“Music is a secret exercise in arithmetic of the soul, unaware of its act of counting.”
Gottfried Leibniz
Music and mathematics may seem like two completely unrelated entities that have no connection to each other. But in reality, the relation between the two entities was found back in the ancient Greek era. Philosophers like Pluto and Aristotle have studied and researched the bond between math and music in great detail. Since then, it has slowly started getting attention for its unique and intriguing appeal. While it has been a controversial topic that has been fueling debates, it is necessary to first understand how it really works before coming to judgements and conclusions.
Let’s start with the very basics- rhythm.
Musical pieces are read much like you would read math symbols. The symbols represent some bit of information about the piece. Musical pieces are divided into sections called measures or bars. Each measure embodies an equal amount of time. Furthermore, each measure is divided into equal portions called beats. These are all mathematical divisions of time.
Fractions are used in music to indicate lengths of notes. In a musical piece, the time signature tells the musician information about the rhythm of the piece. A time signature is generally written as two integers, one above the other. The number on the bottom tells the musician which note in the piece gets a single beat (count). The top number tells the musician how many of this note is in each measure. Numbers can tell us a lot about musical pieces.
Each note has a different shape to indicate its beat length or time. Notes are classified in terms of numbers as well. There are whole notes (one note per measure), half notes (two notes per measure), quarter notes (four notes per measure), eighth notes (eight notes per measure), and sixteenth notes (sixteen notes per measure). These numbers signify how long the notes last. That is, a whole note would last through the entire measure whereas a quarter note would only last ¼ of the measure and thus there is enough time for four quarter notes in one measure. This can be expressed mathematically since 4 x 1/4 = 1.
It is important for musicians to understand the relationships and values of fractions in order to correctly hold a note.
Moving on to a more complicated concept now, let’s look at how frequencies are used to create harmonies and melodies.
Sound progresses as a wave through the air, and sound cannot be produced without an atmosphere. A sound wave creates minute areas of higher and lower air pressure, and all the sounds we hear are caused by these pressure changes. With music, the frequency at which these areas strike your ear controls the pitch that you hear.
A basic rule is that higher-pitched notes have a higher frequency. For example, the note Middle G has a frequency of about 392 Hertz, while the note Middle A has a frequency of about 440 Hertz.
No two musical notes fit together better than those which are exactly one octave apart. Pairs of notes like Middle C and High C. Or Middle G and High G. Notes which are one octave sound like a new variation on the same note.
Going up one octave is always the same as doubling the frequency. Notes one octave apart are always bound together by this basic relationship.
While octaves do sound well together, the result is sort of hollow. Let’s look at other intervals that are a lot more interesting.
Consider the notes, Middle C and Middle G. They are a perfect fifth apart.
This graph shows the sin graph for the notes.
We see that every second arrival for Middle C lines up perfectly with every third arrival for Middle G every 0.003823 seconds. The two wave patterns fit well together. The two notes complement each other, rather than clashing. This is commonly known as consonance.
By contrast, the notes Middle C and MiddleF-Sharp do not fit together well.
We can see that their graphs rarely ever line up. They have no simple relation to each other. The fit just isn’t there. This is known as dissonance.
Combining the notes C and G produces a sound which is fine, but not very exciting. To get a really pleasing sound, let’s add a third note – E. Middle E is practically between middle C and middle G. When all three notes are played together, they form the “C major chord”, which is a sweetly harmonious, happy sound. It is the basis for music as diverse as Row, row, row your boat, and the symphonies in C Major of Mozart and Beethoven and Schubert.
Why do these three notes – C, E, and G – sound so sweet together? Let’s have a look.
We see that the graphs of the three notes almost line up perfectly at 0.015 seconds.
So, even though certain of the areas do not line up for these three notes, quite a number of them do. This combination of variety and consistency is just what is required to produce the C Major chord – one of the most pleasing sounds known to humanity, and the basis of multitudes of tunes from Mendelssohn to Metallica.
So, humanity’s quest for beautiful music amounts to finding creative and interesting ways to line up the air pocket X’s of different musical notes.
Now you may ask, how is this used in music? Well, composers use consonance and dissonance to create tension in their music. The “off-sounding notes” that are not very pleasant to the human ear are used to create tension and the sense of impending doom, while the notes that do sound well together are used to resolve tension. It is by contrasting this dissonance with consonance, that composers add the unquantifiable elements of emotion and creativity to the certainty of mathematics.
The Math of Music 