Phys 1240 Fa 05, SJP 5-1
Chapter 6: The ear, and its response (Loudness)
This chapter has a lot of cool stuff in it. I'm not going to cover it all in lecture - if you're interested in
the connection of physics and physiology and psychology, I encourage you to read it! I'll talk about the
ear in class, at a level slightly simpler than section 6.1. Don't memorize the names of these various
parts (unless you're into that kind of thing), what I want you to take away is a sense of the physical
purpose of a few key elements, as outlined below
1) The OUTER EAR: is the channel and structures that help bring sound waves to the eardrum. This
part of the ear ends at the eardrum - a thin membrane that vibrates when there is excess (or under-)
pressure on the outside. (The inside, i.e. the middle ear, is suitably "separated" from the outer ear that
changing room air pressure causes the eardrum to vibrate. )
2) The MIDDLE EAR: Connects to the outside world through the Eustachian tube to your throat.
(That's why swallowing, or holding your nose and blowing, can help equalize the pressure in your
inner ear and the outside world when you're flying. When would you want to "blow", on takeoff, or on
landing? Why?) The middle ear has this awesome collection of little "levers", three bones (hammer,
anvil, and stirrup, colloquially) which transfer the vibrations of your eardrum into the inner ear. The
principle of the lever is that you can use one to change a small force into a bigger force (the PRICE
you pay is that the bigger force acts over a smaller distance - so you don't get any extra WORK out of
it) The next "link" in the chain is the "oval window", which separates the middle ear from the inner
ear. There's a second key piece of physics at work here: the oval window is very small, much smaller
than the eardrum (20-25 times). So, you're applying even MORE force on an even SMALLER
membrane, so we've really ratcheted up the pressure by this point.
So basically, the middle ear has taken pressure waves from the outside, used these levers and the
different areas on the two ends to make a lot MORE pressure variation on the "oval window". Since
intensity grows like (pressure)^2 (remember?!) we have REALLY ratcheted up the intensity by this
point, it's like we have a little "amplifier". Very cool - it's all mechanical, and it doesn't violate
conservation of energy, we're just "concentrating" the energy... The text argues that all together, this
middle ear business amplifies intensity by about 100 to 1000 times , which is 20-30 dB improvement
in sensitivity.
There's lots of other nifty things going on - e.g. muscles which are designed to prevent damage from
overly loud sounds.
3) The INNER EAR: The cochlea is a coiled up region that contains mechanisms to convert pressure
variations into electrical signals. That's the big idea here - converting the physical signal into
something your brain can interpret! Part of the key is the 20,000 "hair cells" which get disturbed by
motion, and initiate nerve signals. At low frequencies, the fluid in the inner ear can transmit motion all
the way towards the back, but higher frequency sounds tend to cause more response closer to the front
(window) So, this provides a simple physical mechanism to distinguish frequencies.
The text goes into more details, but I think this is what I pulled out as the key physical ingredients.
Quite a remarkable little device. Damaging these little "hair cells" is part of what causes hearing loss
as you grow older, or get exposed to too much (or too loud) sound, by the way. The box on page 95 of
the text has a little more info about this, but some web hunting might be worthwhile if you're
interested!
Phys 1240 Fa 05, SJP 5-2
The next section of the text talks first about the range of hearing. We've discussed this in round
numbers (20-20,000 Hz for normal hearing), but in reality, it's more complicated. For one thing, you
are more sensitive to certain pitches than others (we'll get back to this soon), and esp. at the low end,
you can "detect" low frequencies but it's not clear if you're "hearing" or "feeling" them!
The text goes on to discuss "JND" or "Just noticeable difference". It's a fun psychology question what small differences can normal humans detect in frequency or amplitude? Think a little bit about
how you might set up such an experiment, it might not be the same as what the book talks about! They
discuss a scheme involving "two-alternative forced-choice" where you hear two sounds and choose
which is (e.g.) louder or softer. They give random pairs, and figure out where most people can reliably
determine which is louder (or higher pitch) Fig 6.6 makes the case that a SIL (that's "Sound Intensity
Level", or decibel level) difference of about a couple of dB is pretty much the JND for normal people
at 1000 Hz at a normal (40 dB) level. Changing the pitch will change that graph (you won't be so
sensitive at very high pitches, for example) and changing the "base level" (dB of one or the other tone)
will also change the results. That's what fig 6.7 shows. Pick the "1000 Hz" curve: if your "base tone" is
40 Hz, your JND is a bit over 1 dB, that's how much DIFFERENT it has to be to notice a difference.
Up at 80 dB, the JND has gone DOWN, you are able to tell a difference of only 0.5 dB. This surprises
me, I would have thought that as the base gets really loud, it would become MORE difficult to detect a
small change. Huh! Of course, the graph doesn't go up to painful levels, so maybe it's not crazy. (At
SOME point I claim those curves would have to head back up again? Do you agree with me?)
