SPECIAL FEATURE: SOUND PHYSICS
www.iop.org/journals/physed
Experimenting with
musical intervals
Michael C LoPresto
Henry Ford Community College, Dearborn, Michigan, MI 48128, USA
E-mail: lopresto@hfcc.net
Abstract
When two tuning forks of different frequency are sounded simultaneously
the result is a complex wave with a repetition frequency that is the
fundamental of the harmonic series to which both frequencies belong. The
ear perceives this ‘musical interval’ as a single musical pitch with a sound
quality produced by the harmonic spectrum responsible for the waveform.
This waveform can be captured and displayed with data collection hardware
and software. The fundamental frequency can then be calculated and
compared with what would be expected from the frequencies of the tuning
forks. Also, graphing software can be used to determine equations for the
waveforms and predict their shapes. This experiment could be used in an
introductory physics or musical acoustics course as a practical lesson in
superposition of waves, basic Fourier series and the relationship between
some of the ear’s subjective perceptions of sound and the physical properties
of the waves that cause them.
(Some figures in this article are in colour only in the electronic version)
Introduction
The main physical cause of the human ear’s
perception of musical pitch is the frequency of
the sound wave. Pitch is a subjective sensation
and, although other factors do affect it, frequency
provides its main physical or objective basis
[1]. Relationships between pitches are known as
musical intervals. These intervals are found within
a harmonic series that consists of a fundamental
and overtones that are integral multiples. The ratio
between the frequencies of the first two harmonics
is 2/1, this is known as a musical octave; the ratio
between the second and third frequencies is 3/2,
a musical fifth, then 4/3 makes a musical fourth,
5/4 a third and so on. Note that the interval names
have nothing to do with the frequency ratios. They
come from the number of steps of the musical scale
within the interval. It is necessary to make this
clear to those without musical background. See
0031-9120/03/040309+07$30.00
Table 1. Frequency ratios of musical intervals
produced with a standard set of tuning forks.
Interval
Musical
notes
Frequency
ratio
Unison
Second
Third
Fourth
Fifth
Sixth
Seventh
Octave
C–C
C–D
C–E
C–F
C–G
C–A
C–B
C–C′
1/1
9/8
5/4
4/3
3/2
5/3
15/8
2/1
table 1 for a more complete list of musical intervals
and their frequency ratios1 .
1
The frequencies of a standard set of laboratory tuning forks
are based on the Just Diatonic Scale. There are several other
types of musical scales including the Equal Tempered Scale,
which is currently used in Western Music. See [1, p 171].
© 2003 IOP Publishing Ltd
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M C LoPresto
What follows is a simple experiment on
musical intervals using a set of standard laboratory
tuning forks and microcomputer hardware and
software for an introductory physics or musical
acoustics course. It introduces superposition of
waves and the basics of Fourier synthesis as well as
demonstrating some of the physical causes of the
ear’s perception of the properties of sound waves.
Fundamental tracking
When the human ear hears a musical interval, the
combination of the pitches is perceived as one pitch
with a frequency equal to the fundamental of the
harmonic series to which both belong, whether or
not that fundamental is present [2]! Since this
pitch does not correspond to any of the frequencies
actually sounding, it is often called a ‘virtual pitch’
[1, p 114]. For instance, if pitches with frequencies
of 384 Hz and 256 Hz (384/256 = 3/2, a musical
interval of a fifth) are sounded simultaneously, the
ear hears a virtual pitch of frequency 384/3 =
256/2 = 128 Hz, the missing fundamental. This
ability of the ear to hear a virtual pitch from a
complex tone is known as fundamental tracking.
If it were not for this ability, the ear could not hear
a distinctive pitch from the complex tone produced
by a voice or a musical instrument [2, pp 45–7].
The sound will have a ‘quality’ that is more
interesting than the dull sensation of a pure tone.
This comes from the complex waveform resulting
from the superposition of the two waves. The main
physical cause of the perception of the quality
or, as musicians call it, timbre of a sound is the
sound’s harmonic structure or spectrum, which is
responsible for the form of the wave [1]. When
the ear hears two musical instruments or voices,
even when sounding the same pitch and loudness,
it can still perceive them as ‘sounding’ different.
The waveforms of musical intervals can
be captured and displayed using a microphone,
computer interface and a data collection program2 .
