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Psycho acoustics. The Biology and Psychology of hearing by
Charles Feilding.
Introduction.
Whether you seek a career as a creator of music, producer
of musical content or just seek to enrich your life with the
pleasure of playing you will inevitably be seeking to develop
consistent and predictable results. Live performers practice
in order to be able to play a piece of music with predictable
results. Composers with home studios invest a lot of money
to create an acoustic environment that will yield them
consistent results. Builders of concert halls invest hundreds
of millions to create halls that will allow live players to
approach a concert with confidence of getting the sound they
have developed.
Paradoxically the one thing that the ear will almost never
provide you with is consistent results. Nature designed the
ear to be a highly flexible device capable of protecting itself
against sudden extremes of volume and constantly adapting
itself to best interpret the constant streams of audio with
which it is bombarded. It is necessary therefore to develop
some understanding of what the ear is doing in order to be
able to guarantee consistent results in the music world. This
study is called psychoacoustics.
Psychoacoustics is not the study of sound. It is the study of
how we perceive sound. It is a complex analysis of both the
biology of the ear (by now quite well understood) and the
psychology of sound perception which is an ongoing study.
1. The biology of hearing
Sound pressure waves are funneled through the acoustic
meatus of the external ear (E) to the tympanic membrane (in
green), and mechanically transduced through the middle ear
(M) into a fluid-filled chamber called vestibulus. The inner
ear (I) organs branch off from this chamber. The inner ear is
the most complex portion of the peripheral auditory system.
It hosts sensory hair cells that signal head accelerations and
sound-induced vibrations to the brain.
The auditory system of mammals is housed in a labyrinthic
foramen deeply embedded in the temporal bone, the hardest
and most complex bone of the skull.
The temporal bone hosts a large central chamber permeated
by physiological fluids, called vestibulus. All of the inner ear
organs branch off from this chamber. On one side is the
snail shaped cochlea, where the acoustic interface of the
nervous system is found, on the other are the semicircular
canals, hosting the sensors of head acceleration, which are
not relevant for the auditory function.
Soon after the discovery of the hearing organ by Alfonso
Corti in 1851, other anatomists, among which Reissner,
Deiters, Lowenberg, Hensen, Hasse, contributed to the
clarification of the cochlear structure. The first relevant
contribution to the understanding of the processes
underlying sound transduction by the ear is due to Hermann
von Helmholtz. The eminent German scientist pointed out
the function of the ossicle chain in the middle ear. In an
exemplar description, von Helmoltz explained that the
mechanical coupling provided by the middle ear optimizes
the transfer of energy from the air that sets the tympanic
membrane into motion, to the fluid filling the cochlea.
QuickTime™ and a
GIF decompressor
are needed to see this picture.
Von Helmholtz also provided the first mechanical model of
the cochlea. Ignoring the hydrodynamical effects of the fluid,
he represented the basilar membrane as an elastic strip
radially clamped across the cochlear duct with different and
graded tension coefficients in the radial directions. Assuming
a negligible tension in the longitudinal direction, this basilar
membrane model is equivalent to a set of harmonic
oscillators tuned to different frequencies. Imagining that each
portion of the basilar membrane senses a force proportional
to the stapedial input, the basilar membrane response to a
tone of given frequency (say 2 kHz) should be similar to that
shown in the movie above, although with a much smaller
amplitude. Accordingly, the cochlea was thought of as a sort
of spectrum analizer providing a frequency-position map of
sound Fourier components.
The idea that the frequency analysis of sound is provided by
such a simple mechanism survived until 1927. Realizing the
importance of hydrodynamic interactions, Georg von Békésy
built a brass and glass frame with two uniformly tapered
chambers, filled with water and separated by a strip having
graded elastic properties. Cochlear travelling waves were
discovered! Von Békésy’s experiments on freshly dissected
human temporal bones confirmed the observations
performed on his physical model.
QuickTime™ and a
GIF decompressor
are needed to see this picture.
The movie here above illustrates, with exaggerated
amplitude, the basilar membrane response to a tone of
about 2 kHz. At variance from the resonator-bank model
imagined by von Helmholtz, the inertial effects of the comoving fluid impart a phase delay of a few cycles to the
basilar membrane oscillation.
