Harmonics, Timbre & The Frequency Domain

Real Instruments

• Real instruments do not normally produce pure tones

• Instead the sound produced by hitting a single note has:

• a fundamental frequency• some extra frequencies

Harmonics

• Frequencies can be harmonically related • These are called harmonics• They are related in whole number multiples

Harmonics

• If extra frequencies are harmonics the sound produced will be perceived as a single pitch at the fundamental frequency

• One fundamental frequency and several harmonics

Harmonics

• The harmonics will be multiples of the fundamental frequency

• The fundamental is the largest common divisor of the harmonics

2nd Partial

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Composite Waveform

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Fundamental

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3rd Partial

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Harmonics

• The composite wave has the same frequency as the fundamental

• The fundamental is frequency at which the entire waveform vibrates

• The brain perceives the composite waveform as a sound that has the same pitch as the fundamental

Fundamental Tracking

• The ability of human brain to track the fundamental frequency of a sound

• Occurs even when the fundamental waveform is not present

• This is because the wave will repeat at the fundamental frequency and the brain detects this

f0 = 1/2f1 = 1/3f2

time

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f1 = 2 and f2 are integer multiples of f0

• t0 = 2t1 = 3t2

• substitute in t = 1/f to get:

• f0 = 1/2f1 = 1/3f2

• So: f1 = 2f0 and f2 = 3f0

For Example

• Say: f1 = 440Hz

• 1/2f1 = 1/3f2 : f1 = 2/3f2 : 3f1 = 2f2 : f2 = 3/2f1

• So: f2 = 3/2 * 440 = 660Hz

• f0 = 1/2f1 = 220Hz

• Which is the highest common divisor of 440 and 660

Harmonic Series

Harmonic Series

Frequency Domain Representation

• Used to sound waves being represented in terms of time and amplitude

• Known as Time Domain Representation• Frequency Domain Rep shows frequency and

amplitude

Frequency Domain Representation

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A 440Hz sine wave shown in the time domain (above) and the frequency domain (below).

Frequency Domain Plots

• Both real instruments and synthesisers normally produce more complex waves

• This means additional frequency components or overtones

• All of these frequencies (including the fundamental) are called partials

Frequency Domain Plots

• The frequency domain content of a wave is represented by plotting each partial on the x-axis

• The height of each line indicates the strength of each frequency component

amplitude

Trumpet

frequency

frequency

amplitude

Clarinet

Trumpet

Timbre

‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’

Oxford English Dictionary

‘A common timbre groups tones played by an instrument at different pitches, loudnesses, and durations. No matter what note it plays, for example, we can always tell when a piano is playing. Human perception separates each instrument’s tones from other instrument tones played with the same pitch, loudness, and duration. No one has much trouble separating a marimba from a violin tone of the same pitch, loudness and duration. Of course a single instrument may also emit many timbres, as in the range of sonorities obtained from saxophones blowed at different intensities.’

Roads (1996, p 544)

Harmonics Again

• Remember: a harmonic is a sound that is an integer multiple of the fundamental frequency

• So: while the fundamental carries out one cycle (or period), a harmonic of this will carry out an exact number of whole cycles

2nd Partial

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Harmonics Again Still

• So a fundamental plus several harmonics produces a composite waveform that is?

• Periodic: it repeats itself exactly• If the added waveforms were not harmonics

the waveform would not be periodic

Timbre

‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’

Oxford English Dictionary

Harmonics and Musical Instruments

• Most instruments produce overtones• Generally the overtones are nearly

harmonic but not quite• Because they are not exact harmonics the

sound wave produced is not periodic (but quasi-periodic)

Harmonics and Musical Instruments

• The trumpet has strong harmonics (brash, full and brassy)

• The clarinet has weaker ones (pure, smooth flute like)

• In-harmonic frequencies are non-periodic(metallic and percussive)

amplitude

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Clarinet

2nd Partial

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Amplitude Variations

• Timbre is also strongly affected by variations in amplitude over time

• This is why amplitude envelopes are so commonly used in electronic sound synthesis

Trumpet

Formants

• A formant is a peak of energy in an absolute frequency region

• Responsible for the timbre of the human voice and many real instruments

• Formant peaks stay the same when pitch of sound is changed

Wave Shapes

• Wave shapes like triangle waves, and sawtooth waves don’t have formants

• This is because they have a shape that always remains the same

• They have identical relationships among their frequency components no matter what pitch they are

Triangle Wave

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Triangle Wave

• Sound produced will be mostly between 1st and 7th harmonics

• So one of 70Hz will have it’s most prominent frequencies 70 - 490Hz

• And one of 500Hz will have it’s most prominent frequencies 500 - 3500Hz

Sound Sources with Formants

• Like the human voice (e.g. saying ah) will change shape depending on their frequency

Humans Saying “Ahh”

• On average the following frequencies are emphasised when a man says “ahh”: 730Hz, 1090Hz, and 2440Hz

• No matter what the pitch the man says ah at • Female “ahh”s are on average: 850Hz, 1220Hz

and 2810Hz

Formants Are Caused by Physical Characteristics

• The formants of a particular instrument or voice are determined by resonance chambers

• Human voice formants vary according to nasal, oral and pharyngeal cavities

Formants Are Caused by Physical Characteristics

• A guitar has formants based on shape character and dimensions of resonance chamber

• Formants are what make a human voice recognisable or give an instrument a particular sound