harmonics, timbre & the frequency domain. real instruments real instruments do not normally...
TRANSCRIPT
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Harmonics, Timbre & The Frequency Domain
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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
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Harmonics
• Frequencies can be harmonically related • These are called harmonics• They are related in whole number multiples
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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
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Harmonics
• The harmonics will be multiples of the fundamental frequency
• The fundamental is the largest common divisor of the harmonics
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2nd Partial
-4
-2
0
2
4
Composite Waveform
-4
-2
0
2
4
Fundamental
-4
-2
0
2
4
3rd Partial
-4
-2
0
2
4
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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
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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
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f0 = 1/2f1 = 1/3f2
time
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5pressure
t1
t2
t0
f0 f2
f1
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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
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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
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Harmonic Series
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Harmonic Series
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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
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Frequency Domain Representation
1
- 1
0
dB
seconds
1
- 1
0
dB
frequency100 200 300 400 500 600
A 440Hz sine wave shown in the time domain (above) and the frequency domain (below).
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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
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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
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amplitude
Trumpet
frequency
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frequency
amplitude
Clarinet
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Trumpet
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Timbre
‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’
Oxford English Dictionary
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‘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)
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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
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2nd Partial
-4
-2
0
2
4
Composite Waveform
-4
-2
0
2
4
Fundamental
-4
-2
0
2
4
3rd Partial
-4
-2
0
2
4
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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
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Timbre
‘Characteristic quality of sounds produced by each particular voice or instrument, depending on the number and character of overtones.’
Oxford English Dictionary
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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)
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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)
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amplitude
Trumpet
frequency
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frequency
amplitude
Clarinet
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2nd Partial
-4
-2
0
2
4
Composite Waveform
-4
-2
0
2
4
Fundamental
-4
-2
0
2
4
3rd Partial
-4
-2
0
2
4
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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
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Trumpet
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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
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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
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Triangle Wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Am
plitu
de
frequency
1 31 7151 11191 131
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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
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Sound Sources with Formants
• Like the human voice (e.g. saying ah) will change shape depending on their frequency
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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
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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
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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
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