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Science of Music

1.3 The Ups and Downs

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Things that go back and forth Pendulum Mass on Spring

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Another Graph .. a biggy!

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25

Time (seconds)

dis

turb

an

ce H

eig

ht

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Important Definitions

Tf

or

periodfrequency

1

1

The PERIOD, T is the time it takes to go from one condition to the next time that exact condition is repeated.The frequency, the number of oscillations per second, is given by:

Example:

If T=2 seconds

F=1/2 (sec-1)=0.5 per second

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Question

What is a tone and how do you prove it??

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Hemholtz

In physiology and physiological psychology, he is known for his mathematics of the eye, theories of vision, ideas on the visual perception of space, color vision research, the sensation of tone, perception of sound. In physics, he is known for his theories on

the conservation of force, work in electrodynamics,

chemical thermodynamics, A mechanical foundation of

thermodynamics.

1821 - 1894

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Siren … a scientific instrument

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The Graph

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

Time in milli-seconds

"air

spee

d"

- re

lati

ve

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100 Bottles of beer on the wall …

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Resonance (later)

Rotational Speed

(Turns/second)

Loud

ness

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Helmholtz Resonators

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Helmholtz’s ResultsNote from Middle C Frequency

C 264

D 297

E 330

F 352

G 396

A 440

B 496

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We need to try to remember a few things from High School Physics

Definitions of acceleration and force. Newton’s Laws Directional Thingys … vectors.

Relax … it ain’t that hard.

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Velocity (speed) Velocity is the distance you have

traveled divided by the time it took you to travel the distance.

If you travel the 20 miles from Orlando to Lake Buena Vista in 20 minutes, you have traveled speed = 20 miles/20 minutes = 1 mi/min

or 60 miles per hour. On I-4 you would also get a ticket if you

could drive that fast.

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Acceleration Something that is moving at one

speed and then starts moving at a faster (or slower) speed is said to be accelerating. Your car accelerates when you start

driving it or when you enter a highway (except for I-4 on occasion).

You accelerate if you jump out of the window of a tall building. The acceleration continues until the splat.

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Definition If you start at a speed of vinitial and finish at a

speed of vfinal and it takes a time “t” to accomplish this, then

t

vva

onaccelerati

initialfinal

it) do it took to (timet

(speed)y in velocit change

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Sir Isaac Newton (1642-1727)

Didn’t really happen but let’s talk about it.

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Interesting Facts about Sir Issac

He lived a long and fruitful life. He died a virgin. He invented the calculus. So did Leibnitz. They fought about it. Nobody one. The apple story is probably untrue but if it

IS true, Newton also must have invented applesauce.

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Newton’s Simplified Three Laws An object moving at a constant velocity will

continue moving at that same constant velocity if NOT acted upon by an external force.

If an external force acts on an object it will accelerate in proportion to the force. F=ma the mass is the proportionality constant When you push on something, that something will

push back on you with the same force that you are pushing with. (Action = Reaction)

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Who cares about Newton??

If Newton’s laws were not correct, there would not be any music!!!

We will use this stuff in many places but we will not get to deep into it.

See Bolemon (reference list) for more information if you want it.

For more … any College Physics Text will suffice.

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Now …. lets look at the MONOCHORD but with an important detour.

Dr. Koons will play the monochord for us soon

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The Spring

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Spring Force Equatiom

F=-kx

The “-” sign indicates that the force and the displacement are in opposite directions.

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Springs Oscillate

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Graph

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Important Result for a Spring:

m

kf

kxF

2

1

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So …. m

kf

2

1

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Concept … Tension

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The Musical String

Force = F

x

Linitial T

T TThe Bigger the angle themore T points UP!

The distance “x” is the samesort of thing as the x in F=-kx.

ANGLE

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The Guitar Strings

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Consider Two Situations

For the same “x” therestoring force is doublebecause the angle is double.

The “mass” is about halfbecause we only havehalf of the stringvibrating.

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So…

For the same “x” therestoring force is doublebecause the angle is double.

The “mass” is about halfbecause we only havehalf of the stringvibrating.

m

kf

kxF

2

1

k doubles

m -> m/2

f doubles!

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Guitar Pressing the fret that is in the middle of the

string doubles the frequency~ Walla … the octave

In general … the frequency is proportional to the length of the string.

Next time we will examine the monochord and Dr. Koons will show us how we develop (a) musical scale(s).

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Octave 0.001 0.002 0.003 0.004 0.005

-1

-0.5

0.5

1

0.001 0.002 0.003 0.004 0.005

-1

-0.5

0.5

1

0.001 0.002 0.003 0.004 0.005

-1.5

-1

-0.5

0.5

1

1.5

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Take a wire (String on Guitar ..)Linitial

Lstretchl

Lfinal

F

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It has been shown that …

k""constant spring a like Looks

area sectional-cross theisA

material afor constant a is E

withpullyou force theis F

initial

stretchinitial

L

EA

LL

EAF

More about this when we do the string thing.

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HAVE FUN!