The Overtone Spectrum

Alexis Baskind
The Overtone Spectrum
Alexis Baskind, https://alexisbaskind.net

Alexis Baskind
Course series
Fundamentals of acoustics for sound engineers and music producers
Level
undergraduate (Bachelor)
Language
English
Revision
January 2020
To cite this course
Alexis Baskind, The Overtone Spectrum, course material, license: Creative Commons
BY-NC-SA.
Full interactive version of this course with sound and video material, as well as more
courses and material on https://alexisbaskind.net/teaching.
Except where otherwise noted, content of this course
material is licensed under a Creative Commons Attribution-
NonCommercial-ShareAlike 4.0 International License.

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Outline
1. What is the overtone spectrum?
2. Physical generation of overtones
3. Shaping of the overtone spectrum in the
instrument
4. Designing the overtone spectrum by playing
5. Conclusion

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What is a spectrum?
=> Any sound is usually represented by two means:
a waveform…
(time representation)
…or a instantaneous spectrum
(frequency representation)
frequency
(log scale)
level
time

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What is a spectrum?
=> Any sound is usually represented by two means:
frequency
(log scale)
level
time
The spectrum ist the decomposition of an excerpt of the
sound according to its frequency content

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What are Overtones?
• According to (not verifiable) legends, Pythagoras (6th
century BC) discovered that all harmonic sounds are
composed of several tones that share simple
frequency ratios
• This observation probably lead to the so-called
harmonic series in music
• Joseph Fourier (1768-1830) gave a modern
mathematical background to this discover
• The corresponding theory is called Fourier analysis
• It’s been used not only for harmonic but also for
inharmonic sounds

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What are Overtones?
original waveform
(square wave)
1st harmonic
3rd harmonic
5th harmonic
7th harmonic
9th harmonic
reconstructed
signal until order
N = sum of the
first N
harmonics
The Fourier decomposition in the time domain

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What are overtones ?
Overtones are all frequency peaks in a spectrum
except its fundamental frequency
frequency (Hz)
(linear scale)
level (dB)
Fundamental frequency Overtones

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Frequency structure of overtones
Fundamental frequency
(=> most of time, provides
the pitch of harmonics sounds)
Overtones
=> also called « harmonics »
for harmonic sounds
F 2F 3F 4F …
Harmonic sounds: overtone frequencies are
multiples of the fundamental frequency
frequency (Hz)
(linear scale)
level (dB)
The frequency difference between neighboring
overtones is always the same
Partials = Fundamental Frequency + Overtones

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Beware of the representation: harmonics don‘t look
equally spaced on a logarithmic frequency scale,
although they actually are
Frequency (Hz)
(linear Scale)
level (dB)
Frequency (Hz)
(log Scale)
level (dB) 
F 2F 3F 4F 5F 6F 7F 8F
F 2F 3F 4F ...5F

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Example: picked
double bass, G2
frequency
(log scale)
level
10 Hz 100 Hz 1 kHz 10 kHz
0 dB
-60 dB
-120 dB

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98 Hz 196 Hz 294 Hz 392 Hz 490 Hz …
Fundamental
frequency = 98 Hz
frequency
(log scale)
level
0 dB
-60 dB
-120 dB
Example: picked
double bass, G2

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Inharmonic sounds: overtone frequencies are
not multiples of the fundamental frequency
Fundamental frequency
Overtones
=> the frequency
structure is not regular
frequency (Hz)
(linear scale)
level (dB)
Partials = Fundamental Frequency + Overtones

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Example: crash
cymbal hit with a
stick
frequency
(log scale)10 Hz 100 Hz 1 kHz 10 kHz
0 dB
-60 dB
-120 dB
level

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Example: crash
cymbal hit with a
stick
105 Hz 120 Hz 154 Hz 210 Hz 276 Hz …
frequency
(log scale)10 Hz 100 Hz 1 kHz 10 kHz
0 dB
-60 dB
-120 dB
level

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Why is the overtone spectrum
important for music?
The frequencies and amplitudes over the overtones
determine largely the tone color
=> The overtone spectrum is an important attribute
of a sound, and consequently of an instrument, of a
playing technique...

