fundamentals of acoustics. the nature of a sound event n sound consists of vibrations of air...
TRANSCRIPT
Fundamentals of Acoustics
The Nature of a Sound Event
Sound consists of vibrations of air molecules Air molecules are analogous to tiny superballs Sound occurs when air molecules are disturbed and made to ricochet off of
each other
The Nature of a Sound Event
The ricochets cause the density of the air molecules to oscillate
Normal CompressedRarefied
The Nature of a Sound Event
The ricochets cause the density of the air molecules to oscillate back and forth
ª›•Œ QuickTime˛ ©M°ßGIF°®∏—¿£¡Yµ{¶°
®”¿Àµ¯≥oπœµe°C
Wave TypesSound consists of longitudinal waves
The wave’s oscillation is in the same direction as its propagation
propagation
oscillation
Water waves are transverse waves
The wave’s oscillation is perpendicular to the direction of its propagation
propagation
oscillation
ª›•Œ QuickTime˛ ©M°ßGIF°®∏—¿£¡Yµ{¶°
®”¿Àµ¯≥oπœµe°C
ª›•Œ QuickTime˛ ©M°ßGIF°®∏—¿£¡Yµ{¶°
®”¿Àµ¯≥oπœµe°C
Sound PropagationSound waves propagate in a sphere from the sound source (try to imagine a spherical slinky).
Note that the molecules themselves are not travelling. What spreads is the energy of the wave.
Sound Perception
When sound waves reach the eardrum, they are transduced into mechanical energy in the middle ear
The mechanical motion is transduced into electrical current in the inner ear. The auditory nerves interpret the current as sound
Speed of sound (in air):
1128 ft./sec (344 m/sec)
Sound Wave Plots
Sound waves are typically represented with molecular density as a function of time
molecular density
compressed
normal
rarefied
time
Music vs. NoiseMusical sounds are typically periodic – the wave repeats regularly
Noise is aperiodic – there is no repeating pattern
Sine wave
Though they don’t exist in nature, sine waves are often useful for demonstrating properties of sounds
Noise
repeats
Properties of a Musical Event
A musical event can be described by four properties.
Each can be described subjectively, or objectively (in terms of measured properties)
Subjective Objective
Pitch Frequency
Volume Amplitude/Power/Intensity
Timbre Overtone content
Duration in beats Duration in time
Frequency/PitchFrequency is measured in cycles per second, or Hertz (Hz)
one secondf = 2 Hz
Wavelength (), the distance between corresponding points on the wave, is the inverse of frequency.
= c f
= 1000 ft./sec.
2 cyc./sec.= 500 ft./cyc.
Frequency/PitchMiddle A = 440 Hz
= 2.3 ft.
frequencies audible to humans
<20 Hz < 20,000 Hz (20 kHz)
= 50 ft. = 0.05 ft.
Sound wavelengths are significantly larger than light wavelengths
Waves reflect from a surface if its height/width is larger than the wavelength
Waves refract around surface if the surface dimensions are smaller than the wavelength
This explains why we can hear sound from around corners,but cannot see around corners:
Light wavelengths are far too small to refract around anyvisible surface
Our Pitch Perception is LogarithmicEquivalent pitch intervals are perceived according to an equivalent change in exponent, not in absolute frequency
For example, we hear an equivalent pitch class with every doubling of frequency (the interval of an octave)
55 x 20 55 x 21 55 x 22 55 x 23 55 x 24 55 x 25 55 x 26
Frequencies of successive octaves of concert A
55 110 220 440 880 1760 3520
Our Pitch System is Based on Equal Division of the Octave
12 Tone Equal Temperament – the octave is divided into twelve equal increments
We can describe an octave by:
n/12 for n = 0 to 11• multiply it by 2
• choosing a starting frequency
A
220
x 22200
12
A#
233
x 22201
12
B
247
x 22202
12
C
261.6
x 22203
12
C#
277
x 2220412
D
293.6
x 22205
12
D#
311
x 22206
12
E
329.6
x 22207
12
F
349.2
x 2220812
F#
370
x 22209
12
G
392
x 22201012
G#
415.3
x 22201112
Higher octaves may be created by doubling each frequency
Lower octaves may be created by halving each frequency
PhasePhase = “the position of a wave at a certain time”
If two waveforms at the same frequency do not have simultaneous zero-crossings, we say they are “out of phase”
Two waves at the same frequency but different phase
Wave 1
Wave 2
Wave 1 + Wave 2
In terms of sound perception, phase can be critical or imperceptible,as we’ll see...
