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THE PHYSICS OF MUSIC AND
MUSICAL INSTRUMENTS
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D
AVID
R. L
APP
, F
ELLOW
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RIGHT
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ENTER FOR
I
NNOVATIVE
S
CIENCE
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DUCATION
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UFTS
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NIVERSITY
M
EDFORD
, M
ASSACHUSETTS
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TABLE OF CONTENTS
Introduction
Chapter 1: Waves and Sound
Wave Nomenclature
Sound Waves
ACTIVITY: Orchestral Sound
Wave Interference
ACTIVITY: Wave Interference
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8
15
18
19
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5
Chapter 2: Resonance
Introduction to Musical Instruments
Wave Impedance
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20
Chapter 3: Modes, overtones, and harmonics
ACTIVITY: Interpreting Musical Instrument Power Spectra
Beginning to Think About Musical Scales
Beats
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37
38
27
Chapter 4: Musical Scales
ACTIVITY: Consonance
The Pythagorean Scale
The Just Intonation Scale
The Equal Temperament Scale
A Critical Comparison of Scales
ACTIVITY: Create a Musical Scale
ACTIVITY: Evaluating Important Musical Scales
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45
47
50
52
55
57
40
Chapter 5: Stringed Instruments
Sound Production in Stringed Instruments
INVESTIGATION: The Guitar
PROJECT: Building a Three Stringed Guitar
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66
70
61
Chapter 6: Wind Instruments
The Mechanical Reed
Lip and Air Reeds
Open Pipes
Closed Pipes
The End Effect
Changing Pitch
More About Brass Instruments
More about Woodwind instruments
INVESTIGATION: The Nose flute
INVESTIGATION: The Sound Pipe
INVESTIGATION: The Toy Flute
INVESTIGATION: The Trumpet
PROJECT: Building a Set of PVC Panpipes
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Chapter 7: Percussion Instruments
Bars or Pipes With Both Ends Free
Bars or Pipes With One End Free
Toward a “Harmonic” Idiophone
INVESTIGATION: The Harmonica
INVESTIGATION: The Music Box Action
PROJECT: Building a Copper Pipe Xylophone
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References
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Contents
Back
“Everything is determined … by forces over which we have no
control. It is determined for the insects as well as for the star.
Human beings, vegetables, or cosmic dust – we all dance to a
mysterious tune, intoned in the distance by an invisible piper.”
– Albert Einstein
INTRODUCTION
T
HIS MANUAL COVERS the physics of waves, sound, music, and
musical instruments at a level designed for high school physics. However,
it is also a resource for those teaching and learning waves and sound from
middle school through college, at a mathematical or conceptual level. The
mathematics required for full access to the material is algebra (to include
logarithms), although each concept presented has a full conceptual foundation that
will be useful to those with even a very weak background in math.
M
ODES OF
P
RESENTATION
Solomon proclaimed that there is nothing new
under the Sun and of the writing of books there is no
end. Conscious of this, I have tried to produce
something that is not simply a rehash of what has
already been done elsewhere. In the list of references I
have indicated a number of very good sources, some
classics that all other writers of musical acoustic
books refer to and some newer and more accessible
works. From these, I have synthesized what I believe
to be the most useful and appropriate material for the
high school aged student who has neither a
background in waves nor in music, but who desires a
firm foundation in both. Most books written on the
topic of musical acoustics tend to be either very
theoretical or very cookbook style. The theoretical
ones provide for little student interaction other than
some end of the chapter questions and problems. The
ones I term “cookbook” style provide instructions for
building musical instruments with little or no
explanation of the physics behind the construction.
This curriculum attempts to not only marry the best
ideas from both types of books, but to include
pedagogical aids not found in other books.
This manual is available as both a paper hard
copy as well as an e-book on CD-ROM. The CD-
ROM version contains hyperlinks to interesting
websites related to music and musical instruments. It
also contains hyperlinks throughout the text to sound
files that demonstrate many concepts being developed.
As the student reads through the text, he or she
will encounter a number of different presentation
modes. Some are color-coded. The following is a key
to the colors used throughout the text:
Pale green boxes cover tables and figures
that are important reference material.
Notes
C
i
D
E
F
G
A
B
C
f
Frequency
interval (cents)
0
204
408
498
702
906
1110
1200
Table 2.8: Pythagorean
scale interval ratios
Light yellow boxes highlight derived
equations in their final form, which will be used for
future calculations.
T
m
f
1
=
2L
†
1
Contents
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I
NTRODUCTION
NTRODUCTION
consider. Investigations are labs really, often requiring
students to make measurements directly on the
photographs. Solutions to the “Do you get it?”
boxes, Activities, and Investigations are provided in
an appendix on the CD-ROM. Finally, projects
provide students with some background for building
musical instruments, but they leave the type of
musical scale to be used as well as the key the
instrument will be based on largely up to the student.
Tan boxes show step-by-step examples for
making calculations or reasoning through questions.
Example
If the sound intensity of a screaming baby were
1¥ 10
-2
W
2
at 2.5 m away, what would it be at
m
6.0 m away?