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Physical generation of overtones
Classical model of a music instrument:
String instruments,
Piano: strings
Brass: lips
Woodwinds: reed,
airflow (flute)
Drums: skin (drumhead)
…
String instruments:
body and neck
Piano: body, resonance
board,
Winds: pipe, horn
Drums: body
…
The Radiation
pattern depends
among others on
frequency and on
the note being
played
Tone
generator
Resonator Radiation

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Basic vibration pattern for a bowed string
Example:
oscillation of
bowed string for a
violin, shown in
slow motion
Source: ViolinB0W, Creative
Commons Attribution 3.0 license

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• The vibration of the string can be
decomposed in simpler vibrations, each of
them at a single frequency (see 1. part)
• Each sinusoidal vibration is called Mode of
vibration (or simply „Mode“)
frequency
(linear scale)
level
F 2F 3F 4F
• The fundamental frequency “F”
depends on the length, the diameter
and the tension of the string
Mode 1
Mode 2
Mode 3
Mode 4
• Each mode corresponds to
the fundamental or to one
overtone
=
+
+
+
+ ...
Position
Bow direction

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• The time evolution of
the oscillation of the
string, measured at
one point, is also
harmonic regarding
the time axis
• Therefore, the
waveform on the time
axis can be
decomposed in
fundamental
frequency and
overtones
Position
Measurement point
time
Animation generated with Matlab-Code from the University of Wyoming

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• In wind instruments,
overtones originate from
the coupled oscillation of
lips (for brass) / reeds
(woodwinds) and of the
tube
• The oscillation frequency
of the lips/reed is
determined by the
oscillation of the air flow
in the tube and depends
on its length
Wind Instruments
(Source: IWK - Music acoustic Vienna / Matthias Bertsch)

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Example:
vibration of a
cymbal hit with a
stick, shown in
slow motion
Inharmonic Instruments

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• Inharmonic oscillations can
also be decomposed in
oscillation modes
frequency
level
Inharmonic Instruments
• Like for harmonic
oscillations, each mode
corresponds to a partial (=
fundamental frequency or
overtone)
• The frequencies of the
overtones are not on an
harmonic series (i.e. they
are not multiple of the
fundamental frequency)
Animations
from Olex
Alexandrov

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Shaping of the Overtone Spectrum
String instruments,
Piano: strings
Brass: lips
Woodwinds: tongue,
airflow (flute)
Drums: skin (drumhead)
…
String instruments:
body and neck
Piano: body, resonance
board,
Winds: pipe, horn
Drums: body
…
The radiation
pattern depends
among others on
frequency and on
the note being
played
Tone
generator
Resonator Radiation
• The overtones created in the generator are shaped in the
resonator (=instrument body
• The (body and air) vibrations in the resonator are also
made of modes

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The whole body also has vibration modes
example: some measured vibration modes (exaggerated) for the body of a guitar
Mode 1: “bending” mode (≈60 Hz)
Source: Dan Russell
(the red plate represents the motion of air in the soundhole)

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Mode 2: “breathing” mode (≈100 Hz), synchronized with the Helmholtz resonance (see later)
Source: Dan Russell
(the red plate represents the motion of air in the soundhole)

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Mode 3 (≈190Hz)
Mode 4 (≈200Hz)
Mode 5 (≈220Hz)
Mode 6 (≈230Hz)
Mode 7 (≈260Hz)
Mode 8 (≈315Hz)
Mode 9 (≈380Hz)
Mode 10 (≈480Hz)
Mode 11 (≈750Hz)
Source: Dan Russell

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• The geometry of the body is carefully designed to fine-
tune the modes in amplitude and frequency
• For instances, guitar top plates (as well as soundboards
on piano) have braces on the back
Source: Neville H. Fletcher, Thomas D. Rossing, The Physics of Musical Instruments

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• Those vibrations entail resonances (= formants) in the
overtone spectrum which are characteristic of the instrument
• The formants are independent of the pitch
example: some measured vibration modes (exaggerated) of the top and back of a violin
457 Hz 545 Hz 723 Hz 850 Hz
Frequenz
(linear)
Pegel
Animations by Terry Borman

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The Human Voice
In human voice,
vowels are
determined by a
precise controle of
the resonances in the
mouth, the nasal
cavity and the throat
Source: J. Meyer, Akustics and the Performance of Music
Frequency

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The Human Voice
The vowel („i“)
remains unchanged,
only the fundamental
frequency changes
Formants are independent of the fundamental frequency!
(this is true for all instruments, not only for the voice)
Formants

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The Human Voice
Example: Overtone singing
0 dB
-60 dB
-120 dB frequency
(log scale)
level
Overtone singing implies a very precise control of the
formants, i.e. the resonances of the mouth and the throat,
independently from the vocal folds
=> The resonance is
so strong in the
1 kHz-2,5 kHz zone
that a second pitch is
perceived above the
fundamental

