1. ABSTRACT
Consonance vs. Dissonance - A Physical Description. BENJAMIN D. SUMMERS
(Louisiana State University, Baton Rouge, LA, 70803)
Fourier synthesis has been applied to the comprehensive set of just and equal
temperament acoustic intervals and triads to uncover unique, objective
relationships between constituents of consonant chords. Beat frequencies have long
been considered a significant contributing factor to the categorization of consonant
and dissonant harmonies, but not absolutely so. Music theorists have generally
concluded that compositional convention is largely responsible for molding human
perception of what sounds "pleasing". This computational approach to acoustic
waveform analysis has found that beat frequency relationships alone can be utilized
to effectively define consonance and dissonance; beat frequencies themselves must
be considered as an additional pitch relative to the specific frequencies comprising
diatonic intervals and triads.
3. Consonance VS. Dissonance
A Physical Description
2
Consonance is defined as a harmonious sounding together of two or more notes, that is
with an 'absence of roughness', 'relief of tonal tension' or the like.! Dissonance is a discord or any
sound which, in the context of the prevailing harmonic system, is unstable and must therefore be
resolved to a consonanc
The purely musical definitions of consonance and dissonance offer guidelines by which to
group intervals into two distinct classes, but fall short of fulfilling their purpose as they are both
victim to the subjectiveness of individuals' perceptions and opinions. A more objective definition
may be found by approaching the issue from a different perspective, looking at the physical
phenomenon that makes up sound and consequently music. Although music is an art form, its
theory is deeply grounded in mathematical roots. Exploring those roots and understanding them
has and will continue to lead to the evolution of music as an even more effective means of
1 Claude V. Palisca and Natasha Spender, "Consonance," New Grove Dictionary of Music and
Musicians, ed. Stanley Sadie (Washington, DC: Grove's Dictionaries of Music, Inc., 1980),
4:668.
2 Claude V. Palisca and Natasha Spender, "Dissonance," New Grove Dictionary of Music and
Musicians, ed. Stanley Sadie (Washington, DC: Grove's Dictionaries of Music, Inc., 1980),
5:496.
.2
4. 3
expressing the emotions of humanity. The concept of consonance and dissonance is one of those
roots and the continued pursuit of understanding its role in tonality leads us to a fuller
comprehension of the nature of music. In an attempt to understand the mechanisms of tonality,
we must first look at what sound is and cover the basics of simple harmonic oscillations.
Sound is defined as a perceived disturbance stimulating the organs of hearing.3 For
humans, that disturbance is the change of density and pressure of a particular medium, usually
air, between the rates of 20 and 20,000 repeated cycles per second. These density and pressure
changes are the propagation of the energy transferred from an analogously vibrating source. It
probably comes as no surprise that the motion of this source has unique characteristics, which set
it apart from many other types of motions. The term simple harmonic oscillator describes an
object exhibiting this type of motion defined by a strict set of criteria that can be expressed
quantitatively. The criteria by which a motion is defined as simple harmonic are as follows: the
motion must repeat itself at regular intervals in time such that the object's displacement from its
equilibrium position is expressed as a function of time. The degree to which the object moves in each
cycle is defined as amplitude, and how often the repeated motion occurs per a given unit of time is
defined as frequency. Amplitude is usually measured in units of distance, while frequency is
measured in cycles per second, Hertz. Mathematically speaking, a simple harmonic motion, or
oscillation, is expressed by sinusoidal functions and accompanying constants. In the most
fundamental case, simple harmonic motion (SHM) is described by the expression:
3 "Sound 1," The American Dictionary of the English Language, ed. William Morris (Boston:
Houghton Mifflin Company, 1976), 1234.
5. 4
x(t) =X sin(w*t + p)
{x(t) is the position of the particle relative to its equilibrium position (usually
defined as the x-axis) for a given moment in time; X is the amplitude, or the
magnitude of the oscillations; sine) is the oscillating function; w is 2*Pi*frequency
[This is necessary because sinusoidal functions complete one cycle every 2*Pi
radians (360 degrees) and multiplying this by the frequency gives the appropriate
number of oscillations per unit time.]; t is time; and p is the phase shift relative to
the sin(w*t) at t=O [phase is the angular position of the function for a given moment
in time.]} 4
The mechanical energy that is manifest as simple harmonic oscillations is transferred
throughout the source, for our purposes the musical instrument, and surrounding medium, air.
