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Duke University
Emotional Sound Waves and The Math of Music
Aaron Paskin
Math of the Universe – 89S
Hubert Bray
8 December 2016
Introduction
Music can invoke a wide range of emotions. While some songs can feel happy
and positive, others might be sad, paranoid, disappointing, or unsettling. It would be
reasonable to wonder whether songs have these characteristics because we have
learned to feel certain emotions when hearing a certain type of music or because there
is a fundamental quality of music that can determine tone. While nurture surely has an
impact on how people interpret tone, it also turns out that there are relationships among
musical notes that the brain naturally reacts to in different ways. Understanding the
mathematical and psychological relationships among certain groups of notes is
essential in answering the question of why music invokes so many various emotions
and feelings.
Note: Several audio files will be used throughout this paper. Listen to them by
double-clicking on their icons, or follow the links above or below the icons to listen to
them online.
Sound, Pitch, and Notes
Before we talk about music, we must understand the fundamental nature of
sound and how it can produce singular musical notes. Sound is the energy of vibrations
that emanate from a source. When something moves, it moves the matter immediately
surrounding it. This begins a domino effect of matter moving matter in a series of
vibrations. For example, when a person claps their hands, the impact of their hands
vibrates the nearest air molecules, which in turn vibrate their neighboring air molecules.
The vibrations travel in waves through the air until they reach another person’s head,
where they impact audio receptors in that person’s ears. The same idea applies to
sounds in any medium; sound waves can travel through gases, liquids, and even solids.
(5, 7)
The rate at which vibrations travel as sound waves is called frequency. Many
sounds, such as those that are produced by clapping or drumming, have frequencies
that are too inconsistent and jumbled to calculate. Sometimes, however, a sound’s
frequency is consistent and measurable. When this is the case, we perceive a clear
pitch, a quality that allows people to distinguish different sounds as “higher” or “lower”
than each other: Sounds with greater frequency sound “higher” while sounds with lesser
frequency sound “lower”. Frequency is most commonly measured in Hertz.
Pitch isn’t the only quality of sound that allows us to identify the noises of the
world. Timbre is different a way we perceive sound, as it allows us to distinguish
between sounds of potentially the same volume and frequency, such as thunder and a
lion’s roar, or hundreds of different human voices. If timbre can change while volume
and frequency are held constant, what determines timbre? It turns out that a sound of a
specific frequency consists of waves not only of that frequency but also of integer
multiples of that frequency. In other words, a sound wave with a frequency of 100 Hz
travels with sound waves of 200 Hz, 300 Hz, 400 Hz, and so on. The root frequency
(which is 100 Hz in the previous example) determines the pitch while the multiple
frequencies, called overtones, determine timbre. Two different singers producing
sounds of 100 Hz with equal volumes sound different because the volumes of their
overtones vary. The varying volumes of their overtones are not significant enough to
change the overall volume of the sound, but they are enough to affect the timbre. A
fundamental or root frequency and its overtones together are called a harmonic series.
Perceiving timbre has numerous evolutionary benefits, including the thunder and
voices examples previously given. Thus, it is not surprising that the brain is quite good
at recognizing and differentiating harmonic series. This idea will be important later in
understanding chords. The brain is also good at recognizing relative frequencies. We
have the inherent ability to qualitatively and relatively recognize just how different the
frequencies of two notes are. These two skills of the human brain are fundamental to
the construction of music. (6)
Constructing A Musical Notation System
By definition, every sound with a distinct frequency has a unique pitch. For the
purpose of making music, it would be sensible to assign notations to sounds with
specific frequencies. Let’s assign the letter “X” to all sounds with a frequency of 261.63
Hz. This frequency seems almost comically specific, which raises the following
question: if we want to designate a symbol for every sound we would possibly want to
use in music, wouldn’t we have to designate an impossibly large amount of symbols to
span the thousands of Hz that the human ear is capable of hearing? Luckily for us,
there are noticeable patterns that arise when exploring sounds of different frequencies,
which allows us to reuse notations. The most obvious pattern comes from doubling the
frequency of a given sound, which results in a pitch strikingly similar to the original pitch.
In fact, this pattern arises when multiplying any given frequency by integer powers of
two.