LoudnessLoudness is related to three measurements:
• Pressure
• Power
• Intensity
All three are related to changes in sound pressure level (molecular density)
Molecular Motion is Stationary
As sound travels, molecules are not traveling with the sound wave
What is traveling is an expanding sphere of energy that displaces molecules as it passes over them
How strong is the force behind this energy wave? The more force is contained in a sound wave, the
greater its perceived loudness.
PowerPower = the amount of time it takes to do work (exert force, move something)
Power is measured in watts, W
The range of human hearing encompasses many millions of watts.
Sound power level is also relative, not absolute. Air molecules are never completely motionless.
Given these two difficulties, sound power levels are measured on a scale that is comparative and logarithmic, the decibel scale.
There are two difficulties in measuring sound power levels.
Logarithmic ScaleLogarithm = exponent(an exponent is typically an integer, a logarithm not necessarily)
102 = 100 log10100 = 2
103 = 1000 log101000 = 3
102.698 = 500 log10500 = 2.698
102.875 = 750 log10750 = 2.875
Logarithms allow us to use a small range of numbers to describe a large range of numbers
The Decibel Scale
The decibel scale is a comparison of a sound’s power level with a threshold level (the lowest audible power level of a sine tone at 1 kHz).
W = 10 watts0-12
L (dB) = 10*log (W/W )010W
Threshold (W0):
Power level of a given sound in watts, LW(dB):
DecibelsTypical power levels:
Soft rustling leaves 10 dBNormal conversation 60 dBConstruction site 110 dBThreshold of pain 125 dB
Halving or doubling sound power level results in a change of 3 dB.
For example, a doubling of the threshold level may be calculated:
10log104×10−12
2×10−12
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ =10log10 2⎛
⎝ ⎜
⎞
⎠ ⎟ =3.01 dBLW(dB) =
Thus, a power level of 13 dB is twice that of 10 dB. A power level of 60 dB is half that of 63 dB, and so on.
Pressure changes
Maximum change in sound pressure level
The amplitude level fluctuates with the wave’s oscillation.Thus, power is the cause, pressure change is the result
(more generally: in a vibrating system, the maximum displacement from equilibrium position)
The degree of fluctuation present in a vibrating object
Peak pressure level:
Pressure changes
Pressure level is measured in Newtons per square meter (N/m ) 2
Threshold: 2 x 10 N/m (p )-5 2
0
Also may be described as changes in sound pressure level (molecular density).
For any propagating wave (mechanical, electric, acoustic, etc.) the energy contained in the wave is proportional to the square of its pressure change.
Pressure changes are also expressed in decibels, but in a way that describes an equivalent change in power level:
L (dB) = 10*log10(W/W0)W = 10*log10(p/p0)2 = 20*log10(p/p0)This is how pressure is measured
logmn = nlogm
There is a direct relationship between pressure and power levels:
Pressure changesIn audio parlance, “amplitude” (the degree of pressure change) is often equated with “loudness.”
The reason is that modifications to volume are made by adjusting the amplitude of electrical current sent to an amplifier.
But perceived loudness is actually based on power level plus the distance of the listener from the source.