Von Békésy’s findings stimulated the production of
numerous cochlear models that reproduced the observed
wave shapes, but were in contrast with psychophysical data
on the frequency selectivity of the cochlea. Beginninnig in
the ‘70s with Rhode’s results and more definitively in the
‘80s, with the improved measurement by Sellick, Patuzzi and
Johnstone (figure at right), it was realized that the vibration
of the basilar membrane was non-linear and more sharply
tuned, at low sound pressure levels, than previously thought.
In 1983, this prompted Davies to state that it was necessary
to accept a revolutionary new hypothesis concerning the
action of the cochlea namely, that an active process within
the organ of Corti increases the vibration of the basilar
membrane.
Between 1985 and 1986 William Brownell, Bechara Kachar
and their collaborators used video microscopy to
demonstrate low-frequency electro motile responses in
isolated outer hair cells. In 1987 Jonathan Ashmore used the
patch-clamp technique to elicit electro motile responses from
outer hair cells (movie below) at kHz frequencies and
measured the cell fractional length change as a function of
trans membrane potential.
QuickTime™ and a
GIF decompressor
are needed to see t his picture.
2. Wave Form Characteristics
Before we begin talking about sound we need to define
some commonly accepted terms for the discussion of sound.
Every waveform has characteristics that distinguish it from
other waveforms. These are:
1. Frequency
2. Amplitude
3. Velocity
4. Wavelength
5. Shape
6. Phase
7. Harmonic content
8. Envelope
Frequency
In the diagram below the value of the waveform starts at 0 at
time t=0, increases to a maximum value in the positive
direction, decreases through zero to a maximum value in the
negative direction then returns to zero at t=2 and then
repeats the process again. One completion of this path is
called one cycle of the wave form. A cycle can begin at any
point on the waveform but to be complete it must pass
through the zero line and end at a point moving in the same
direction (positive or negative) having the same value as the
starting point. Thus the wave from t=0 to t=2 constitutes a
cycle and the wave from t=1 to t=3 is also a cycle. The
number of cycles which occur end to end (that is cycle two
begins at the cycle one end point) in one second is called
the frequency of the waveform and is measured in the unit
hertz (Hz). The term Hertz is equivalent to “cycles per
second”.
Amplitude
In the case of sound waves the positive and negative
excursions of the diagram represent increases and
decreases in the atmospheric pressure of the air caused by
the sound source and perceived by the listener’s ear with the
zero line representing normal atmospheric pressure. The
distance above or below the line is called the amplitude of
the waveform at that particular instant of time. The maximum
positive and negative excursions are called the positive and
negative peaks respectively.
Velocity
The velocity of a wave is the speed with which it travels
through a medium and is given by the equation:
V = d/t2-t1
Where
V is the velocity of propagation in the medium
d is the distance from the source
t is the time in seconds.
In the case of sound waves the medium is commonly air
mollecules, for electricity the medium is commonly electrons.
The wave velocity determines how fast a particular cycle of a
waveform will travel a certain distance. At 70 degrees
Fahrenheit the speed of sound waves in air is approximately
1130 feet per second. This speed is temperature dependent
and increases at a rate of about 1.1 feet per seconds for
each degree of Fahrenheit of temperature.
Wavelength
The wave length (∆) of a wave is the actual distance in the
medium between the beginning and the end of a cycle and is
equal to:
∆ = V/f
where:
∆ is the wavelength in the medium
V = the velocity in the medium
F is the frequency in Hertz
Thus the wavelength of one cycle of a 30Hz sound wave is
1130 ft/sec divided by 30 Hz or 37.66 feet. Although we
speak of wave velocity, the particles of the wave medium do
not move far. Sound travels as a compression wave. The air
in one spot is compressed by the sound source and it
compresses the air next to it as it expands back to normal.
The location of the compression moves at the speed of
sound but the air molecule only move to the extent that they
are pushed together and then returned to their normal
spacing. The molecules do not travel with the wave. The
action of electrical waves in a wire is the same except that
electrons rather than air molecules are compressed.
