5: PRELAB - SYNTHESIS
INTRODUCTION
In previous labs you have looked at and listened to a number of oscillations
called sine oscillations. The sine oscillation is the most basic of all oscillations,
and is often called simple harmonic oscillation. It turns out that any kind of
oscillation can be produced by an appropriate combination of sine oscillations of
different frequencies and amplitudes. Indeed, the study of oscillations often
begins by analyzing the oscillation into its sine components. This is particularly
true in the area of sound, where the number and type of sine components
determine the timbre or tone quality of the sound.
PHASE
To construct a given complex oscillation from sine components, it is important to
know the relative amplitudes and frequencies of the components, and also to
know their PHASES relative to each other. Phase has to do with where the
oscillation begins. Below are three oscillations that are otherwise identical, but
differ in phase:
¦
0
§
¨
t
The function in diagram 1 is zero when t = 0 and is called the sine function.
The function in diagram 2 is identical except that is starts 1/4 cycle later. This
particular phase relationship has a special name: cosine.
The curve in 3 is a sine function starting 1/2 cycle late and is said to be “180
degrees out of phase with 1” or simply “out of phase.” If you look carefully at
these graphs you will realize that 3 can be thought of either as 1 out of phase or
as 1 with a negative amplitude. In the next lab out of phase oscillations will be
referred to as having negative amplitude.
5: Synthesis Prelab–1
HARMONIC SERIES
ANY oscillation which is periodic (repeats exactly over and over again) can be
produced by combining a very particular set of sine oscillations known as the
Harmonic Series. The harmonic series starting on 100 Hz is:
Harmonic Series on 100 Hz
First Harmonic
100 Hz
Second Harmonic
200 Hz
Third Harmonic
300 Hz
Fourth Harmonic
400 Hz
...etc.
The lowest frequency in the series (the first harmonic) is called the fundamental.
Each possible fundamental frequency has its own harmonic series, which is
obtained by multiplying the fundamental by 1, then by 2, then by 3, etc. In
music, the frequency of the fundamental determines the perceived pitch of the
note, whereas the relative amplitudes of the other harmonics (the “overtones”)
determine the timbre. The harmonic series starting on 123 Hz is:
Harmonic Series on 123 Hz
Name
Symbol
Frequency
f1
123 Hz = 1 × 123 Hz = 1f1
1st Harmonic
2nd Harmonic
f2
246 Hz = 2 × 123 Hz = 2f2
3rd Harmonic
f3
369 Hz = 3 × 123 Hz = 3f3
4th Harmonic
.
.
.
f4
492 Hz = 4 × 123 Hz = 4f4
nth Harmonic
.
.
.
fn
___ Hz = n × 123 Hz = nfn
The fundamental
The analysis of any periodic oscillation that happens 123 times every second will
reveal only these frequencies. Ever. 150 Hz, for example, will never be present.
This extremely powerful fact was first studied by Euler and by Bernoulli, who
first published the mathematical equations of the series in 1728. Extensive later
work by Fourier resulted in the name “Fourier Analysis” for this kind of study of
periodic functions. It is important to realize that while Fourier Analysis is exact
for periodic oscillations, real life oscillations are, at their least complex, only
approximately periodic. Musical sounds, in particular, are usually much more
5: Synthesis PreLab–2
complex. Treating them as approximately periodic is, nonetheless, often an
excellent way to begin an analysis, which can later be made more subtle.
ADDING OSCILLATIONS
In this lab you will be studying oscillating voltages. An oscillating voltage is a
voltage that changes in time. You can add two oscillating voltages simply by
adding the two voltages at each point in time. Consider the two oscillating
voltage signals, the two thin lines "1" and "2," shown below. You can find their
sum by using a four-step process.
A: Find the times where Signal 2 crosses through V=0. Circle these points. For
each of these times, place a dot on top of Signal 1.
B: Find the times when Signal 2 has a maximum and mark these by up-arrows.
For each of these times, place a dot the same distance above Signal 1 as the
length of these arrows.
C: Find the times when Signal 2 has a minimum and mark these by downarrows. For each of these times, place a dot the same distance below Signal 1
as the length of these arrows.
D: Connect the dots with a smooth curve, representing Signal 1 + Signal 2 (in
this case, the thick line).
4
3
2
1
↑
voltage 0
[V]
-1
-2
-3
-4
0
1
2
3
4
5
6
time [s] →
7
8
9
10
5: Synthesis Prelab–3
Answer following questions:
1. Determine the frequencies of the first six harmonics of 342 Hz.
2. If 227 Hz is the fundamental, which harmonic is 1589 Hz?
3. Sometimes, the amplitude of some harmonics will be zero. These harmonics
are “missing.” It is even possible for the fundamental to be missing. What is
the highest frequency that could be the fundamental for a harmonic series
containing the following frequencies: 500 Hz, 750 Hz, 875 Hz, and 1000 Hz.
(Of course, 1 Hz could be the fundamental for a series containing these
harmonics, but that’s not really what we’re after. So just find the highest
frequency which could be the fundamental.)
4. Which harmonics (including fundamental) in the series in question 3 are
missing?
5: Synthesis PreLab–4
5. Add the voltages shown below (use a color other than black.)
4
3
2
1
↑
voltage 0
[V]
-1
-2
-3
-4
0
1
2
3
4
5
6
time [s] →
7
8
9
10
5: Synthesis Prelab–5