chart of a “Farabi-Bach Equal Temperament” C-major scale in the middle of the piano. Equal temperament is an approximation to another system that has been practiced for centuries in east: Just Intonation. In Just Intonation, notes within the scales form simple ratios of the first note in that scale called magical ratios. The third column shows these ratios in a C-major scale.
Despite the fact that equal temperament makes the music world much easier to work with (for instance because every pair of adjacent notes has an identical frequency ratio), I personally have found its eastern sister quite insightful in explaining why we enjoy music in our own way. Below I give you one example of a music figure that we like.
If you have taken some music lesson in your life, you might still be able to play simple chords on your instrument since they are usually easy to play and sound surprisingly good. When you look at a, say, C major chord, it includes C, E, and G. C major is the most important chord in C major scale. You may observe that we have picked the note with the same name of the scale and the first two notes that are each one note away from the last note that was chosen in the scale. But this is not all. It is easy to verify that C major is the answer to the following optimization problem:
Pick three notes n1, n2, and n3 in C major scale such that they minimize the denominator of n1+n2+n3. (I mean summing up the fractions. Forgive me for the lazy notation!) At this point, you may wonder whether this is a phony argument and one could come up with other objectives that C major chord optimizes. In fact, I will show in the next issue that this objective function or rather a class of objective functions including this one has a physical meaning and is compatible with other music figures that we tend to like.

Copyright Nelia Jafroodi