3.3 Waves and Stuff
Science of Music2007
Last Time
Dr. Koons talked about consonance and beats.
Let’s take a quick look & listen at what this means ….
Listen First
Recall what a single frequency tone sounds like
Play on Sound Generator A=440 Hz. The Graph:
0.005 0.01 0.015 0.02
-1
-0.5
0.5
1
440 Hz. and 450 Hz. Compared
0.005 0.01 0.015 0.02
-1
-0.5
0.5
1
0.005 0.01 0.015 0.02
-1
-0.5
0.5
1
440 Hz.
450 Hz.
440 Hz. and 450 Hz. Together
0.005 0.01 0.015 0.02
-1
-0.5
0.5
1
440 Hz. and 450 Hz. Togetherfor a longer time (0.1 second)
0.02 0.04 0.06 0.08 0.1
-1
-0.5
0.5
1
440 Hz. and 450 Hz. Togetherfor an even longer time (0.5 second)
0.1 0.2 0.3 0.4 0.5
-1
-0.5
0.5
1Tbeat
From this graph we see that Tbeat = 0.1 seconds. fbeat = 10 Hz..
Why??
0.02 0.04 0.06 0.08 0.1
-1
-0.5
0.5
1
In phase out of phase in phase again
The Concept Two sounds start together (in
phase). After Tbeat seconds they get
back together again. One of the waves must have
gone through N cycles while the other went through (N+1) complete cycles.
They are therefore together again!
0.02 0.04 0.06 0.08 0.1
-1
-0.5
0.5
1
The *(*@)$# math
121221
21
21
211
21
211
21
21
2211
2111
11
)(
)1(
ffTTTT
TTf
TT
TTT
TT
TTNT
TT
TN
TTTN
TNTN
beat
beat
The beat frequency between two simultaneous tones is equal toThe difference between the frequencies of the two tones!
Back to
The frequencies
Lm
T
LL
vvf
vf
L
/2
1
2
2
11
11
1
These are the frequencies at which theString RESONATES.
RESONANCE
STRINGS HAVE MORE THAN ONE RESONANT FREQUENCY
The RESONANT frequencies
L
vf
L
2
2
12
1
2
2
ff
SOL
f
BUT
Standing Wave Produced by TWO waves traveling along a
string in opposite directions. Each wave reflects at the end of the string
and then goes the other way. Both waves travel with the same velocity
over the same length of string. Many pairs of waves may travel along the
string at the same time. More than one set of standing waves is
possible on the string at the same time.
actual string shape
fundamental
first overtone, second harmonic,second octave
second overtone, third harmonic,fifth above second octave
third overtone, fourth harmonic,second octave.
etc.
Resonance of Strings
Multiple Frequencies Determine the Timbre
What does sound look like?
Repeats
Music
For short periods of time, a musical sound is PERIODIC.
It has a weird shape. How do we produce a strange looking
but periodic shape? Answer: FOURIER
Fourier’s Theorem Any periodic signal can be broken
down into a sum of simple sine waves at different frequencies and sizes (amplitudes).
This theorem allows us to understand why different instruments sound different even when playing what we perceive as the same tone.
Square Wave
How do we create weird periodic shapes??
FOURIER THEOREM
For Example
The FOURIER spectra for each of these consists of asingle frequency.
When you strike the string All standing wave modes are excited
at the same time. “Non-Standing” waves die out quickly. This is a Fourier thing. If we could initially shape the string
exactly to one of these modes, then this would be the only one that would be excited.
But we can’t do this .. can we???
We can! Sorta..
Two different plucks would require different sets of harmonics to create the shape.
These will produce somewhat different instrument sounds.
Modes
Place finger near thecenter of the stringand the strike it. Theodd overtones should be suppressed.
Plucked atMidpoint
Touched atMidpoint
So – How does the guitar work?
Each string that is plucked will vibrate in one of its fundamental modes. The shape of the initial string stretch determines which modes will be excited.
Each mode is established by waves bouncing back and forth along the instrument.
More ..
The sound from the instrument depends on how and where along the string it is plucked.
The strings by themselves emit little sound. The connection to the bridge causes the sound box to move and resonate, a topic we will discuss later.
So now you know!
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