The propagation of this energy through an elastic medium from its origin is called waves and is
perceived as sound. When these waves encounter an inelastic medium, they are reflected, unable
to travel through a medium that cannot flex into a waveform. As would be expected, waves carry
the same characteristics as the motion that produced them. They are composed of frequency,
wavelength, or the distance of propagation per one complete wave cycle, amplitude and phase
and are described mathematically very similarly to SHM. This is due to the fact that sound waves
are actually the sum of the SHM experienced by each infinitesimally small element of matter along
the path of the wave. Sound travels along this path at approximately 343 meters per second if the
4 David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics Extended, 4th
edition} (New York: John Wiley & Sons, Inc., 1993),382-384.
6. 5
oscillating matter happens to be 20 degree Celsius air. The speed of sound is constant within any
particular medium (yet may vary from medium to medium depending on the properties of each
medium) and as with all waves is equal to the product of its frequency and wavelength. We will
see later how frequency can be manipulated by using this fact that the speed of sound is constant
and dependent upon wavelength. Like speed, amplitude also depends on the properties of the
medium in which sound travels. For a given energy, amplitude can vary greatly depending on
the elasticity, density, and other factors associated with the medium in which it travels. The
amplitude of the oscillating medium due to a sound wave is perceived as volume while the
frequency is perceived as pitch. The phase of a sound wave, even though imperceivable to the
human ear, is relevant when multiple waves become superimposed resulting in interference.
(Physically speaking, superimposed sound waves would be those which are sounded
simultaneously within audible proximity of each other.) Interference is the resultant wave induced
by the addition of the displacement, orthogonal to the direction of propagation, of two
superimposed waves. If the phase of two superimposed oscillating functions differs by 180
degrees and their frequencies are the same, a net cancellation will result in their sum; this is called
destructive interference. For all other cases, if two superimposed waves are of slightly different
frequency, the resultant wave has periodic oscillations in amplitude called beats. The beat
frequency is the number of periodic amplitude oscillations per unit time. If the one frequency is
double that of the other, however, the beats are lost in that the beat frequency perfectly overlaps
the natural oscillations of the lower frequency resulting in constructive interference of the lower
frequency. As the integer multiple of the lower frequency increases, beats begin to become
noticeable and increasing. Any number of waves of infinite variance can interfere to produce
extremely complicated beat patterns. This resultant pattern can be quantified, however, by
Fourier Analysis, the process of taking the sufficient sum of sinusoidal functions required to
7. 6
exactly produce any desired complex pattern. As with all waves, the characteristics of sound are
dictated by the physical properties of the materials in which it travels. The manipulation of these
properties is the basis for the production of musical sound.
Now that the fundamentals of sound have been discussed, we can begin to apply these
conditions in a discussion of how particular sounds are produced. All that is needed to produce a
particular pitch is some form of matter oscillating at the particular frequency of the desired pitch.
This is achieved in many different ways. The most common oscillating sources used in music are
strings, as in string instruments, membranes, as in the timpani, and columns of air as in wind
instruments. The most simple of these cases is that of a vibrating string with each end tethered to
an inelastic node as in a guitar. The factor determining the loudest, or fundamental, frequency of
the string is its length. The wavelength of the fundamental frequency of oscillation is equal to
twice the length of the string and completes one half cycle each length of the string. This is due to
the fact that when the string is vibrated, the inelastic boundaries are immobile and quickly damp
all the resulting waves in the string, except for those of frequencies that naturally pass through
one of their equilibrium, or immobile positions at both nodes. That is, if the two points that cross
the equilibrium position in each cycle of a sinusoidal wave naturally fall at the endpoints of the
string, the negative wave will be reflected back on to itself at the endpoints 180 degrees later. The
result is that the half cycle of the fundamental and all the cycles of integer multiple frequency are
superimposed onto each other due to the reflections at the nodes. The reflected waves of same
frequency and phase cause constructive interference and the result is a series of what are called
standing waves. Each bound end is called a node, and the oscillating center is called the envelope.