Shape
The diagrams up to this point have shown only one shape of
wave form, the sine wave, but the eight characteristics apply
to simple waveforms of all shapes.
These are called simple waveforms because they are
continuous and repetitive. One cycle of a square wave looks
exactly like the next and they are all symmetrical round the
zero line. Complex waves are waves that do not necessarily
repeat and are not necessarily symmetrical around the zero
line. An example of complex waveform would be that created
by speaking a word. Since complex waveforms often do not
repeat it is difficult to divide them into cycles or categorize
them as having a frequency. Fortunately all possible simple
and complex waveforms can be constructed through the use
of combinations of sine waves of different frequency, phase
and amplitude plus noise. The discussion here focuses on
sine waves.
Phase
Since a cycle can begin at any point on a waveform it is
possible to have two wave generators producing waveforms
of the same shape, frequency, and peak amplitude which will
have different amplitudes at different point in time. These
wave are said to out of phase with respect to each other. A
cycle can be divided into 360 degrees and the sine wave (so
named because its amplitude follows the trigonometric sine
function) is usually considered to begin at 0 degrees with
zero amplitude, increase to a positive maximum at 90
degrees, decrease to 0 at 180 degrees, decrease to a
negative maximum at 270 degrees and returns to 0 at 360
degrees. In the following diagram the fist wave A can be
considered in phase with the ideal sine curve because their
amplitudes match at every point on the X axis. The second
wave reaches its maximum positive amplitude 90 degrees
before the first and is considered to be 90 degrees out of
phase with the first because it led it by 90 degrees. The third
wave C begins decreasing from zero 180 degrees before the
first and is 180 degrees out of phase. The fourth D leads the
first wave by 270 degrees and is also out of phase.
Phase cancellation.
Waveforms can be added by adding their signed amplitudes
at each instant of time. When two waveforms which are
completely in phase (0 degrees phase difference) and of the
same frequency, shape, and peak amplitude are added the
resulting waveform is of the same frequency phase and
shape but will have twice the original peak amplitude.
If two waves are the same as the ones described except that
they are completely out of phase (phase difference = 180
degrees they will completely cancel each other out when
added resulting in a straight line of zero amplitude. If the
second wave is only partially out of phase (not exactly 180 or
(2n-1)x180 out of phase it would interfere constructively in
some places resulting in a more positive amplitude than in
the first wave and interfere destructively at other points
resulting in a more negative amplitude at those points in time
than in the first wave.
Harmonics
A waveform having a frequency that is an integral multiple of
the frequency of a second waveform is called a harmonic of
that second waveform. For example a 1000 Hz wave is a
harmonic of a 500 Hz wave because it is two time the 500
Hz frequency. In this case the 500 Hz wave is called the
Fundamental (or first Harmonic) and the 1000 Hz wave is
called the Second Harmonic because its is the result of
multiplying the fundamental frequency by two. The third
harmonic would be 1500 Hz.
Envelopes
The envelope of a waveform describes the way its intensity
varies over time. For a sound wave it describes changes in
loudness. The envelope is composed of three or four
sections depending on whether the sound sustains or not.
Attack: The time necessary for a sound to rise from silence
to it’s maximum amplitude.
Decay: The time required for a sound to drop from its
maximum amplitude to its sustaining level (if there is one) or
to silence if there is not one.
Sustain: the level at which a sound sustains (if it does
sustain)
Release: The time required for a sound to drop from its
sustaining level to silence after the sustaining energy is
discontinued.
3. Characteristics of ears
1. Perception of related frequencies
Beats
Combination tones
Masking
2. Perception of dynamic range
3. The hot spot
4. Perception of spectral content
5. Perception of sound pressure levels
6. Perception of direction
7. Threshold of hearing
8. Threshold of feeling
9. Threshold of pain
10. The phon
11. The sone
3.1 Perception of related frequencies
The ear perceives frequencies that are even multiples of
each other to be specially related, and this relationship is the
basis of the musical octave. For example: since concert A is
440 Hz the ear hears 880 Hz as having a special relationship
to concert A. it is the next tone higher than concert A which
sounds most like concert A. The next note above 880 Hz
that sounds most like concert A would be 1760 Hz.
