Trigonometric functions and sound

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Samson McGee
5 years ago
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1 Trigonometric functions and sound The sounds we hear are caused by vibrations that send pressure waves through the air. Our ears respond to these pressure waves and signal the brain about their amplitude and frequency, and the brain interprets those signals as sound. In this paper, we focus on how sound is generated and imagine generating sounds using a computer with a speaer. of oscillation We need to describe oscillations that occur many times per second. The graph of a sine function that oscillates through one cycle in a second loos lie: A function that oscillates times per second will loo more lie this: We say that the oscillation is Hertz, or cycles per second. Generating sound with a computer speaer A speaer usually consists of a paper cone attached to an electromagnet. By sending an oscillating electric current through the electromagnet, the paper cone can be made to oscillate bac and forth. If you mae a speaer cone oscillate times per second, it will sound lie a pure A note. Clic here to listen. If you mae a speaer cone oscillate 88 times per second, it will sound lie an A, but one octave higher. Clic here to listen. We ll call this A. On a later page, there are graphs of the location of the speaer cone as a function of time, each for one twentieth of a second. Raising the frequency to 76 Hertz raises the pitch another octave to A3. Changing the amplitude of oscillation, that is, how high and how low the graph goes, or how far forward and bacward the speaer cone goes, changes the volume of the sound. The middle graph on the next page shows the amplitude of an oscillation of 76 Hertz rising from to. Clic here to listen to this increasing volume A3 four times. Chromatic and major scales The chromatic scale increases the frequency of oscillation by steps from one octave to another. Starting at A, the frequencies of the chromatic scale would be,,, K = 88 Each of the notes in between has a name of its own; they are: A, A#/B, B, C, C#, D, D#, E, F, F#, G, G#, A. The A major scale starts at Hertz, then increases over 8 steps to A, along the notes A (/), B (/), C# (/), D (5/), E (7/), F# (9/), G#

2 (/) and A (/). A graph of how the speaer cone moves in an A-major scale can be found on a later page. Clic here to listen to the A-major scale four times. Chords and superposition of sounds A chord is formed by playing multiple notes at once. You could play a chord with three notes by putting three speaers side by side and maing each oscillate at the right frequency for a different note. Or, you can add together the functions for each frequency to mae a more complicated oscillatory function and mae your speaer cone oscillate according to that function. For example, if you want to play an A C# - E chord, you can separately mae three speaers oscillate according to the functions: sin( πt ),sin( πt ),sin( πt ) If you want one note louder or softer than the others, you can multiply the whole function by a constant to increase or decrease the volume of that note. Each speaer will mae pressure waves in the air, and these pressure waves from different speaers will overlap as they move toward your ear. By the time they are at your ear, you will be unable to tell which speaer they came from; the pressure waves will have been superimposed on one another, or added to one another. Your ear is amazing at being able to respond to the different frequencies separately to perceive the three notes being played at once. This explains why you can simply add the three functions above and mae the speaer cone of a single speaer oscillate following the sum: sin( πt ) + sin( πt ) + sin( πt ) Here again, if you want the different notes to have different volumes, you can multiply each sine wave by a constant. Clic here to hear an A C# - E chord. Clic here to hear an A C# - E A chord. The graphs on the following page show what happens when you add the three functions to mae the A C# - E chord and the A C# - E A chord. 7 7

3 Speaer location A ( Hz) Time (seconds) A (88 Hz) A (76 Hz) getting louder A C# E chord (, 55, 659 Hz) A C# E A chord (, 55, 659, 88 Hz)

5 spectrum The sounds we have generated so far are very simple, being sine functions or sums of a few sine functions, and they sound very much computer-generated. Real sounds are more complex, and it isn t entirely clear that sine functions have anything to do with them. However, it can be shown that any continuous sound (that is, a sound that is constant, or unchanging over time) can be reproduced as a sum of sine functions of different frequencies and amplitudes. That is, if a speaer is playing a continuous sound by maing the speaer cone follow some function L( over the time interval from to second, then we can write L( as a sum of sine functions this way:, L( = a sin( π = The number is the frequency, and for sounds that humans can hear, we should use frequencies from Hertz to about, Hertz. As we age, we can t hear sounds at, Hertz very well anymore. The numbers a are called the amplitudes for frequency. It is easy to find the values of a using integrals: a = sin( π L( dt. (Actually, this isn t the whole story for this to be exactly correct, you either need to use cosine functions or you need to be able to shift each sine function horizontally on the t axis, but this is close enough to the truth that you can learn from it!) The point is that you can tae a sound and thin of it in terms of the different frequencies and amplitudes that it is made up of. The graph of a versus is called the frequency spectrum of the sound. It shows graphically which frequencies are present in the sound. On the next page, there are graphs of the frequency spectrum for a variety of sounds that we have seen so far, and some new ones. First, below the graph of the pure Hz A is the graph of the frequency spectrum. The amplitudes a are zero except when is, the frequency of the oscillation. Next, below the graph of the A C# - E chord is its frequency spectrum, concentrated at the three frequencies present in that chord. Some frequencies near the C# and E frequency are also present due to the numerical technique that finds the frequency spectrum. The next four graphs are the frequency spectra of Dr. Craig Zirbel saying the continuous sounds long A, long E, long O, and OO. These were all spoen at the same pitch, so they all have frequency spies at similar frequencies. The difference between these sounds is in the relative heights of the peas. If you wanted to mae a computer recognize and differentiate between these sounds, you could train it to pay attention to the relative heights of the different peas. Clic here to listen to the long A, long E, long O, and OO.

6 Pure A A C# E chord Speaer cone location Speaer cone location Time Time 5 Pure A 5 A C# E chord Spoen a.8.6 Spoen e Spoen o.8.6 Spoen oo

Seeing Music, Hearing Waves

Seeing Music, Hearing Waves NAME In this activity, you will calculate the frequencies of two octaves of a chromatic musical scale in standard pitch. Then, you will experiment with different combinations