Musikality Net

Today, I’ll talk about a subject that can be a little be confusing to beginners in digital music: Fourier transform and harmonics. A harmonic is defined by wikipedia as follows

In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the fundamental frequency is f, the harmonics have frequencies f, 2f, 3f, 4f, etc. The harmonics have the property that they are all periodic at the fundamental frequency, therefore the sum of harmonics is also periodic at that frequency. Harmonic frequencies are equally spaced by the width of the fundamental frequency and can be found by repeatedly adding that frequency. For example, if the fundamental frequency is 25 Hz, the frequencies of the harmonics are: 25 Hz, 50 Hz, 75 Hz, 100 Hz, etc.

Harmonics are what give to an organ its dinstict timbre. If you strip an instrument of all of its harmonics and leave only the fundamental frequency, it will sound the same on all instruments.

Various factors contribute to the different harmonics produced by each instrument. The end result is that two notes never sound the same on two different instruments.

A piano waveform (let) and a violin waveform (right)

But, how can we know what harmonics are included in an instrument?

To achieve this we use Fourier Trasnform.

Fourier transform is defined by wikipedia as follows

In mathematics, the Fourier transform is an operation that transforms one complex-valued function of a real variable into another. The new function, often called the frequency domain representation of the original function, describes which frequencies are present in the original function. This is in a similar spirit to the way that a chord of music can be described by notes that are being played. In effect, the Fourier transform decomposes a function into oscillatory functions. The Fourier transform is similar to many other operations in mathematics which make up the subject of Fourier analysis. In this specific case, both the domains of the original function and its frequency domain representation are continuous and unbounded. The term Fourier transform can refer to both the frequency domain representation of a function or to the process/formula that "transforms" one function into the other.

Alright, what all this means is the following.

A waveform contains many harmonics. Each harmonic can be represented by a mathematical function. The simplest wave function that can be is the sine wave. A sine wave contains only one frequency.

A sine wave along with some mathematical properties

What we do with fourier transform, is to take the function of a waveform and make it simpler, by finding the correspondent sine waves. So, we can derive the harmonic content of a waveform, and then represent it on a spectrum analyser. In the spectrum analyser we actually have each frequency that is present in the waveform and its corresponding amplitude, like shown in the picture..

Not difficult, ah?

In future articles we will write more things about spectrum analysers and fourier transform. I am also planning to include an article on Native Instruments’ Absynth, which allows you to control each individual harmonic of a waveform.

This entry was posted on Saturday, January 10th, 2009 at 7:16 am and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.