None of the standing waves' energy is lost to the bound ends of the string allowing sustained
vibrations of the envelope. This phenomenon, called resonance, can be expressed mathematically:
8. yet) =X*sin(n*k*x+w*t) + X*sin(n*k*x-w*t) =2*X*sin(n*k*t) * cas(w*t)
{yet) is the vertical position of the wave at a particular time t; x is the horizontal
position; k is 4*Pi/string length, n is the integer corresponding to each of the multiple
frequencies. The function between the equal signs shows the addition of the initial,
or incident, waves and the waves reflected by the nodes. Since each incident and
reflected wave are of the same frequency and in phase, the resultant standing wave,
expressed by the function to the extreme right, illustrates a doubling of the
amplitude due to constructive interference. These two functions are equal by a
form of trigonometric manipulation. The result is a magnification in the
fundamental frequency and its integer multiples as the nodes absorb the energy of
all extraneous frequencies.} 5
7
The waves of these resonant frequencies continue to be reflected back and forth until they die out
due to resistance within the string itself. (In order for a string to resonate at its maximum
potential, however, the relationship between its tension and density must also be conducive to
vibrating at the fundamental frequency. The math involved in this relationship is not important
for our purposes as it does not affect the waves of integer multiple frequency relative to the
fundamental.) The phenomenon of resonance is exhibited in all other instruments in ways
perfectly analogous to those just described, but it is not necessary to go into the mechanics of them
now for the purposes of our discussion.6 The fundamental frequency of any resonating source
5 Halliday, Resnick, and Walker, Fundamentals afPhysics Extended, 491-495, 514-516.
6 Halliday, Resnick, and Walker, Fundamentals afPhysics Extended, 491-495, 514-516.
9. 8
and all of its integer multiples are called harmonics. The fundamental is the first harmonic, while
twice the frequency of the fundamental is the second harmonic, and so on and so forth. For
example, if a particular fundamental frequency is 100 Hz, the second harmonic will be 200 Hz,
while the third harmonic will be 300 Hz and so on and so forth. Since the speed of sound is
constant for a particular medium and equal to the product of its wavelength and frequency as
mentioned before, the inverse relationship exists between the wavelengths of each harmonic. So if
the wavelength of the fundamental is 60 meters, the second harmonic will have a wavelength of 30
meters, while the third will be 20 meters and so on and so forth. As was mentioned earlier, when
waves of frequencies that are multiple integers of each, or harmonics, are superimposed, beats
begin to be produced in the resultant interference pattern only as higher harmonics are reached
because of the large difference in frequencies of the fundamental relative to its first few harmonics.
For any particular resonating source, the intensity of each superimposed harmonic dies off
significantly as the harmonic number increases, while the beat patterns begin to become more
pronounced. The beats induced by higher harmonics in a single resonating source are hardly
perceptible, however, because they are drowned out by the intensity of the lower harmonics.7
Due to the fact that the human ear perceives continuous sounds to be more soothing than
intermittent ones, intervals producing beatless waveforms are defined as most consonant.
Because of this the set of consonant intervals are those comprised of two pitches whose
frequencies are that of a fundamental and one of its first few harmonics. For example, the interval
between a fundamental frequency and its second harmonic is called an octave. Another way of
expressing this relationship is to say that the ratio of the frequency of an octave to its
7 Hermann L. F. Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory ofMusic,
(New York: Dover Publications, Inc., 1954), 182,226.
10. 9
fundamental is 2: 1. The next most consonant interval is the fifth; the frequency ratio
of it relative to its fundamental is 3:2. This comes from the fact that the third
harmonic is three times the frequency of the fundamental, which produces the interval
of a twelfth, 3: 1. Converting this compound interval to its primary counterpart, the
fifth, requires doubling the fundamental to close the gap of the interval by an octave
resulting in the 3:2 ratio. The Fourier Analysis of the fifth, however, gives the greatest
peak in the graph reoccurring at a frequency of approximately half that of the
fundamental, or an octave below. If this interval is played in the lowest register of
perceivable sound, these spikes cause irregularity in the resultant sound because they
are of frequency too low to be perceived as tone. The result is that this interval could
pass as dissonant in the lowest register. The third most consonant interval is the
fourth consisting of the fundamental and its fourth harmonic, the double octave, with
the frequency of the fundamental being multiplied by three to produce the primary
interval resulting in a 4:3 ratio. The Fourier Analysis of the fourth illustrates the
vague outline of a beat frequency with its max reoccurring at approximately one third
the frequency of the fundamental, a twelfth below. This explains why a fourth is
considered dissonant when its fundamental is in the bass. In most cases, when a
fourth is rooted in the bass, a twelfth below the fundamental usually borders being
out of audible frequency range resulting in an irregularity in the resultant sound.