Power combined with distance is intensity, I, measured in watts per square meter (W/m ).2
IntensityPower corresponds to the sphere of energy expanding outward from the sound source
The power remains constant, spread evenly over the surface of the sphere
Perceived loudness depends primarily on the sound power level and the distance from the sound event
Intensity is also measured in decibels:
L (dB) = 10*log (I/I )010I I = 10 W/m0-12 2
TimbreThe perceived difference in sound quality when two different instruments play at the same pitch and loudness
Sine waves are useful as demonstrations because they are a wave with one frequency only, thus they are often termed pure tones
Natural sounds are composed of multiple frequencies
To understand how a wave can be composed of multiple frequencies, we can consider the behavior of a wave in a bounded medium, such as a string secured at both ends (or air vibrating within a pipe)
TimbreWhen we pluck a string, we initiate wave motion
The wavelength is twice the length of the string
The perceived pitch is the fundamental, the speed of sound divided by the wavelength
Timbre
This curved shape represents the string’s maximum deviation
It’s more accurate to think of it as a series of suspended masses (kind of like popcorn strung together to hang on a Christmas tree).
TimbreEach suspended mass can vibrate independently.
Thus, many simultaneous vibrations/frequencies occur along a string.
When a string is first plucked, it produces a potentially infinite number of frequencies.
TimbreEventually, the bounded nature of the string confines wave propagation and the frequencies it can support
Only frequencies that remain in phase after one propagation back and forth can be maintained; all other frequencies are cancelled out
Only frequencies based on integer subdivisions of the string’s length, corresponding to integer multiples of the fundamental, can continue to propagate
Timbre
…etc.
NOTE:These frequencies are equally spaced
Therefore, they do not all produce the same pitch as the fundamental
Therefore, other frequencies are introduced
These frequencies are called harmonics
Timbre
Harmonics are well known to many instrumentalists– Strings– Brass
Timbre
The first six harmonics are often the strongest:
220
Fundamental
440
Octave
660
Perfect fifth
880
Octave
1100
Major third
1320
Perfect fifth
People can learn to “hear out” harmonics
Timbre
Instruments and natural sounds usually contain many frequencies above the fundamental
These additional frequencies, as part of the total sound, are termed partials
The first partial is the fundamental
Timbre
The first partial is the fundamental Other terms are also used Overtones are partials above the
fundamental (the first overtone is the second partial)
Harmonics are partials that are integer multiples of the fundamental
The Spectrum
Jean Baptiste Fourier (1768-1830) discovered a fundamental tenet of wave theory
All periodic waves are composed of a series of sinusoidal waves
These waves are harmonics of the fundamental Each harmonic has its own amplitude and phase The decomposition of a complex wave into its
harmonic components, its spectrum, is known as a Fourier analysis
The SpectrumIt is often more useful to represent complex waveforms with a spectral plot as opposed to a time domain plot
=
time domainamplitude as a function of time
spectral domainamplitude as a function of frequency
Sound in Time
Our perception of sound and music events is determined by the behavior of frequency and loudness over time
Sound in Time
All instruments can be characterized by changes in amplitude over time (the envelope)
time
loudness
trumpet bowed violin harp
Changes in amplitude often correspond with changes in frequency content...
Sound in Time
Most instrument’s sound begins with an initial transient, or attack, portion
The transient is characterized by many high frequencies and noise
Example: the scraping of a bow or the chiff of breath
An instrument’s distinctiveness is determined primarily by the transient portion of its sound
Sound in Time
Following the transient, instruments usually produce a steady-state, or sustained, sound
The steady state is characterized by– Periodicity– Harmonic spectrum
The SpectrogramMost natural sounds (and musical instruments) do not have a stable spectrum.
Rather, their frequency content changes with time.
The spectrogram is a three-dimensional plot:
Vibraphone note at 293 Hz (middle D)
1) time
2) frequency
3) power of a given frequency (darkness level)
The instrument’s sound is characterized by the fundamental at 293 Hz and the fourth harmonic at 1173 Hz.The attack also contains noise below 2 kHz, the tenth harmonic at 2933 Hz and the seventeenth harmonic at 4986 Hz.Once the steady state portion sets in, the highest harmonic fades first, followed by a fading of the fundamental.