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• Depending on the playing technique, Dynamics and
radiation pattern, the overtones can be amplified or
in contrary silenced
• This influences two main features of the sound
significantly:
1. The Pitch
2. The Tone Color
Designing the overtone spectrum by playing

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Selecting overtones
(the slide of the
trombone does not
move, only the lips
are more and more
tightened and the
blowing pressure
higher and higher)
0 dB
-60 dB
-120 dB frequency
(log scale)
level
Example 1: natural harmonics glissando on a bass
trombone (fundamental: 59 Hz = Bb1)
=> The frequency of the overtones remains constant, only the
amplitude is changing (in red: overtone frequencies until 1,7kHz)

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Selecting overtones
Example 2: natural string harmonics (“flageolets”) on
a double bass based on open string D2
0 dB
-60 dB
-120 dB
level
=> The frequency of the overtones remains constant, only the
amplitude is changing (in red: overtone frequencies until 2,2kHz)
The string is gently
touched but not pressed
on the fingerboard
Frequency (log)

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Overtones depend on dynamics
In most instruments, the tone color changes
drastically with dynamics
Example 1:
crescendo with
bass trombone
The volume of low-frequency overtones rises only slightly, while
more and more high-frequency overtones appear during the
crescendo, thus making the sound brighter and louder (= “brassy”)
0 dB
-60 dB
-120 dB frequency
(log scale)
level

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Overtones depend on dynamics
Example 2:
crescendo roll on
cymbal with
wool mallets
In most instruments, the tone color changes
drastically with dynamics
0 dB
-60 dB
-120 dB frequency
(log scale)
level
The volume of low-frequency overtones rises only slightly, while
more and more high-frequency overtones appear during the
crescendo, thus making the sound brighter and louder

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Release: Time evolution of overtones
In the resonance, high-frequency overtones decay
faster than low-frequency overtones in most of
acoustic instruments
Example 1:
picked double
bass, G2
0 dB
-60 dB
-120 dB frequency
(log scale)
level

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Release: Time evolution of overtones
In the resonance, high-frequency overtones
decay faster than low-frequency overtones
Example 2:
crash cymbal hit
with a stick
0 dB
-60 dB
-120 dB frequency
(log scale)
level

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< 450 Hz (omnidirectional)
• Low frequencies
radiate in all
directions 650 Hz
Radiation pattern of a trombone
Example: angular width of the regions of main radiation (between -3 dB
and 0 dB relative to the maximum level in the band) for a trombone for
various frequency zones
500 Hz
1 kHz
Between 2 and 5 kHz
Between 7 and 10 kHz
(highly directional)
• Except for the
650 Hz zone (the
first formant),
directivity increases
with frequency

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Radiation pattern of a trumpet
The trumpet has a similar
radiation pattern than the
trombone, but shifted up
in frequency
The limit for
omnidirectional radiation
is around 500 Hz
(source: J. Meyer)

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Radiation pattern: trumpet
0 dB
-60 dB
-120 dB frequency
(log scale)
level
Example: trumpet miked with a TLM-103:
1 – on-axis

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Radiation pattern: trumpet
Example: trumpet miked with a TLM-103:
2 – 90° off-axis
0 dB
-60 dB
-120 dB frequency
(log scale)
level

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Conclusion
• The overtone spectrum is an essential component of the
specific tone color of an instrument
• It is related to fundamental aspects of the perception of a
sound, like:
- Pitch
- Harmonicity
- Dynamics
• Its characteristics depends dramatically on the type and
construction of the instrument, as on the playing technique
• Those properties are used by the perception in order to
recognize the instrument and the playing technique

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But...
• The sound of an instrument does not only consist in its overtone
spectrum!
• The perception of the timbre is not limited to the tone color. It‘s
much more complex and involve more features, that are equally
important:
– The Noise component, i.e. the part of the spectrum, which
cannot be modelled with sine waves
– The time evolution of frequencies: for instance Vibrato
– The time evolution of the amplitudes in the spectrum: for
instance Tremolo, Attack, Release...
=> The overtone spectrum is a very important, but not unique
component of the sound of a music instrument

Alexis Baskind
To go further...
• Jürgen Meyer, Acoustics and the Performance of Music,
Springer-Verlag New York (2009)
• Neville H. Fletcher, Thomas D. Rossing, The Physics of
Musical Instruments, Springer-Verlag

The Overtone Spectrum

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