Therefore 880 is said to be one octave above concert A and
1760 Hz is said to be two octaves above concert A. In all
probability the ear is able to recognize octaves due to the
absence of beats when they sound together
Beats
Two tones which differ only slightly in frequency and have
approximately the same amplitude will produce beats
(periodic changes in amplitude) at the ear equal to the
difference between the two frequencies. Beats are the result
of the ear’s inability to separate two closely pitched notes.
For example, if A440 and A442 are sounded together the ear
will hear a pitch which is the average of the two pitches
(440+442)/2 = 441, and the beating will modulate at a rate
equal to the difference between the two frequencies (442440 = 2Hz.
Combination tones.
Another important psycho acoustic phenomenon associate
with related frequencies is the production of combination
tones. Whenever two waves of whose frequencies differe by
more than 50Hz are sounded together two additional
frequencies are generate by our hearing mechanism: one
which is the sum of the two frequencies and one which is the
difference between the two frequencies.
For example:
If a 2,000 Hz tone is sounded with a 2500 Hz tone will
additionally produce
1) A sum tone at 4500 Hz
2) A difference tone at 500
Masking
Masking is a phenomenon by which loud sounds prevent
the ear from hearing soft sound. The greatest masking effect
occurs when the frequency opf the sound aqnd the
frequency of the masking noise are close to each other.
For example a 4kHz will mask a softer 3.5 kHz tone but have
little effect on a 100 Hz tone. Masking can also be caused by
harmonics of a masking tone so a 1kHz tone with a strong
second harmonic could mask a 1900 kHz tone.
3.2 Perception of dynamic range
The ear operates over an energy range of 1^12:1
(1,000,000,000,000:1) and compresses the perceived
intensity level in order to protect itself. The loudness of a
sound is perceived by the ear as varying approximately in
proportion to the logarithm of its energy. As a result
increasing the power output of an amplifier by 10 watts from
10 to 20 watts gives a significantly greater volume increase
than increasing power output from 60 to 70 watts. To get the
same increase in loudness the 60 watt output would have to
be doubled to 120 watts.
When we compare this information with the measurements
by Sellick we can easily see why it is folly to mix at high
volume. The ear’s sensitivity to sound changes decreases
logarithmically as the gain increases
3.3 Hot spots and distortions
The ear has its greatest sensitivity in the range 1 to 4 kHz.
This means that a 1 kHz sine wave that produces a given
sound pressure will sound louder than a 10 kHz sine wave
which produces the same sound pressure. In addition the
nature of the ear causes it to produce harmonic distortion of
sound waves above a certain volume level. Harmonic
distorion is the production of harmonics of a waveform which
do not exist in the original sound. Thus the ear can cause a
loud 1 kHz tone to be heard as a combination of 1kHz, 2 kHz
3kHz etc tones.
3.4 Perception of Harmonic Content
Harmonics are very important with respect to musical
instruments because their presence and relative intensities
in the sound waves produced enable the ear to differentiate
between instruments playing the same fundamental tone.
For example a violin has a set of harmonics differing in
degree and intensity from that of a viola. The overtone
stucture is called the timbre of an instrument.
According to Hamm: (1973)
The primary color characteristic of an instrument is
determined by the strength of the first few harmonics. Each
of the lower harmonics produces its own characteristic effect
when it is dominant or it can modify the effect of another
dominant harmonic if it is prominent. In the simplest
classification the lower harmonics are divided into two tonal
groups. The odd harmonics (3rd and 5th) produce a stopped
or covered sound. The even harmonics (2nd 4th and 6th)
produce choral or singing sounds. Musically the second is an
octave above the sound making it fuller. The third is termed
a quint or musical twelfth, It produces a sound that many
musicians refer to as blanketed. Instead of making the tone
fuller a stron third actually makes the tone softer. Adding a
fifth to s strong third gives the sound a metallic quality that
gets annoying in character as its amplitude increases. A
strong second with a strong third tends to open the covered
effect. Adding the 4th and 5th to to this changes the sound to
an open horn like character the higher harmonics above the
7th give the tone edge or bite. Provided the bit is balance to
the basic musical tone it tends to reinforce the fundamental
giving the sound a sharp attack quality. Many of the edge
harmonics are musically unrelated pitches such as the 7th 9th
and 11th. Therefore too much edge can produce a raspy
dissonant quality. Since the ear seems very sensitive to the
edge harmonics controlling their amplitude is of paramount
importance. The study of a trumpet tone shows that the edge
effect is directly related to the loudness of the tone. Playing
same trumpet note loud or soft makes very little difference in
the amplitude of the fundamental and lower harmonics.