Fig 6.8 shows how changing the pitch impacts your sensitivity to pitch. E.g., at 80 dB (solid curve) as
the frequency goes UP, you steadily become less sensitive to pitch (your JND in pitch goes up, which
means you need MORE of a change to reliably notice it) Do you see how the curve TELLS you this?
(This says if you play a bit "off key" at really high pitches, apparently people are less likely to notice!)
Do you see why JND in fig 6.8 is given in Hz, but it's in dB in fig 6.9? They measure the "just
noticeable difference" of two DIFFERENT quantities: loudness in Fig 6.7 (dB), and pitch in Fig 6.8
(Hz) You could imagine all sorts of variations on this - what happens to your JND of loudness as the
pitch goes up? (That's what the various curves in 6.7 are trying to show, do you understand how?)
Are you making sense of what these curves are trying to tell you? Look at them, try to tell yourself a
little story about what the curve means. (So e.g., what does it mean that the 200 Hz line is HIGHER
than the 1000 Hz line in Fig 6.7? What is that saying about human hearing? If you can't answer this at
all, ask someone - it's part of what I mean by assigning you to look at graphs like this!)
This is good fun, figuring out what difference you can detect, but doesn't answer the question that lots
of engineers, ear doctors, and music listeners are perhaps more interested in: how MUCH different are
two different sounds? The answer is "it all depends". It's complicated! I'd like to say that "perceived
loudness" just depends on dB. But no, different frequencies may seem louder or softer, even if they're
the same dB level. And indeed, notes that last a short time (transients) may generate a different
"perception" than ones that last longer. And then there's TIMBRE, a catch-all phrase that describes
"the waveform". Remember, when you have a periodic wave, it's easy (no matter how COMPLEX that
repeating wave looks) to pick out the basic frequency and amplitude. But different shapes SOUND
different, even if the pitch and loudness are the same. That's the difference between instruments, e.g. a
violin and a sax playing the SAME note at the SAME (basic) loudness. It goes the other way too: a sax
and a violin can play the same frequency and put out the same measured intensity, but one may sound
louder than the other, because we interpret the different waveform (timbre) differently. Sorry, I wish
there were simple rules! Fig 6.9 tries to show schematically all the different connections between
measurables (on the left) and psychological impressions (on the right). The numbers on the arrows are
the section numbers from the book if you want to investigate that "connection" further!
Phys 1240 Fa 05, SJP 5-3
Section 6.4 of the text talks about the connection between loudness and intensity. This is basically
what you expect: the more intense the wave (i.e. more energy per second entering your ear) the
LOUDER it should seem. (More overpressure means more motion of the hair cells means more
electrical signal.)
The decibel scale was designed to get closer to how it "feels" than amplitude does - increasing the
amplitude by a factor of 10 does not FEEL like it's 10 times "louder". You can try asking people:
"here's a sound, now tell me how much louder the next one feels". Of course, the answer will vary a
lot, but generally, there's a pretty good consensus. For a given frequency, as the SIL goes up, the
loudness goes up. Generally, (VERY roughly!) increasing the SIL (that's "the decibels") by 10 dB will
double the "loudness" you perceive. That's why dB are useful. Turn the knob on your stereo another 10
dB, people think the music is twice as loud. (And in fact, the intensity is 10 times as much!)
So if you have a choir with 10 people (that's 10 times the intensity), they will sound TWICE as loud as
the soloist. To sound twice as loud as this, you need need 100 people. A huge choir is thus only 4
times as loud as a soloist.
But if you keep the intensity level fixed and crank up the frequency, people will ALSO say it sounds
louder. (That's why the curves in Fig 6.10 are different)
Section 6.5 talks about how it's NOT true that pitch and frequency are exactly the same. (Which we've
been loosely arguing all term). Once again, the story is a little more interesting. The octave is a very
strong "base" for comparing pitch, people are pretty darn good at identifying an octave, which is
defined as being twice the frequency, and is perceived as being twice the pitch. So people are indeed
much more reliable and consistent with pitch/frequency than with loudness/amplitude. But for smaller
intervals, more variability exists (and you can "fool" people at different loudness or with sounds that
are not pure sin waves too)
Section 6.6 talks about this last point - how pitch and loudness kind of "mix together" in our
perceptions, to make how it sounds not determined by a simple formula based on physics
measurements! There are two graphs in this section, and I really want you to stare at 6.12 and try to
make some sense of it. That's basically the crux of this section, and contains a lot of interesting ideas.
