Fourier series:
In the 18th century, the French mathematician Jean Bastiste Joseph Fourier made an extraordinary discovery; as a result of his investigation into partial differential equations modeling vibration and heat propagation in bodies, he found out that every function could be represented by an infinite series of sines and cosines (the basic trigonometric functions). For example the sound signal produced by any musical instruments such as piano, drums, etc. could be decomposed into its trigonometric constituents which reveals the fundamental frequencies that are combined to produce a specific sound. This idea of decomposing signals lies at the heart of modern electronic music; such as a synthesizer combines sine and cosine tones to
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It seems that the graph of S_n (x) is approaching the graph of f(x), with the exception when x=0 or x is an integer multiple of π.
The Fourier series of all kinds of signals and waveforms cannot be calculated. The function must satisfy some specific conditions known as the Dirichlet condition for convergence of Fourier series. These conditions are then used to define and prove the Fourier Convergence Theorem which gives the sum of the Fourier series. These conditions are that a periodic signal must be a piecewise continuous function, have finite number of extrema (maxima and/or minima) and have finite number of discontinuities (points in the function’s domain where the function is not continuous i.e. has a break).
Before going to prove the Convergence Theorem, some definitions of terms needs to be clarified. Periodic Function – A function f(x) is periodic with period T if for all the values of x, f(x+nT)=f(x), where T is a positive constant and n is an integer. For example, the sine function sinx is periodic with the least period 2π and other periods such as -2π,4π,6π,
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Also at these points where f is discontinuous, the sum of Fourier series is the average of the left and the right limits, that is
1/2[f(a^+ )+f(a^- )]
And at any point where the function f is continuous,
1/2 [f(a^+ )+f(a^- )]=f(x)
It is quite evident that the Fourier Convergence theorem is quite effective and easier as it gives the partial sum of Fourier series without plotting the graph. For the above example (Fourier series of the square wave function) we observed that f(0^+ )=lim┬(x→0^+ )〖f(x)〗=1 "and" f(0^- )=lim┬(x→0^- )〖f(x)〗=0
The square wave function has discontinuities and the average of these left and right limits is 1⁄2, so for any integer n the Fourier Convergence Theorem says that
1/2+∑_(k=1)^∞▒〖2/((2k-1)π) sin〖(2k-1)〗 x〗={█(f(x) "if" n≠nπ@1/2 "if "
At any point in the air near the source of sound, the molecules are moving backwards and forwards, and the air pressure varies up and down by very small amounts. The number of vibrations per second is called the frequency which is measured in cycles per second or Hertz (Hz). The pitch of a note is almost entirely determined by the frequency: high frequency for high pitch and low for low .
Last but not least, for Ritika,Preeti and Prem (2013), they are using Freeduino board and solder the component on that board. They are using compiler Arduino 1.0 for uploading the codes to the board, MAX 232 Line Driver, ATmega328 and other components. The reason for they are using Freeduino is that Freeduino is an open-source electronics prototyping platform based on flexible, easy-to-use hardware and software and Freeduino projects can be standalone or they can communicate with software running on a computer (e.g. Flash, Matlab, Processing, MaxMS ). The wave plot also can be in form of sine and square wave form.
RENÉ DESCARTES by career being a Mathematician carried his interest of entering into the philosophy realm. At a very young stage, he decided that nature is to be explained with certainty as Mathematics. Mathematics in itself is very numerical, where the nature cannot be expressed numerically but is bound in a neat and clear cut way. Thus, his philosophy about everything in nature is very mechanical and machine-like.
Physical science is the study of non-living matter which I will be talking about the physics of music, and how pitch and frequency play a role together in science. Physics is known to be the most fundamental science, based on the principle and concepts. It deals with matter, motion, force, and energy, (Shipman-Wilson Higgins, 2013). In this report I will be dealing with pitch, frequency sound and waves which are also dealing with physical science and how they our related.
Unlike water waves for example, Feynman emphasised that when electrons are fired through the slits one at a time, an interference pattern would be produced. He then famously said that this phenomenon “has in it the heart of quantum physics [but] in reality, it contains the only mystery.”
...obvious areas like rain on a tin roof, the croaking of frogs, the trees whistling in the wind, birds chirping in the early morning, hands clapping together, somebody humming or whistling and tapping on something with a deep tone. As the years have gone by people have made a lot of progress getting the sounds to come together to make music as we know it today.
...the amount of time that passes from the beginning of a cycle to the beginning of the next cycle, measured in degrees. Phase shift describes the difference in timing between two otherwise identical periodic signals.
Sound is essentially a wave produced by a vibrating source. This compression and rarefaction of matter will transfer to the surrounding particles, for instance air molecules. Rhythmic variations in air pressure are therefore created which are detected by the ear and perceived as sound. The frequency of a sound wave is the number of these oscillations that passes through a given point each second. It is the compression of the medium particles that actually constitute a sound wave, and which classifies it as longitudinal. As opposed to transverse waves (eg. light waves), in which case the particles move perpendicular to the direction of the wave movement, the medium particles are moving in the same or opposite direction as the wave (Russell, D. A., 1998).
waves are further divided into two groups or bands such as very low frequency (
These simple forms of music expanded and the need to invent new instruments were created through knowledge of sounds. Modern science tells us that sound is simply energy created by vibrations from various medium. However in early human history, sound was believed to have mysterious power. These new instruments created forms of music that did not rely heavily on human voices.
The history of the computer dates back all the way to the prehistoric times. The first step towards the development of the computer, the abacus, was developed in Babylonia in 500 B.C. and functioned as a simple counting tool. It was not until thousands of years later that the first calculator was produced. In 1623, the first mechanical calculator was invented by Wilhelm Schikard, the “Calculating Clock,” as it was often referred to as, “performed it’s operations by wheels, which worked similar to a car’s odometer” (Evolution, 1). Still, there had not yet been anything invented that could even be characterized as a computer. Finally, in 1625 the slide rule was created becoming “the first analog computer of the modern ages” (Evolution, 1). One of the biggest breakthroughs came from by Blaise Pascal in 1642, who invented a mechanical calculator whose main function was adding and subtracting numbers. Years later, Gottfried Leibnez improved Pascal’s model by allowing it to also perform such operations as multiplying, dividing, taking the square root.
Between 1850 and 1900, the mathematics and physics fields began advancing. The advancements involved extremely arduous calculations and formulas that took a great deal of time when done manually.
Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclicalphenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.