This is done by striking two tuning forks and
allowing them to vibrate simultaneously while
being held in front of the microphone (figure 1).
Figure 2 shows the waveform of a musical fifth
produced by two tuning forks of the above
2
Logger Pro and the Lab-Pro Interface by Vernier Software
(www.vernier.com) were used. Any comparable software
and hardware could be used, such as Data Studio and
Waveport with the Science Workshop 750 Interface by Pasco
(www.pasco.com).
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Figure 1. Capturing the waveforms.
Figure 2. The captured waveform of a musical fifth
(top) and a plot of the equation y = cos 3x + cos 2x.
frequencies. Using the time axis of the display
to measure the repetition period of the complex
waveform and then reciprocating it will give
the frequency of the combination tone. The
calculated frequency of the wave in figure 2 is
f = 127.4 Hz, very close to the expected 128 Hz,
the fundamental of the harmonic series of which
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Experimenting with musical intervals
Table 2. Comparison between expected ‘missing’ fundamental frequencies of musical intervals produced by
tuning forks and those calculated from their waveforms.
Interval
Second
Minor third
Third
Fourth
Fifth
Sixth
Seventh
Octave
Frequency
ratio
Fundamental
frequency (Hz)
Calculated
fundamental (Hz)
288/256 = 9/8
384/320 = 6/5
320/256 = 5/4
341.3/256 = 4/3
384/256 = 3/2
426.6/256 = 5/3
480/256 = 15/8
512/256 = 2/1
288/9 = 256/8 = 32
384/6 = 320/5 = 64
320/5 = 256/4 = 64
341.3/4 = 256/3 = 85.3
384/3 = 256/2 = 128
426.6/5 = 256/3 = 85.3
480/15 = 256/8 = 32
512/2 = 256/1 = 256
32.4
63.7
64.6
84.1
127.4
84.96
31.9
259.7
256 Hz and 384 Hz are the second and third
harmonics. Using a standard set of laboratory
tuning forks these complex tones can be captured
and ‘missing’ fundamental frequencies can be
determined and compared with what is expected
from the frequency ratio of the tuning forks. See
table 2 for a comparison between expected and
measured fundamentals for each interval tested.
Waveforms
Individual tuning forks produce simple sine or
cosine waves. This can easily be verified with
the data collection programs and is the reason for
their ‘dull’ sound quality. The expected shape of
the waveform of an interval can be predicted by
plotting the function that should correspond to the
interval. For instance, y = cos 3x + cos 2x, is
the equation for a fifth; see figure 2. The plots
can be made very easily with graphing-calculator
software3 by using the numbers in the ratio as
the coefficient of the trigonometric function’s
argument. This amounts to a simple hands-on
introduction to superposition of waves or even the
basics of Fourier synthesis4 . Figure 3 shows the
expected waveform compared with the captured
one for the interval of 512/256 = 2/1, a musical
octave.
Amplitude and phase differences
Although the captured waveforms have close to the
expected fundamental frequencies (see table 2),
they may look different than expected. This is
3
Waveforms were prepared with Graphing-Calculator by
Pacific Tech. (www.pacificT.com). Data Studio by Pasco has
a graphing calculator as well.
4 If graphing software is not available then [3] shows the
waveforms of the common musical intervals.
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Figure 3. The captured waveform of a musical octave
(top) and a plot of the equation y = cos x + sin 2x.
because of differences in the amplitude and phase
of the waves produced by the individual tuning
forks. The loudness of a sound is mostly controlled
by the wave’s intensity, which is proportional to
the amplitude of the pressure variations that cause
the vibrations of the eardrum [1]. Although every
effort can be made to strike each tuning fork
equally hard, producing the same loudness, small
differences in amplitude that will affect the form
of the combination tone are unavoidable. This
is also true for differences in the phase of the
individual waves as they reach the microphone.
