Tuesday, March 5, 2013

The Science of Music: Part 2

by kathz, This post originally appeared at Function of a Rubber Duck.

This is a long overdue post but as the saying goes, anything worth it is always worth waiting for,  right? In my previous blog post, I explored the frequencies of tones and the relationships between notes that sound “good” together. In the following paragraphs I will talk about overtones, something that might not be so familiar to those who do not personally play a musical instrument.

To start off, what are overtones? Overtones are also known as harmonics, artificial harmonics, or partial tones, depending on the instrument. (Although music professionals might say that these terms can imply drastically different meanings as well.) They are notes that you hear when another note is played; for example, when we play a low G, we may also hear at the same time a higher G two octaves up, even though we did not play that note. In fact, for a trumpet player, overtones are essential to the music they play – how else can they create a spectrum of notes with only three keys to press? Finally, on a flute, we can observe similar trends. Using the fingering for middle C, one can play other notes such as C (in other octaves,) G, and E simply by blowing harder.  From personal experience as an ARCT flautist, I notice that the higher notes obtained from this manner sounds richer and far more complex than the notes obtained by the regular fingerings. Overtones exist for all instruments, whether strings, piano, woodwinds, or brass, but they are most easily perceived by woodwind and string players.

How can we predict which notes will result as overtones of the note we are playing?
Daniel Bernoulli, a mathematician from the 1700s, noticed a pattern while studying guitar strings and how they vibrate. He first found out that the fundamental vibration, or first harmonic,  on a string can be written as sinπx. He then discovered that the next overtone’s vibration length is sin2πx, the one after that sin3πx, and so on with each overtone. This trend can be easily graphed and understood, and I encourage you to try to derive the following graph yourselves:
So how does the string actually look like? It is able to vibrate at all these frequencies naturally but can only take on one form. Bernoulli then derived that the general vibration of the string would be a₁sin(πx) + a₂sin(2πx) + a₃sin(3πx) + a₄sin(4πx) +……. for some coefficients a₁, a₂, a₃, a₄, etc.
It would take the brilliant minds of many other mathematicians, such as Euler, to come up with a way to find the actual value of these coefficients – but that’s another adventure that would require a much higher level of math and music to explain and comprehend.

I hope you can now see more sides to music. It can be at times, just as technical as your physics textbook or a professor’s lecture, but be warned! Don’t let these technicalities behind it hinder you from enjoying music in your life. For music is foremost, an art, a freedom, and a language that will live on as long as life does. Thanks for reading!
About this contributor: Kathyz: An idoyncratic gr10 student who loves playing the piano, flute and violin and enjoys reading historical fiction and Edgar Allan Poe. She completes jigsaw puzzles in her spare time and aspires to learn Latin as well as publish a children's book in the near future.

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