8. Pitch, Loudness and Timbre

The mechanism of human hearing does not operate as a perfect scientific instrument. In this chapter we relate a few subjective measurements of sound (things people report after hearing a sound) to objective, scientific measurements (measurements made in a laboratory using scientific instruments). The three subjective quantities of pitch, loudness and timbre are related to laboratory measurements of a sound wave's fundamental frequency, amplitude and waveform, respectively.
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9A. Sound Texture
This simulation explores the aural texture of four basic periodic waveforms: sine, triangle, square, and sawtooth. The sine waveform has a single frequency and is the building block of other periodic waves by summing harmonics in a Fourier Series as we will see in the next section. The richness of the sound is called the timbre (defined in the previous chapter) and is determined by the amplitude of the harmonics in the Fourier sum.
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9B. Fourier Series
The French Mathematician Jean Baptiste Joseph Fourier showed any periodic function can be formed from an infinite sum of sines and cosines. This is very convenient because it means that everything we know about sines and cosines applies to a periodic function of any shape. Although the sum is infinite in theory, in many cases using just a few terms may be close enough to provide a good approximation.
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10A. The Ear and Perception

We look at how the ear turns vibrations into the perception of sound. Some of the exact details of this process are still not completely understood but the general picture of how we hear is fairly well established. Once again we will see that the human hearing mechanism gives us experiences that do not correspond exactly to laboratory measurements.
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10B. Beats
The phenomena of beats occurs when two notes are close together in frequencies and we perceive one note which varies in loudness. A guitar string can be tuned by comparing a note with a known pitch and tuning the string until the beats disappear. What happens if the two frequencies get further and further apart?
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11A. Driven String and Resonance
In this simulation a string is driven at one end by an oscillating driver. The result is that a wave will eventually form on the string.  At certain frequencies the wave will become large and we refer to this resonance phenomena as a standing wave.
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11B. Plucked String
In this section a set of initial conditions for a vibrating string is shown. The first is the fundamental frequency of the string, the second is the second harmonic. The third and fourth initial conditions simulate plucking in the center and at a location one fourth of the way along the string.
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11C. Vibrating Plates Simulation
The Vibrating Plates simulation examines vibrational modes on a rectangular surface. The surface is fixed at the edges so the nodal lines occur in different places compared to a rectangle with free edges. The model assumes that the surface is very thin and very flexible; real surfaces which are stiffer will have slightly different nodal lines and anti-nodes.
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12.A Sanding Waves in a Tube
This section shows a simulation that compares the fundamental, second, third and forth harmonics of standing waves on a string with standing waves in a tube. Notice that for a tube open on both ends the displacement nodes occur where the string has nodes and the displacement anti-nodes in the tube occur where the string has displacement nodes. The pressure nodes in a tube open at both ends occur in the same place as the string nodes.
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12B. Reflection of Waves at a Boundary
Waves on a string form standing waves because the wave reflects from each end of the string where there is a fixed node. How do standing waves form in a tube full of air?  This section shows that waves reflect from the end of a tube and that this reflection can be of two types, depending on whether the boundary is 'soft' or 'rigid'.
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12C. Impedance
But why do sound waves reflect from the open end of a tube or when the size of tube changes abruptly? The resistance to the movement of a wave crossing a boundary from one medium into another is called impedance and occurs for waves on a string, sound waves in air and electronic signals in a circuit. When a wave tries to travel from a medium with one impedance to a region where the impedance is different, there will be a partial reflection.
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13. Percussion and Drumheads
This section allows you to see and manipulate the modes for a square drum head. You can change the modes using the sliders to change the mode numbers n and m. For a membrane there are nodal lines which do not vibrate similar to the nodes we saw on the string but now in two dimensions. You can rotate and enlarge the surface by dragging the mouse over the image. Just like the case for a vibrating string, more than one mode can be present on the two dimensional surface at the same time.
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14. The Human Voice

The human singing voice is also a vibration which is amplified by resonance. The vocal chords are the vibrating part and the throat, mouth, nasal cavities and bronchial tubes constitute the resonance cavities that amplify these vibrations into sound. Because every person's combination of throat, mouth, nasal cavities and bronchial tubes is slightly different, we all sound slightly different.
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15. Musical Scales
The notes on musical instruments are organized into scales and we would like to have a scale where we get the greatest number of combinations that sound good together. We also would like to standardize the scale in such a way that if we build other instruments, two instruments playing together can play the same pitch. This turns out to be more difficult than it would seem. The choice of scale is arbitrary and different cultures have chosen differently.
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16. Acoustics

The study of what happens to sound in an enclosed space or as the result of interactions with large objects such as buildings is called acoustics. Humans have been trying to improve the acoustics of auditoriums and other public spaces since the time of the ancient Romans.  Reflection, refraction, path difference, diffraction and interference will govern how sound behaves inside rooms, auditoriums and concert pavilions.
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