Phase differences can result in slightly different
waveforms of the combination tone [4]. For
instance, since the sine and cosine functions are
90◦ out of phase, the function y = sin x + sin 2x
has a different waveform than the same expression
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M C LoPresto
Figure 4. The Fourier Synthesizer from NTNU Virtual Physics Laboratory. Reproduced with permission of the
webmaster.
with cosine functions. The forms generated with
sine functions or combinations of sines and cosines
may resemble those of the captured intervals more
closely than those generated from cosines. This
was the case with the interval of an octave in
figure 3. The combination that most closely
resembled the captured waveform was y = cos x +
sin 2x. The plots of different combinations of
sines and cosines can be checked to see which
best resembles the captured waveform. It is
generally accepted that although they are not
totally imperceptible, unless the phase differences
cause a drastically different waveform, they are
usually difficult for the ear to detect [1, p 160]
(see also [3, p 189] or [4, p 124]). This is the
reason why the harmonic structure of a tone is
considered the reason for the timbre rather than the
waveform; slightly differing waveforms, which
sound the same, can be produced by the same
harmonic spectrum [5].
The NTNU Virtual Physics Laboratory (from
www.phy.ntnu.edu.tw/java/sound/sound.html, see
figure 4) has a Fourier synthesizer, which displays
waveforms as the user combines sine and cosine
waves of different harmonics and amplitudes
while listening to the resulting sound. Adjusting
the amplitudes until a waveform matches the
one produced by the tuning forks provides an
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even better way to determine the equations of
the captured waveforms. This tool can also
be used to test whether or not the ear can
detect variations in waveforms caused by phase
differences. For instance, various combinations of
sines and cosines of second and third harmonics
(an interval of a fifth), although varying the
waveform, did not seem to noticeably change the
perceived quality of the sound.
Although graphing-calculator software is
easiest for producing expected waveforms for
comparison, a cathode-ray oscilloscope with input
from a variable frequency function-generator or
an actual Fourier synthesizer could also be used.
Either of the latter, with speakers, could be
used to perform the above test on whether phase
differences perceptibly affect sound quality.
Consonance and dissonance
Since the time of Pythagoras, intervals that
consist of pitches that have small whole-number
frequency ratios have been considered, when
sounded simultaneously, to be more ‘pleasing’ to
the ear or, in musical terms, consonant. If the
ratios are between larger numbers, the interval
is perceived as less ‘pleasing’ or dissonant [1,
p 55]. Although there have been many attempts
to quantify consonance [1, p 156; 2, pp 167–8;
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Experimenting with musical intervals
Figure 5. Left: The expected waveform of a musical sixth (bottom), an imperfect consonance, compared with the
captured waveform (top). Right: The expected waveform of a musical second (bottom), a dissonant interval,
compared with the captured waveform (top). Note that the more dissonant waveforms are more complex. The
intervals of a fifth and an octave, shown in figures 2 and 3, are consonances.
5, pp 91–3, 299–307; 6] it can still be somewhat
subjective and could depend on musical training,
i.e. what the ear is ‘used’ too [2, p 78].
The intervals discussed thus far, the octave
and fifth, are known as perfect consonances. As
can be seen in figures 2 and 3, their waveforms are
relatively simple. This is also true of the musical
fourth, a ratio of 4/3. The intervals of a third (5/4),
a minor third (6/5) and a sixth (5/3) are called
imperfect consonances; their forms are a bit more
complex. Intervals of a second (9/8) and a seventh
(15/8) are even more complex and are considered
dissonant. See figure 5 for examples. The more
dissonant intervals have more complex waveforms
[3, p 193; 7], as if they are more difficult for the
ear to ‘sort out’.
A-426.6 Hz tuning fork simultaneously with an
A-440 Hz. The frequency of the perceived pitch,
the ‘fine structure’ [2, p 48] of the waveform, is
436.2 Hz, compared with an expected 433.3 Hz,
which is the average of the frequencies of the two
tuning forks. The beat frequency of the envelope
is 13.8 Hz compared with the expected 13.4 Hz,
the difference between the frequencies of the two
tuning forks5 .
The reason why there are two different
frequency tuning forks labelled as the musical note
‘A’ is because of the existence of different musical
scales [1; 2, p 17; 5, p 50; 6, p 243]. A-426.6
is ‘A’ in the Just Diatonic Scale based on C-256,
meaning that it is a musical sixth 53 × 256 = 426.6
above C-256. A-440 is the pitch standard of the
equal tempered scale6 .
Beats
Note that the waveforms of the less consonant
intervals in figure 5 begin to resemble the familiar
pattern of beats. This is to be expected, as beats
are considered very dissonant. When musicians
‘tune’ instruments they are attempting to play at
the same frequency as one another. Beats are
what they are listening for when they attempt
to determine whether or not they are in tune.