We can explore this by using our sound “X” from earlier. Below are three
separate sounds: The first (1) is “X”, a sound with a frequency of exactly 261.63 Hz; the
second (2) is a sound with a frequency of 261.63 Hz * 21 = 523.25 Hz; the third (3) is a
sound with a frequency of 261.63 Hz * 22 = 1046.50 Hz.
https://en.wikipedia.org/wiki/Audio_frequency
1.
2.
3.
This fundamental relationship is the basis for all of the common musical notation
systems used today. In general, a sound of a specific frequency is designated with a
letter. As another example, let “A” be a sound of frequency 100 Hz (we’ll call this an A
note from now on). We designate all sounds of frequency 100 Hz * 2x, with “x” being any
integer, as A notes as well, so sounds of 25, 50, 100, 200, 400, 800 Hz and so on are
all A notes. The frequency range between two subsequent A notes, called an octave, is
then divided into a certain number of sections. We will be focusing on a twelve-tone
temperament, a system that divides each octave into twelve notes, labeled in order of
increasing frequency A, A# (Bb), B, C, C# (Db), D, D# (Eb), E, F, F# (Gb), G, G# (Ab).
Notes with the symbol “#” are called “sharp” while notes with the symbol “b” are called
“flat”. A# and Bb have the same frequency, as do the other notes with two different
names. (1)
There are many different types of twelve-tone temperaments, or methods by
which an octave can be split into twelve notes, but we will only focus on equal
temperament. The equal temperament is what is used to construct most pianos and
other instruments, and it is the most mathematically consistent temperament for
understanding chords and scales. It subdivides octaves by keeping a constant ratio of
about 1.05946 (the twelfth root of two) between the frequencies of successive notes.
Thus, the ratio of the frequency of A# to that of A is 1.05946, as is the ratio of B to A#, C
to B, and so on. These increments from one note to the next are called half-steps.
Narrowing down our notation system one more time, let’s give the A note a
frequency of 440 Hz and create all other notes using a twelve-tone equal temperament.
We have now described what is likely the most common complete musical notation
system and the one that we will use to understand chords and scales: the A440 twelvetone equal temperament. A table of the frequencies of nine octaves in such a system
can be found below. (3, 6)
Figure 1: Frequencies of the notes of an A440 twelve-tone equal temperament (in Hertz) (http://www.bloudoff.com/2012/11/patentfor-sale.html)
Triads, Chords, Scales, and Keys
What Are Triads and Chords?
Notes that sound nice when played simultaneously, such as a group of notes
separated by whole octaves, are said to have harmony. We can start exploring more
complex groups of notes with harmony, or chords, by analyzing certain types of triads,
which are combinations of three notes. Not all triads have harmony or are meaningful,
but the ones that do can be classified by the number of half-steps separating the three
notes. The most popular classifications of triads are Major and Minor triads.
A Major triad is any group of three notes in which the first two notes are
separated by four half-steps and the second and third notes are separated by three halfsteps. The chord is named by the first of the three notes. For example, an A Major triad
contains the notes A – C# – E, and a C Major triad contains the notes C – E – G. The
relative frequencies of the notes in all Major triads are described by the ratio 4 : 5 : 6.
A Minor triad, on the other hand, is any group of three notes in which the first two
notes are separated by three half-steps and the second and third notes are separated
by four half-steps. For example, an A Minor triad contains the notes A – C – E, and a C
Minor triad contains the notes C – Eb – G. The relative frequencies of the notes in all
Minor triads are described by the ratio 10 : 12 : 15.
Why do these calculated and precise triads produce such characteristic sounds?
In order to discover what is so special about the 4 : 5 : 6 and 10 : 12 : 15 ratios, we can
derive the Major and Minor triads using only the information that has been presented so
far. (1, 6)
Deriving the Major Chord
If we want to create a pleasant chord from scratch, where should we start? We
know that the brain has a unique skill of recognizing harmonic series. This skill might
explain why playing notes separated by whole octaves sounds so nice, as such notes
are in a harmonic series together (their frequencies are equal to a given frequency
multiplied by an integer power of two). If this is true, then it would be reasonable to
believe that simultaneously playing all overtones of a given note would sound nice too,
as the brain would recognize the harmonic series. Recall that the overtones of a note
include notes with all integer multiples of the given frequency, not just those with a
factor of two to an integer power.
Let’s test this hypothesis by using an A note with a frequency of 440 Hz. From
now on we will use scientific pitch notation, which specifies the note and the octave.
A440 is designated as A4, the A note in the “fourth” octave. The “fifth” octave begins
with the C note above A4, designated as C5.