However harmonics above the 6th increase and decrease in
amplitude in almost direct proportion to the loudness. The
edge balance is a critically important loudness signal for the
human ear.
Although the ear may receive the overtone structure of a
violin if the listening level is loud enough the ear will provide
additional harmonics and change the perceived timbre of the
instrument. This means that sound monitored at very loud
levels may sound quite different when played back at very
low levels. To make things even more difficult the frequency
response of the ear changes with the loudness of perceived
signals. The loudness compensation switch found on many
hi fi pre amps is an attempt to compensate for the decrease
in the ear’s sensitivity to low frequency sound at low levels.
Sound Pressure Levels
The curves in the following diagram are the Fletcher Munson
Equal Loudness Contours and they indicate average ear
response to different frequencies at different levels. The
horizontal curves indicate the sound pressure levels (spls) at
different frequencies that are require to produce the same
perceived loudness. Thus to equal the loudness of a 1.5 kHz
tone at a level of 110 dB spl, (the typical level create dby a
trumpet type car horn at a distance of three feet) a 40 Hz
tone has to be 2 dB greater in sound pressure level. A 10
kHz tone must be 8 dB greater in sound pressure than the
1.5 kHz tone to be perceived as being as loud. At 50 dB spl,
(the noise level present in the average private business
office) the level of a 30 Hz tone must be 30 dB greater and a
10 kHz tone must be 14 dB greater than a 1.5 kHz tone to be
perceived at the same volume. Thus if a piece of music is
monitored so that the signals produce a sound pressure
level of 110 dB and it sounds well balanced it will sound both
bass and treble deficient when played back at 50 dB spl.
Loudness and pitch
The loudness level of a tone can affect the the pitch the ear
perceives. For example if the intensity of a 100 Hz tone is
increased from 40 to 100 dB spl the pitch will decrease by
about ten percent. You have only to listen to recordings of a
ships hortn as it leaves harbour to hear the flattening induce
by huge sound pressure levels. At 500 Hz the pitch changes
about 2% fo the same increase in sound pressure level.
Important features to note when looking at Fletcher-Munsen
equal-loudness curves: Notice that the curves flatten out at
higher loudness levels. For example in the graph above, the
range from highest to lowest in physical decibel level is
about 70 dB; while for the medium loudness level, the range
is 37 dB; and for the highest level, 28 dB.
Notice that there is an increase in sensitivity to frequency
intensity around 3500 - 4000 Hz. This is due in part to the
first resonance of the ear canal (which is about an inch long).
There is a second resonance at twice the first that can be
seen in the medium loudness level curve at around 8000 Hz.
Perception of Direction
One ear cannot discern the direction from which a sound
comes but two ears can. This is called the binaural effect
and is made possible by four cues received by the ears.
1) relative intensity
2) time of incidence of the sound
3) phase
4) complexity of the wave form.
Relative intensity
Refers to the fact that a sound coming from the right will
reach the right ear at a higher intensity level than the left ear.
There are two reasons for this:
1) the intensity of a sound wave is inversely proportional to
the square of the distance between the source and the
listener. The intensity 6 feet from the source is one fourth
that at three feet. Since the left ear is farther away from
the source than the right it receives a lower intensity
signal.
2) The left ear is in an acoustic shadow cast by the head,
The head blocks the direct sound waves and allows only
reflected sound to reach the left ear at the blocked
frequencies thus reducing the intensity of the sound
perceived at the left ear. This effect is insignificant for low
tones because they bend around the head easily but it is
considerable for tones above 500 Hz because the effect
increases as the frequency rises.