Figure 6 shows beats produced by sounding an
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Dyads and triads
When two musical pitches are sounded simultaneously the resulting tone is called a dyad; when
three are sounded it is called a triad. The intervals
5
Reference [8], an accompanying laboratory manual to
Vernier Software’s equipment, has an experiment on beats on
p 21-1.
6 Reference [8] also has an experiment on the frequencies of
the equal tempered scale, Mathematics of Music, on p 23-1.
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M C LoPresto
Figure 6. The pattern of beats between tuning forks of frequencies 440 Hz and 426.6 Hz.
Table 3. Comparison between expected ‘missing’ fundamental frequencies of triads produced by tuning forks
and those calculated from their waveforms.
Triad
Musical
notes
Frequency
ratio
Fundamental
frequency (Hz)
Calculated
fundamental (Hz)
Major
Minor
C–E–G
E–G–B
256/320/384 = 4/5/6
320/384/480 = 10/12/15
256/4 = 320/5 = 384/6 = 64
320/10 = 384/12 = 480/15 = 32
63.7
31.9
that have been discussed thus far are all dyads.
When three pitches of frequency ratio 4/5/6 are
sounded together this is known as a major triad;
the ratio of 10/12/15 is a minor triad. These two
common types of ‘chords’ are the basis for much
of the harmonic structure of Western music. Since
a third tuning fork was involved, it was no surprise
that it was harder to capture the expected complex
forms of triads than dyads, but the expected fundamental frequencies could, however, still be calculated with just as much accuracy as the dyads.
Figure 7 shows the expected and actual shapes of
a major triad of frequencies, 384/320/256, fundamental 384/6 = 320/5 = 256/4 = 64 Hz. The
measured fundamentals in table 3 are very close
to what was expected7 .
Conclusion
The exercise of capturing the combined waves,
observing their forms, calculating the fundamental
frequencies and making comparisons with the
expected waveforms and frequencies can provide
a hands-on and practical lesson in superposition
of waves and even an introduction to the
7
Reference [3, p 195] also has waveforms for triads if
graphing software is not available.
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Figure 7. The expected (bottom) and captured (top)
waveforms of a major triad.
basics of Fourier composition. Inspecting the
effects of amplitude and phase and exploring
consonance and dissonance can help to provide
understanding of the relationships between the
subjective properties of sound perceived by the ear
and the objective ones produced by the waves.
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Experimenting with musical intervals
Note. Five Logger-Pro files that were used in
the experiment are available from the electronic
journal (stacks.iop.org/0031-9120/38/309).
Acknowledgments
I thank HFCC honor’s student Leisanne Smedala
for asking the questions that originated this project
and Honor’s Program Director Nabeel Abraham
for allowing her to work with me an extra semester.
I also thank Dr Leroy S Williams, retired Professor
of Music from Edinboro University of PA, and
HFCC music instructors Kevin Dewey and Randy
Knight for providing essential perspective from the
musical side of things.
[3] White H E and White D H 1980 Physics and
Music: The Science of Musical Sound
(Philadelphia, PA: Saunders College
Publishing) p 192
[4] French A P 1971 Vibrations and Waves (New
York: Norton) p 25
[5] Sethares W A 1999 Tuning, Timbre, Spectrum,
Scale (New York: Springer) p 16
[6] Rigden J S 1985 Physics and the Sound of Music
2nd edn (New York: John Wiley) p 239
[7] Helmholtz H 1954 On Sensations of Tone (New
York: Dover) p 196
[8] Vernier et al 2000 Physics with Computers
(Beaverton, OR: Vernier Software and
Technologies)
Received 14 February 2003, in final form 12 May 2003
PII: S0031-9120(03)59582-X
References
[1] Rossing T D 1990 The Science of Sound 2nd edn
(New York: Addison-Wesley) p 80
[2] Roederer J G 1995 The Physics and
Psychophysics of Music—An Introduction
3rd edn (New York: Springer) p 44
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Michael C LoPresto is chair of the
Physics Department at Henry Ford
Community College in Dearborn,
Michigan and is an amateur trombone
player and vocalist. He is also the author
of an astronomy laboratory manual and a
textbook supplement ‘Cycles in the Sky’.
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