It's a new KIND of graph, called a "contour graph", so we have to think about it!
It's called the "Fletcher-Munson" graph, and is quite famous.
Let's start with the big picture. Let's define a new thing, which is totally subjective, that measures
"perceived loudness" (whatever that might really MEAN, we'll just let PEOPLE define it). That's the
"phon". So the more "phons" a sound is, the louder it seems. And going up by 10 phons should be a
certain amount louder... So going from 10 to 20 phons should feel twice as loud. We're trying to
quantify perceptions, here, so clearly we're on murky ground, but fortunately people seem moderately
reliable in this, so it's not as bad as it might seem.
What we've been saying so far is "SIL (in decibels) tells you loudness" (roughly). So I would expect
that human subjective loudness would MATCH decibels. Indeed, to "calibrate" our "phon" scale, we're
going to say that, for 1000 Hz sounds, that's exactly right, "phon" and "dB" are basically the same. (It's
now an experimental question to verify that people really agree with this, and they pretty much do.
Any deviations from this are the distinction between Phon and "sone", which we're not going to hassle
with!)
Phys 1240 Fa 05, SJP 5-4
But now the point of Fig 6.12 is that if we all AGREE on this loudness scale at 1000 Hz, we will
disagree at other frequencies!
Let's start starting at that figure. First of all, let's focus on the "10 phons" curve. Realize that "10
phons" means a nice, quiet sound. EVERY point on that curve sounds equally loud to a "typical
person". At 1000 Hz, the curve is vertically at 10 dB, so this is how loud a 10 dB 1000 Hz tone
"sounds". (Pretty darn quiet, by the way!!) But the curve wiggles up and down, what does that mean?
The x-axis is frequency (a physically measurable quantity) and the y-axis is SIL (in dB, also a
physically measurable quantity) So what the curve says is that different combinations of frequency
(pitch) and intensity will sound "equally loud". For example, to sound like 10 Phons, if the sound is
LOW pitched (look at, say, 20 Hz), then the curve is quite high up, you need 80 dB to "notice" the
sound. Since the curve is way up high in the y-axis there, it means you're pretty insensitive to those
low frequencies, you NEED lots of energy hitting you (80 dB tells you that) to feel like it's even a
quiet sound. Move over to 2000 Hz, but stay on the 10 phon curve. Now you're down to about 5 dB.
So you hardly need any energy hitting your ear at this frequency to think it's just as loud, 10 Phons.
You are very sensitive at this frequency. You're MOST sensitive where the curve dips the lowest,
around 4000 Hz. At that point, you can detect just over 0 dB (i.e. a TINY amount of energy, 10^-12
Watts/m^2) and it sounds "10 Phons loud". As the pitch goes up from there, you need more energy to
notice/care about it at this same level. There's a second dip around 13,000 Hz, where you get a bit
more sensitive again (but not as much as down at 4000 Hz!)
This line, the "10 Phon" line, is called a "contour". It tells you the collection of "pairs" of intensity and
frequency that combine to give the SAME loudness level. This is a "contour map". Figure 6.11 is
showing you something similar, but they're only showing you two of the contours, and now they use a
"third dimension" (the z-axis) to show you the loudness. (If you find Fig 6.11 hard to interpret, we'll
talk about it in class a little, but it's not such a big deal, I don't think you really need it)
Now let's move up to the next curve in Fig 6.12, the "20 Phon" contour. Of course, it's shifted UP,
because no matter what, for a sound at any given frequency to seem louder, it must be more dB. But
it's not the SAME number of dB at every frequency! It's shifted up by different amounts as you move
left or right on the graph. Down at the low (left) end, we see a surprise. Although it seemed that we
weren't very sensitive down there (because we needed a whopping 78 dB to think it's 10 Phons loud)
you'll notice that only a couple of dB more makes it sound twice as loud, 20 Phons. So maybe I should
NOT say you're "insensitive" down there, it's deliciously complicated! You do need a lot of dB to
notice the sound, but a small increase in dB will yield a large perceived increase in loudness! (So
"sensitivity" is a vague word here!) If you go over to the minimum of the curve, around 4000 Hz, you
see that you have to move up the page about 10 dB to double the perceived loudness, to go from 10 to
20 Phons. (That's what I claimed in the previous sections - every 10 dB is 10 times more energy/sec,
and is perceived as "twice as loud". But this curve shows us that's not true at all frequencies!)
