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Lecture 3

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University of Toronto Scarborough
Mark Schmuckler

PSYC56 Music Cognition Lecture 3 Basic Dimensions of Sound The Physics of Sound  All sounds are a result of a number of interesting and interacting processes. Sounds begin with the vibration of an object.  Blowing through a mouth piece, expelling air through the vocal cords, pulling a string are all examples of producing a sound.  The air vibrating within a tube or the vibrating string must transfer its sound energy to a medium, some sound body. This sound body amplifies vibrations and transfers the vibrations to the air.  The air picks up these vibrations and the air moves vibrations, which is passed to our ear.  This is the process through which all sound occurs.  We can take any vibration and analyze vibration into component vibrations, each component vibration moves independently.  Simple harmonic motion is caused by elastic restoring force, take some object and move it away from its equilibrium, and then we let go of the object, there is a force that acts on that object o try and bring it back to its resting point, equilibrium. The force that acts on that is proportional to the distance that we moved that object from its resting point. The distance is called the displacement, and this is how we produce most of the vibrating systems we are interested in. Simple Harmonic Motion  The ball hits the equilibrium point, however it still has force, and goes past equilibrium amount, it will overshoot by the exact same amount we compressed the string. Once it hits maximum displacement on other side it shoots back past equilibrium point. This is the case if there is no friction involved in the system.  We end up with a curve at the bottom of the diagram, called a sound wave.  The wave repeats itself indefinitely if there is no friction. This means we can look at the number of times this wave repeats itself per some unit time. it will give us characterization of how fast this wave is. This is called frequency. Unit time is the number of cycles that occur every second. The unit for frequency is hertz.  The period is the time it takes to complete one cycle. The period is the inverse of the frequency.  The phase has to do with where that wave starts relative to resting position. The peak of the wave is 90 degrees, halfway back to resting position we are now at 180 degrees. When we get 2/3 of the way to the maximum displacement in the opposite direction now at 270 degrees. When we get to resting position after completing a circle now at 360 degrees or 0 degrees.  (b) Increasing the displacement (doubling it) of the string, the amplitude is going to be twice as much as it was. We have the same weight and we actually have the same frequency, it is still going at the same speed, and it still took the cycle the same time. We put more force, so we get more of a displacement.  (c) Using a stiffer string, using the same displacement we used initially, there is an increase in frequency. The stiffer spring has a faster frequency then the first, same displacement and same mass.  (d) A heavier string is going to make the string move slower, and there is a slower frequency.  The pitch of a musical note doesn’t change as it gets softer because the frequency stays the same. Additivity and Superposition of Sine Waves  (a) Take two waves at identical frequencies. The amplitude of the resulting wave is adding the amplitudes of both waves.  (b) Take waves that differ in their frequencies, when we take a wave 90 degrees of out of phase, the first wave starts at 0 and the other start at 90 (top of its displacement), the resulting wave is addition of the two. We get a wave with the same frequency but differing amplitude.  (c) The dashed wave starts at 0 and dotted wave starts at 180. The two waves actually cancel each other out resulting is a straight line.  (d) Take two waves of different frequencies, dotted wave is slightly faster frequency than dashed wave. The resulting frequency is the lined wave. It is a much more complex wave. The important part of this is that it is still a periodic wave, at some point that wave is going to repeat itself and it will be cyclical as well.  (e) Take two waves of different frequencies. We can take sine waves and add them together and this produces a new wave that may no longer be a sine wave but it is still a periodic wave, it still repeats itself. This notion is interesting because it makes us wonder if we can reverse the process. Can we take a complex wave form and can we break it down? We can do that. Fourier Decomposition or Fourier Analysis  Fourier’s Theory demonstrates that we can take any repetitive complex wave and break it down into a set of sine waves where these sine waves, all they are going to do, is differ in term of their frequencies and their amplitudes and their phases.  The process of breaking the complex wave down is called Fourier Analysis  Fourier also discovered that the frequencies of the wave forms we are breaking them down into are harmonically related.  The complex wave is just the sum of all the sine waves; it is the addition of the sine waves.  The sine waves are harmonically related, the frequencies lie in ratios to one another, they lie in a 2:1, 3:1, or whatever ratio. If the frequency of one wave was 200 hertz then the frequency of the next wave would be 400 hertz, at a 2:1 ratio. *can there be a 1:1 ratio?  The first one is called the first harmonic, and the second is called the second harmonic. There are three systems for labeling the harmonic. We call the first harmonic the fundamental harmonic because it is the first harmonic we hear.  Take note that the 2 ndharmonic is also called the 1 overtone, and also the 2 ndpartial.  The square is called so because it looks like a square. A square wave is the sum of the odd number of harmonics with amplitudes that are proportional to their number in the harmonic series. rd  The 3 harmonic is 1/3 of the fundamental harmonic.  The 5 harmonic is 1/5 the amplitude of the fundamental harmonic.  Once we take the sum of the odd harmonic, it produced a square wave.  A saw tooth wave composes all the harmonics, odd and even.  The fundamental harmonic does not necessarily need to be the loudest. We do not hear all the individual harmonics, but we hear the pitch, and it is normally the pitch of the fundamental harmonic. It is possible to hear the upper harmonics above the fundamental harmonic. th  In an English horn, the 5 harmonic is actually the loudest. In the square wave and saw tooth wave, the first harmonic is the loudest, but looking at the last part of the diagram, in the English horn, the th 5 harmonic is the loudest. The Perception of Pitch  Physiology of the Ear  The c
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