Time of Incidence.
Because the path length to the right ear is shorter than the
path length to the left ear with respect to a sound source on
the right, sudden changes in sound pressure from the source
will be sensed earlier by the right ear. Thus if a wave begins,
changes intensity, changes frequency, changes wave shape
or stops the right ear detects it before the left ear.
Phase
For waves which are continuous the ears can detect the
phase difference between them resulting from the time
needed for a particular portion of the wave to travel from one
side of the head to the other. The phase cue is most
accurate in localizing low frequency tones where the path
length between the ears is a wavelength or less. As the
frequency increases the wavelength decreases and it is
possible the wave may appear in phase at both ears with the
far ear hearing the wave several complete cycles after the
near ear. Thus phase cues are less effective at high
frequencies.
Complexity of the wave form.
The complexity of the waveform acts as a cue for the
localization of compound tones and noises. The sound
shadow created by the head attenuates high frequencies
while low frequencies reach the low ear by bending around
the head. As a result the two ears hear a difference in the
timbre of the waveform. The farther ear hears fewer high
frequency components.
To summarize: the accuracy of cues for localization varies
with the frequency and complexity of the perceived tone.
High frequency tones are located by intensity, low frequency
tone by phase, complex tones and noise by a combination of
intensity, time of incidence, and timbre. If there is no
difference between what the left ear and the right ear hears
the source appears to be the same distance from each ear
and the only places a single source could fulfill this
requirement are directly in front of or behind us. Since the
shape of our ears favors sound from the front the brain
generally assumes that it any unidentified location is in front
of the listener. This phenomenon allows the recording
engineer to place sound not only in the left and right
speakers of a stereo system but also between the speaker.
By feeding the same sound to both speakers the ear hears
the sound identically in both ears and the sound appears to
be directly in front of the listener. By changing the
proportions fed to the two speakers the engineer can create
the illusion that the sound source is anywhere between the
two speakers. This is called panning.
Conclusions for Musicians.
The ear is a highly adaptive and self-protecting organ and
that makes it a less than 100% reliable witness.
Of the eight characteristics of wave forms only three are not
affected by psycho acoustic factors: Velocity, Wavelength
and Shape.
The remaining five are very much influenced by psycho
acoustic factors.
Frequency – Sounds appear flat at high volumes
Amplitude – depends on playback volume differently at
different frequencies.
Phase – Although the phase of a sound is fixed at the point
of origin the perception of phase by the ear is frequency and
distance dependent.
Harmonic content high volume can add harmonics that do
not exist.
Envelope envelopes played at high volume will become
compressed by the natural logarithmic compression of the
ear.
Most musicians first encounter these variables when they
discover that a mix that they had spent most of the night
working on sounds terrible in the morning. This “next day
test” can be massively disappointing and can also be
avoided using the following rules.
The Golden Rule:
Learn to mix at low to medium volume levels unless you are
specifically checking a timbre or trying to impress a client
with a finished mix in which case crank it to 75dB plus..
Gain matching
Match the gains between instruments using the lowest
possible gain. For example if you trying to match the level of
a bass drum against a bass slowly turn the master gain
down to silence and see which one disappears first. Adjust
their respective levels until turning the gain to zero cause
them to disappear both at the same time.
Avoid masking and difference tones.
This is basically done at the arrangement level. Arrange the
parts in your music in such a way that instruments with
similar harmonic content play in different octaves (like guitar
and piano for example). Unless you are seeking a special
effect do not double chord voicing between similar
instruments.
Timbre matching.
If you are making critical adjustments to timbre do so at a
reasonably high volume. (around 60-70dB). Listening to
timbre at low volume will cause you to boost the lows and
highs too much.
Avoid working too long on a single sound.
Do not work on a single sound for long or your ear will adapt.
Alternate between percussive and sustaining sounds when
possible. Do not spend more than 10-15 minutes on any one
sound. If you do your ear will rapidly “protecting” itself from
the sound.
Take frequent breaks and listen to something different.
If you are working on pop listen to Classical and vice versa.
If you have been working on decaying sounds (percussion)
listen to strings.
Avoid over tiredness.