1 * 440 Hz = 440 Hz: The first note in the harmonic series is A4

2 * 440 Hz = 880 Hz: Doubling the frequency of A4 simply yields A5, which we
already know sounds nice when played with A4.

3 * 440 Hz = 1320 Hz: In A440 equal temperament, E6 has a frequency of
1318.51 Hz. A 1.49 Hz difference is virtually negligible, so we can round this to
1320 Hz and include E6 in our harmonic series.

4 * 440 Hz = 1760 Hz: This is A6, which we already know sounds nice when
played with A4.

5 * 440 Hz = 2200 Hz: In A440 equal temperament, C#7 has a frequency of
2217.46 Hz, which may be rounded to 2200 Hz. We intend to construct a Major
triad, so we may stop now that we have three different notes in our harmonic
series.
The notes we have identified as being a part of A4’s harmonic series are A4, A5,
E6, A6, and C#7. For simplicity, we can condense these notes into three distinct notes
contained in a single octave by dividing their frequencies by integer powers of two.

Use A4 (440 Hz) as the root, the first note in the triad.

E6: 1320 Hz / 2 = 660 Hz, which is approximately E5

C#7: 2200 Hz / 22 = 550 Hz, which is approximately C#5
The resulting three notes are, in ascending order of frequency, A4, C#5, and E5.
Finally testing our hypothesis that playing a note and its overtones together produces a
pleasant chord, we find that playing the chord A – C# – E in any octave sounds nice.
Indeed, this chord has the attributes of a Major triad: A is four half-steps below C#
while C# three half-steps below E, and their frequencies (440 Hz, 550 Hz, and 660 Hz)
are described by the ratio 4 : 5 : 6. All Major triads can be constructed by choosing a
note and finding its two closest harmonics (besides itself). The chosen note is called the
root, the second note in the triad is called the Major third, and the third note in the triad
is called the perfect fifth. Using the ratio 4 : 5 : 6, it will be convenient to observe that the
ratio of the Major third to the root is 5/4, the ratio of the perfect fifth to the root is 3/2,
and the ratio of the perfect fifth to the Major third is 6/5. (6)
Deriving the Minor Triad
In deriving the Major triad, we confirmed that such a triad sounds pleasant
because the brain easily recognizes the harmonic series present. But what if we want a
chord that doesn’t sound so comfortable? If we are writing a piece of music that we
want to be sad or unsettling, for example, the uplifting nature of the Major triad just
won’t suffice.
Referring back to the first section, recognizing harmonic series is one of two
special skills the brain has in listening to music. The second special skill of the brain is
that of recognizing and remembering patterns in relative frequencies. We can slightly
alter the Major triad in a way that veers from the harmonic series, but still appeals to the
second special skill of the brain by including most of the frequency ratios found in a
Major triad. The brain will recognize the relative frequencies as it recalls the Major triad,
but it is left searching for the missing harmonics. The result is a chord that sounds
slightly “off” but still thematic. We call this the Minor triad.
So, how exactly can we alter the Major chord so that it maintains most of the
same frequency ratios (though not necessarily in the same order)? If we reduce the
Major third of a Major triad by a half-step, we have a triad consisting of a root, a
harmonic of the root (the perfect fifth), and a note that isn’t quite a harmonic. However, if
we calculate the ratios of frequencies, we find that the new triad consists of relative
frequencies 1 – 6/5 – 3/2. The 5/4 from the Major triad is missing, but all other ratios
from the Major triad are present (recall that 6/5 is the ratio of the perfect fifth to the
Major third). The brain easily recognizes these ratios as describing the relative
frequencies of a note and its two closest harmonics, but it is left searching for the
missing harmonic.
In short, the Minor triad sounds slightly “off” or unsettling, but not dissonant,
because it violates the expected pattern of the harmonic series but recalls the familiar
relative frequencies of the Major triad. (6)
Deriving the Major Scale
Now that we have derived the Major triad, how can we use such chords to create
music? A piece of music is played in a key, which means that the composition
predominantly consists of notes from a specific collection of seven notes. Such a
collection is known as a scale. Like chords, scales and keys have characteristic sounds
and can be used to give music theme and emotion.
We were able to find three notes that sound good together when we derived the
Major triad. In order to find a Major scale, we will need to find different Major triads that
are related. Using the A Major triad again as an example, let’s make a new triad by
setting the root to be the overtone that corresponds with the closest multiple to A. The
first new note we get when multiplying the frequency of A4 by integer multiples is E (3 *
440 Hz = 1320 Hz = E6). Thus, we shall make a new Major triad with E5 as the root,
dividing by two to lower the octave and stay near A4.