Localization
The auditory system localizes events through interaural time delay – the sound wave reaches the nearer ear a few milliseconds before it reaches the farther ear
For stereo systems, using delay for localization is impractical because it requires people to listen from a “sweet spot”
Localization effects are simulated through differences in loudness
Localization
In a multi-speaker system, a sound emanating from one speaker will be localized at that speaker
A sound produced at equal volume from two speakers will be perceived as a “phantom image” placed in space between them
Changing the volume balance between two speakers will cause the phantom image to “drift” towards the louder speaker
Measurement and Perception
Our perception of auditory events is based on all these measurements in combination
And more An auditory event may be more than the
sum of its parts
Measurement and Perception
Changing the phase of components in a steady-state tone produces no perceptible change in sound, although the shape of the wave may change noticeably
Phase
Measurement and Perception
The behavior of components in the attack segment is likely to be far more complex than in the steady state segment
Changing the phase of attack components can change the character of the attack
Solo performance sounds different from group performance because no two players can ever sound at exactly the same time; thus the attack is blurred
Since an instrument’s characteristics are defined primarily by the attack, the phase of attack components is critical
Phase
Measurement and Perception
We have discussed timbre as the result of overtone content It is also judged by the sound’s envelope Research in sound synthesis has shown the envelope shape to
be more definitive than an exact match of overtone content The attack portion is critical—a faster attack can be confused
with “brightness” (more high frequency overtones) Considerable research has gone into the creation of “timbre
space,” a multi-dimensional plot in which timbres are classified according to overtone content, envelope and attack time
Timbre
Measurement and Perception
LoudnessWhile intensity is the measurement most closely correlated to loudness, the perception of volume is based on a number of factors, not all of them entirely measurable.
Measurement and perception
Perceived loudness is frequency-dependent
0
20
40
60
80
110
120 2 x 10
2
2 x 10-1
2 x 10-2
2 x 10-3
2 x 10-4
2 x 10-5
Sound pressure level
(dB)
20 100 500 1,000 5,000 10,000
Frequency (Hz)
120
110
100
90 fff
80 ff
70 f
60 mf
50 p
40 pp
30 ppp
20
10
0
Limit of pain
Loudness level (phons)
Newtons/m2
Threshold of hearing
Equal loudness curves (Fletcher, Munson, 1930s).
Perceived equal loudness of sine tones
This is why many receivers have a Loudness knob
Loudness
Measurement and perception
Perceived loudness is frequency-dependentLoudness
Within close frequency ranges, perceived loudness is proportional to the cube root of intensity
Two violins playing the same pitch will generate twice theintensity of one violin, but will not sound twice as loud
To achieve twice the volume, eight violins are required
Measurement and perception
Perceived loudness is bandwidth-dependentLoudness
Increasing the bandwidth (component frequency content) of a sound makes it sound louder, even if the intensity remains constant
Despite many efforts, no one has suceeded in creating a definitive perceptual scaling system for loudness
Measurement and PerceptionLoudness
Some have argued that estimation of loudness is not automatic (measurable), but depends on a number of higher-level estimations of distance, import, context, etc.
…we are exceedingly well trained in finding out by our sensations the objective nature of the objects around us, but we are completely unskilled in observing these sensations per se; and the practice of associating them with things outside of us actually prevents us from being
distinctly conscious of the pure sensations.
Hermann Helmholtz, On the Sensations of Tone (1885):
Measurement and PerceptionConclusion
Objective measurements can tell us more about sound events
By the same token, they give us insight into what we don’t know
This course will examine music in technical terms
This examination will give us some new insights
It will also give us an idea of where music crosses the barrier from the objective (acoustics) to the subjective (magic?)