These Fourier coefficents are equivalent to the amplitudes of the sine and cosine functions corresponding to k-th harmonic (having frequency $ 2\pi k/T $).īelow are graphs of the Fourier coefficents of the different harmonics for six different musical instruments, all playing the A below middle C (440 Hertz). In Fourier analysis, a complicated periodic wave form, x(t), can be written as The reason for this is that the energy in each of the harmonics is different for the two instruments: the amplitudes of the simpler functions making up the complex wave that a person hears as a single note of sound are different. When two musical instruments play the same note, the notes have the same pitch for both instruments, but the two instruments sound different they have different timbres.
In principle, there are an infinite number of harmonics, but once the frequency of a harmonic is greater than 20,000 Hertz, the human ear cannot detect it, so these do not contribute to the sound that is heard. So, for example, the A below middle C has a fundamental frequency of 440 Hertz, a second harmonic of frequency 880 Hertz, a third harmonic of frequency 1320 Hertz, and the k-th harmonic will have a frequency of k*440 Hertz. We call f2 the second harmonic, f3 the third harmonic and so on. The frequency f1 is called the fundamental, and the frequencies f2, f3, f4, etc. Thus, in addition to f1, there are frequencies f2, which equals 2 f1, f_3, which equals 3 f1, f4, which equals 4 f1, and so on. These frequencies are integer multiples of the fundamental frequency f1. When this vibration occurs in a musical instrument, not only is the frequency f1 produced, but other frequencies are produced as well.
The period T is the length of time before the signal repeats, and the frequency $ f_1 $ equal to 1/T is the fundamental frequency or pitch of the note produced. This vibration causes a periodic variation in air pressure that is heard as sound. Musical instruments produce sound as a result of the vibration of a physical object such as a string on a violin, guitar, or piano, or a column of air in a brass or woodwind instrument. Sound waves are one type of waves that can be analyzed using Fourier series, allowing for different aspects of music to be analyzed using this method.
WAVE OF MUSIC NOTE C SERIES
Fourier created a method of analysis now known as the Fourier series for determining these simpler waves and their amplitudes from the complicated periodic function. Because of their familiarity and usefulness as well-defined functions, mathematicians often use sine and cosine waves as the simple waves, expressing more complicated waves as a sum of sines and cosines with differing amplitudes. In other words, a complicated periodic wave can be written as the sum of a number of simpler waves. The French mathematician Joseph Fourier discovered that any periodic wave (any wave that consists of a consistent, repeating pattern) can be broken down into simpler waves. This tutorial gives an overview of Fourier analysis and how it can be applied to music to account for differences in musical sounds.
Keyword: tutorial, Fourier, Fourier analysis, harmonics By: Maria Bell, proud Member of the Math Squad.
0 kommentar(er)