Root: 1320 Hz (E6) / 2 = 660 Hz  E5

Major third: (5/4) * 660 Hz = 825 Hz  G#5

Perfect fifth: (3/2) * 660 Hz = 990  B4

The result is the E Major triad: E – G# – B
Let’s make a third Major triad by going in the other direction; that is, let’s
designate the root of the chord we started with (the A Major triad) as the perfect fifth of
our new Major triad. This time, we will multiply by two to raise the octave and stay near
A4.

Root: 440 Hz / (3/2) = 293.333 Hz  D4

Major third: (5/4) * 293.333 Hz = 366.666  F#4

Perfect fifth: A4 as designated

The result is the D Major triad: D – F# – A
Combining these three neighboring triads and eliminating the duplicates, we get
the following sequence of notes: A – B – C# – D – E – F# – G#. This is the A Major
scale, constructed with three interlocking Major triads. As expected this sequence of
notes sounds quite nice when played in order. (2, 6)
http://dictionary.onmusic.org/appendix/topics/major-scales
A Major scale
Deriving the Minor Scales
We know that a Major scale consists of three interlocking Major triads. With the
introduction of the Minor triad, there are now many ways to interlock Major and Minor
triads to create different types of Minor scales. The Natural Minor scale, for instance, is
constructed in the same fashion as the Major scale, but with three Minor triads instead.
For example, the Natural A Minor scale interlocks the A Minor triad (A – C – E) with the
D Minor triad (D – F – A) and the E Minor triad (E – G – B). The resulting Natural A
Minor scale consists of notes A – B – C – D – E – F – G. As expected, the Minor scale
has the same feeling as the Minor triad: unsettling and slightly “off”.
http://dictionary.onmusic.org/appendix/topics/minor-scales
Natural A Minor scale
Other types of Minor scales interlock different combinations of Major and Minor
triads. The Harmonic Minor scale, for example, replaces the upper Minor triad in a
Natural Minor scale with a Major triad. Thus, a Harmonic A Minor scale interlocks an A
Minor triad with a D Minor triad and an E Major triad (E – G# – B). The Melodic Minor
scale, on the other hand, interlocks a Minor triad with two Major triads on either end. (2,
6)
Using Keys As Tools
A key, as previously described, indicates the scale to which the notes of a given
piece of music are generally restricted. A song in the key of Eb Major, for instance,
predominantly consists of the notes in the Eb Major scale. Since scales have
characteristic sounds, some uplifting and comfortable, others unsettling or sad, keys
can be used to give pieces of music powerful tones and emotions. This is a fundamental
aspect of music and can be observed in almost all genres.
Examples of the impact keys can have are everywhere in music. We can see this
by comparing songs in the key of A Major with songs in the key of A Minor. Two popular
songs in the key of A Major are “Pompeii” by Bastille and “Here Comes the Sun” by The
Beatles. These pop songs are uplifting and easy to listen to, easily reflecting the
qualities of a Major key.
https://www.youtube.com/watch?v=F90Cw4l-8NY
“Pompeii” by Bastille
Compare this with songs in the key of A Minor, such as “Californication” by Red
Hot Chili Peppers and “Stairway to Heaven” by Led Zeppelin. These songs aren’t at all
positive or uplifting like the songs in A Major.
https://www.youtube.com/watch?v=j0ZygrPsCpY
“Californication” by Red Hot Chili Peppers
https://www.youtube.com/watch?v=oW_7XBrDBAA&spfreload=10
“Stairway to Heaven” by Led Zeppelin
Conclusion
Music is an incredibly powerful force. It invokes a wide range of emotions, brings
people together, and is a form of communication. Musicians throughout history have
used music for all of these purposes, and the most successful ones have done so by
taking advantage of the many mathematical and psychological relationships among the
sound wave frequencies that characterize musical notes. With so many different chords,
scales, and keys and so many different emotions and tones to explore, people will
surely continue to innovate with music for many years.
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