Another example: going from point E to P in Fig 6.12 . Look at what that says: they both are on the
"100 Hz line", so we're talking about the same pitch getting more intense. E is at about 38 dB, and P is
right at 60 dB, so we've gone up by 60-38 = 22 dB. So you might think that's 22 Phons louder, but no,
Phons are not dB, and in fact by sliding over along the curves, you'll notice that E is on the 20 Phon
curve, but P is on the 50 Phon curve (Look! Do you see that?) so it's perceived as 30 Phon louder. (30
may not seem like that much more than 22, but it is, remember that a small change in dB means a BIG
change in intensity.)
Phys 1240 Fa 05, SJP 5-5
There is an interesting and important consequence to all of this, when you try to reproduce sound and
play it back at a different intensity level than it was recorded. Imagine, for instance, that you record a
live, loud, rock concert. The recording is PERFECT, so if you played it back at 'full volume' it would
sound exactly the same in your house as at the concert. (Sweet!) But your parents are around, and ask
you to turn it down. Suppose you turn down the volume on your stereo by 40 dB (which means 10^4
or 10,000 times lower intensity. That ought to keep your parents happy!) Now, a 1000 Hz sound which
had originally been, say, 100 dB (which is also 100 Phons) at the concert is now 60 dB (which is also
100 Phons. Remember, at 1000 Hz, dB and Phons are the same, by definition). But consider a low
pitch, a 100 Hz sound. Supposing that IT had started off also 100 dB at the concert, and so it is now
being played back at 60 dB. Look at Fig 6.12, go over to 100 Hz on the x-axis. If you go up to 100 dB
on the y-axis, you'll see that this is just under 100 dB. But if you go down to 60 dB, you're at point
"P", which is 50 Phons. Weird! So the medium frequency (1000 Hz) sound went down from 100
Phons to 60 Phons, a drop of 40 Phons. But the lower pitch (100 Hz) went down from about 98 Phons
to about 50 Phons, a drop of 48 Phons. So to your perception, the bass has gotten much softer... it's no
longer "equalized" the same way! You're losing the bass sounds. Bummer, your PERFECT recording
is no longer perfect, just because you turned down the volume! To fix this, you would need some kind
of "equalizer" buttons, (which many stereos have!) or at least a separate "treble" and "bass" volumes
(which ALL stereos have, pretty much, but that's also awfully crude. Every frequency needs its own
separate "fix" to make the sound reproduction high fidelity!) Many stereos have tricks to try to take
this effect into account (stare at Fig 6.12, take some time to convince yourself that this is a general
feature: the lower frequencies get "hit" harder when you turn down the volume)
For you to think about: what happens if you turn UP the volume? Imagine you record a quiet concert,
(perfectly) and there is a 20 Hz tone and a 1000 Hz tone which are both playing at 80 dB. How will
each SOUND in terms of phons? Now what happens if you crank your stereo by 30 dB, do you're now
listening at 110 dB. How many phons will each sound be now? (Will they both change by the same
amount? Which gets "overemphasized"?)
Couple of final points for this section:
The text worries about the subtle difference between "loudness level" and "loudness", between
"phons" and "sones", ack. I'm not so concerned about this (so we won't be talking about "sones" vs
"phones", it's just too much detail). The main point is that "how loud it sounds" depends on TWO
things, frequency and intensity, and how it depends on those is a wee bit complicated, but has some
general features that show up pretty clearly in the figure. (The curves have wiggles, they're not
straight lines, but they're not THAT wiggly either)
By the way, it occurs to me that lambda*f = 344 m/s for sound, so 4000 Hz (where most of those
contours reach their minimum SIL) corresponds to lambda = 344 m/s / 4000 Hz = a little under 10 cm,
about the size scale of your inner ear. (An unrolled cochlea is about 4 cm long). I don't think this is an
accident, we might think a little about this...)
________________
The last section is a brief discussion of TIMBRE, or "quality" or "character" of sounds. We'll discuss
this more soon when we talk about harmonics. But the text does mention something interesting that's
somewhat distinct from the "overtones". That's the ATTACK and DECAY of the sound. (How quickly
the sound "starts up" when you play a note, and in what way it moves from zero to it's level, and then
how it moves BACK to zero when you stop playing. ) The times are short (milliseconds, maybe 10's of
milliseconds) but that's still enough for your brain to distinguish big changes from one instrument to
another.