ABSTRACT

Consonancevs.Dissonance-APhysicalDescription.BENJAMIND.SUMMERS

(LouisianaStateUniversity,BatonRouge,LA,70803)

Fourier synthesis has been applied to the comprehensive set of just and equal

temperament acoustic intervals and triads to uncover unique, objective

relationshipsbetweenconstituentsofconsonantchords.Beatfrequencieshavelong

beenconsideredasignificantcontributingfactortothecategorizationofconsonant

and dissonant harmonies, but not absolutely so. Music theorists have generally

concludedthatcompositionalconventionislargelyresponsibleformoldinghuman

perception of what sounds "pleasing". This computational approach to acoustic

waveformanalysishasfoundthatbeatfrequencyrelationshipsalonecanbeutilized

toeffectivelydefineconsonanceanddissonance;beatfrequenciesthemselvesmust

beconsideredasanadditionalpitchrelativetothespecificfrequenciescomprising

diatonicintervalsandtriads.

Consonance vs. Dissonance

A Physical Description

Benjamin D. Summers

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.

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.

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.

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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

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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:

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.

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 of Music,

(New York: Dover Publications, Inc., 1954), 182,226.

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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.

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.

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.

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.

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

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unison 15

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Equal Temperament Equal Temperament

FIFTH 749/500 FOURTH - 3337/2500 MAJOR SIXTH 841/500 MAJOR THIRD 5/4 MINOR THIRD - 2973/2500 MINOR SIXTH - 7937/5000 TRITONE - 7071/5000

MAJOR SECOND 449/400

MAJORSEVENTH - 7551/4000

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Triads (Just Tuning)

Just Triads Tuning

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Triads (Equal Temperament)

Equal Temperament Triads

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Minor (Second Inversion) 15 1

Plot[S (Sin[x] +Sin[2973x/2S00] +Sin[7071x/SOOO]), {x, -30Jr, 30Jr},

12

AspectRatio -+ Automatic, PlotRange -+ {{ -100, 100}, {-lS, lS}}, PlotLabel -+ "Diminished "]

Diminished

-100

-15

- Graphics -

CONvsDIS.mm

Plot [5 (Sin[2973x/2S00] +Sin[7071x/SOOO] +Sin[2x]), {XI -30:rr, 30:rr}, AspectRatio ... Automatic, PlotRange ... {{-100, 100}, {-lS, lS}}, PlotLabel ... "Diminished (First Inversion)"]

-100

- Graphics -

Diminished (First Inversion) 15

Plot [5 (Sin[7071x/SOOO] +Sin[2x] +Sin[2973x/12S0]), {x, -30:rr, 30:rr}, AspectRatio ... Automatic, PlotRange ... {{-lOO, 100}, {-lS, lS}}, PlotLabel ... "Diminished (Second Inversion)"]

-100

- Graphics -

Diminished (Second Inversion) 15 1

-10 -15

13

BIBLIOGRAPHY

Askill" John. Physics of Musical Sounds. New York: D. Van Nostrand Company, 1979.

Backus, John. The Acoustical Foundations of Music. 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 of Physics Extended. 4!b edition.

New York: John Wiley & Sons, Inc., 1993.

Helmholtz, Hermann L. F. On the Sensations of Tone as a Physiological Basis for the Theory of Music.

New York: Dover Publications, Inc., 1954.

"Sound 1, " The American Heritage Dictionary of the English Language. edited by William Morris,

1234. Boston: Houghton Mifflin Company, 1976.