This irregularity is cause for it being labeled dissonant in this case.8
The Fourier Analysis of the next most consonant interval, the major sixth, 5:3,
shows a regular patter of spikes that complete on cycle at a frequency approximately
one third that of the fundamental. The difference between the sixth and the fourth,
8 Appendix 1.
11. however, is that there is not even a hint of a crescendo to each peak in the sixth and
that one cycle consists of two half peaks of high then low pressure. This results in
10
less obvious disturbances of the sound in the lower registers. The major third, 5:4, is
the first interval that gives a clear beat pattern with obvious nodes and envelopes.
These beats occur at a frequency approximately one fourth that of the fundamental,
two octaves below. Again, no audible beats are produced until the interval is played in
registers such that the beats become too low in frequency to be perceived as tone and
create a disturbance in the sound. The next most consonant interval is the minor
third, 6:5. Its beats sound at a frequency one fifth that of the fundamental, two
octaves and a major third below. When the beats are in audible frequency range, the
tone they produce in combination with the fundamental and the minor third above it
forms the major six triad in the parallel minor of the fundamental. This third tone is
difficult to hear, but when it is actually played in conjunction with the minor third
interval, the roughness of the minor third subsides explaining the smoothness of
major triads. The last consonant interval is the minor sixth, 5:8. Its Fourier pattern
is very irregular, with one complete cycle occurring one seventeenth the frequency of
the fundamental, but the dominant characteristic of the graph is that it reaches its
max peak at half the frequency of the fundamental. The result is a moderately
consonant sound with it being supported in use by the fact that it is the inversion of
the more consonant major third. After the minor sixth, the graphs begin to become
continually more irregular and thus less consonant sounding. With the shift from just
tuning to equal temperament, the Fourier Analysis still maintains the crucial
characteristics which define the intervals in the same manner as was just described.
12. 11
The differences are subtle and not significant as long as the intervals are played within
their respective temperaments.9
The combination of consonant and dissonant intervals throughout a musical
work is what creates and relieves the tension of the piece, making it move and
bringing the listener to feel the emotions felt by the composer. Understanding why
particular sounds fit well together leads to a more effective manipulation of these
sounds in the composition process. Even though music is as subjective as any other
art form, the understanding of its mechanics will only lead to the enrichment of its
evolution with an improvement in new compositional techniques.
9 Appendix 1.
13. 12
Appendix I.
The following appendix includes the Fourier Analysis of various musical intervals. The
x-axis is time, and the y-axis is pressure. This analysis is the exact analog of the
pressure variations in air due to sound for each given interval. The numerical values
for each axis are only relevant for scale, not for actual pressure and time values.
14. CONvsDIS.mm
Just Tuning
OCTAVE 2/1
FIFTH 3/2
FOURTH 4/3
MAJOR SIXTH 5/3
MAJOR THIRD 5/4
MINOR THIRD 6/5
TRITONE 64/45
MAJOR SECOND 9/8
MAJOR SEVENTH 15/8
Plot [5 (Sin [x] + Sin [x] ), {x, - 20 lr, 20 lr}, AspectRatio -+ Automatic,
PlotRange-+ {{-70, 70}, {-is, 15}}. PlotLabel-. "Unison"]
m Graphics -
unison
15
-15
Plot [5 (Sin[x] + Sin[2 xl), {x, -20 lr, 20 lr} I AspectRatio -+ Automatic,
PlotRange-+ {{-70, 70}, {-15, 15}}, PlotLabel-+ "Octave"]
- Graphics -
Octave
15
10
-10
-15
1
27. BIBLIOGRAPHY
Askill"John. Physics ofMusical Sounds. New York: D. Van Nostrand Company, 1979.
Backus, John. The Acoustical Foundations ofMusic. New York: W. W. Norton & Company, Inc.,
1969.
Benade, Arthur H. Hams, Strings & Harmony. Garden City: Anchor"Books Doubleday &
Company, Inc., 1960.
Halliday~ David, Resnick, Robert and Walker, Jear!. Fundamentals ofPhysics Extended. 4!b edition.
New York: John Wiley & Sons, Inc., 1993.
Helmholtz, Hermann L. F. On the Sensations ofTone as a Physiological Basis for the Theory ofMusic.
New York: Dover Publications, Inc., 1954.
"Sound 1," The American Heritage Dictionary ofthe English Language. edited by William Morris,
1234. Boston: Houghton Mifflin Company, 1976.
