The Sound of Mathematics
Pythagoras’ Experiment
What I just played is the scale for C major. So where does the harmony of that sound come from? It happens that how your brain interprets sound and even how music is made is influenced by mathematics….
When sound travels a medium like air, the air molecules move back and forth while it travels along. The number of times sound waves move back and forth in one second is called frequency which is measured in hertz. So, this musical sound, the C note,
has a frequency of 261.63 Hz. In short, the frequency of the sound is what defines the musical note
It has been observed that when a frequency is multiplied by 2, the note remains the same. For example, the C note (261.63 Hz) multiplied by 2 is 523.26 Hz, which is also a C note, just one octave above.
If the goal was to lower an octave, it would be enough to divide it by 2. We can then conclude that a note and its respective octave maintain a ratio of ½.
Before we continue let’s go to the past, to ancient Greece. At that time there was a man named Pythagoras. Yep! it’s exactly who you think it is. Pythagoras made important discoveries for music too as I’m going show.
Pythagoras’ experiments
Imagine a stretched rope attached to its ends. When we play this string, it vibrates.
Pythagoras decided to divide this string into two parts and touched each end again. The sound produced was the same, only higher (since it was the same note an octave higher).
Pythagoras did not stop there. He decided to try what the sound would look like if the string was divided into 3 parts:
He noticed that a new sound appeared, different from the previous one. This time, it was not the same note an octave higher, but a different note, which needed to be renamed. This sound, despite being different, combined well with the previous sound, creating a pleasant harmony to the ear.
So, he continued making subdivisions and mathematically combined the sounds creating scales that later stimulated the creation of musical instruments that could reproduce these scales.
Given a scale, how do you move from one musical note to the next. By frequency analysis, it’s been established that by multiplying the frequency of one note by 1.0595, you get the frequency of the next higher note. And by dividing by 1.0595, you get the next lower semitone.
For example, the frequency of the B note is 246.9 Hz and that of the C note is 261.6 Hz.
By multiplying the frequency of the B note by 1.0595, we have:
246.9 x 1.0595 = 261.6 Hz (C note)
Since our goal is to maintain this same distance to the other notes, we will use this procedure to find out which note will come after C. By multiplying the frequency of the C note by 1.0595, we have:
261.6 x 1.0595 = 277.2 Hz (C sharp note)
Repeating this procedure to see what comes after C sharp:
277.2 x 1.0595 = 293.6 Hz (D note)
By following this logic, we can form the entire chromatic scale! That is, after multiplying the frequency of the C note by the number “1.0595” twelve times, we will return to the C note.