The distance from the source of sound is greater by a
factor of
6.0
=
2.4
. So the sound intensity is decreased
2.5
by
†
†
1
(2.4 )
2
P
HYSICS AND
… M
USIC
?
†
=
0.174
. The new sound intensity is:
(1¥ 10
-2
W
m
2
)(0.174)
=
1.74
¥
10
-2
W
m
2
†
Gray boxes throughout the text indicate
stopping places in the reading where students are
asked, “Do you get it?” The boxes are meant to
reinforce student understanding with basic recall
questions about the immediately preceding text. These
can be used to begin a discussion of the reading with
a class of students.
†
Do you get it? (4)
A solo trumpet in an orchestra produces a sound
intensity level of 84 dB. Fifteen more trumpets join
the first. How many decibels are produced?
In addition to the
“Do you get it?”
boxes,
which are meant to be fairly easy questions done
individually by students as they read through the text,
there are three additional interactions students will
encounter:
Activities, Investigations,
and
Projects.
Activities more difficult than the “Do you
get it?” boxes and are designed to be done either
individually or with a partner. They either require a
higher level of conceptual understanding or draw on
more than one idea. Investigations are harder still and
draw on more than an entire section within the text.
Designed for two or more students, each one
photographically exposes the students to a particular
musical instrument that they must thoroughly
“Without
music life would be a
mistake.”
– Friedrich Nietzsche
With even a quick look around most school
campuses, it is easy to see that students enjoy music.
Ears are sometimes hard to find, covered by
headphones connected to radios or portable CD
players. And the music flowing from them has the
power to inspire, to entertain, and to even mentally
transport the listener to a different place. A closer
look reveals that much of the life of a student either
revolves around or is at least strongly influenced by
music. The radio is the first thing to go on in the
morning and the last to go off at night (if it goes off
at all). T-shirts with logos and tour schedules of
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Contents
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I
NTRODUCTION
NTRODUCTION
understand the rationale for the development of the
musical scales one needs a broad foundation in most
elements of wave and sound theory. With that said,
the approach here will be to understand music and
musical instruments first, and to study the physics of
waves and sound as needed to push the understanding
of the music concepts. The goal however is a deeper
understanding of the physics of waves and sound than
what would be achieved with a more traditional
approach.
popular bands are the artifacts of many teens’ most
coveted event … the concert. But the school bell
ringing for the first class of the day always brings
with it a stiff dose of reality.
S
OUND
, M
USIC
,
AND
N
OISE
Do you like music? No, I guess a better question
is, what kind of music do you like? I don’t think
anyone dislikes music. However, some parents
consider their children’s “music” to be just noise.
Likewise, if the kids had only their parent’s music to
listen to many would avoid it in the same way they
avoid noise. Perhaps that’s a good place to start then
– the contrast between music and noise. Is there an
objective, physical difference between music and
noise, or is it simply an arbitrary judgment?
After I saw the movie
8 Mile,
the semi-
autobiographical story of the famous rapper Eminem,
I recommended it to many people … but not to my
mother. She would have hated it. To her, his music is
just noise. However, if she hears an old Buddy Holly
song, her toes start tapping and she’s ready to dance.
But the music of both of these men would be
considered unpleasant by my late grandmother who
seemed to live for the music she heard on the
Lawrence Welk Show. I can appreciate all three
“artists” at some level, but if you ask me, nothing
beats a little Bob Dylan. It’s
obviously not easy to define
the difference between noise
and music. Certainly there is
the presence of rhythm in the
sounds we call music. At a
more sophisticated level there
is the presence of tones that
combine with other tones in
an orderly and ... “pleasing”
way. Noise is often associated
with very loud and grating
sounds – chaotic sounds which
don’t sound good together or
are somehow “unpleasant” to
listen to. Most would agree
that the jackhammer tearing
up a portion of the street is
noise and the sound coming
from the local marching band
is music. But the distinction
is much more subtle than that.
If music consists of sounds
with rhythmic tones of certain
frequencies
then
the
jackhammer
might
be
3
H. L. Mencken writes, “School days, I believe,
are the unhappiest in the whole span of human
existence. They are full of dull, unintelligible tasks,
new and unpleasant ordinances, brutal violations of
common sense and common decency.” This may
paint too bleak a picture of the typical student’s
experience, but it’s a reminder that what is taught
often lacks meaning and relevance. When I think back
to my own high school experience in science, I find
that there are some classes for which I have no
memory. I’m a bit shocked, but I realize that it would
be possible to spend 180 hours in a science classroom
and have little or no memory of the experience if the
classroom experience were
lifeless or disconnected
from the reality of my life.
Middle school and high
school students are a tough
audience. They want to be
entertained … but they
don’t have to be. What they
really need is relevance.
They want to see direct
connections and immediate
applications. This is the
reason for organizing an
introduction to the physics
of waves and sound around
the theme of music and
musical instruments.
It’s not a stretch either.
Both music and musical
instruments are intimately
connected to the physics of
waves and sound. To fully
appreciate what occurs in a
musical instrument when it
makes
music
or
to

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