LECTURE 9 What is sound What is Music by happo7


									                                                 LECTURE 9

                                      What is sound? What is Music?

Introduction                                            The importance of sound
    In this and the following lecture I will deal           In many cases the perception of the outside
with the two basic means by which we perceive           world through sound, i.e. hearing, is even more
and communicate with the world at a distance.           important than that of sight. Certainly it seems to
They are sound and light. Unlike the most               be for many mammals. Some specialists in the
primitive organisms, which respond only to the          evolution of mammals suggest that their highly
transfer of chemicals as in smell and taste, or to      developed sense of sound originated from the
pressure as in touch, these two means of sensing        dinosaur age when it was extremely dangerous to
the outside world give us a considerable                be out during the day and therefore much of our
advantage in survival and reproduction. Our             foraging had to be done at night.
astounding ability to use these phenomena to                Also, there is evidence that there is generally
perceive our world to the very limit imposed by         more traffic along the aural nerve channels (one
the laws of physics is one of the marvels of the        for each ear until they join at the back of the
Universe. Understanding the form into which             neck) then there is along the optics nerve canals
our senses, i.e. our eyes and ears, have evolved to     from the two eyes. Further evidence comes from
deal with sound and light and how our brain             the common observation that old people who go
processes the information these senses provide,         deaf are more difficult to live with then old people
therefore requires an understanding of what             who go blind. The loss of hearing really does
sound, and the particular form of sound called          seem to cut us off from the outside world more
music, actually is.                                     so than does the loss of sight. In other words
    The field of study being introduced here is         hearing seems to be a more basic need than even
called "psychophysics", a very important modern         sight.
branch of psychology. In general this field deals           As you know by now I like to open a lecture
with the sensory organs and how the brain               with a puzzle, hoping that the lecture will resolve
processes input from these organs. Often the            that puzzle for you before its end. In this case the
sub-field dealing with hearing is referred to as        puzzle is called the "Sheppard Staircase", named
"psychoacoustics". The importance of this field         after the scientist who developed it at the Bell
alone is indicated by the fact that about a third of    Telephone Laboratory in the USA. It is a
the scientific articles in the Journal of the           sequence of musical tones that are clearly heard
Acoustical Society of America, a journal dealing        to be going down in pitch successively from one
with all aspects of sound including, of course,         tone to the next but that after a dozen or so tones
sonar detection of submarines, are on                   seem to have gotten nowhere. They are the same
psychoacoustics alone. And some of the most             pitch as when we started! And it seems possible
revealing research in this area is carried out on the   for this to go on forever. To understand this you
perception of music. If you are interested in this      must first understand what sound actually is.
subject then you might consider taking the two          What is sound?
courses at McGill, 198-224 and 198-225
designed for entering the field of Recording                Basically, sound is a pressure oscillation.
Engineering. (And you thought that music was            This oscillation can occur in practically any
just fun!)                                              material; in fluids, such as that surrounding the
    The field of psychophysics is so extensive          basilar membranes in your ears, in steel, in the
that I can only give the barest introduction here,      Earth as shocks from Earthquakes, inside the Sun
and then only to the two specific examples of           and even inside neutron stars. However, the
hearing and sight. I introduce these two because        easiest form to deal with, and the form of most
they are both forms of vibrations. Understanding        importance to you is that of pressure oscillations
sound and sight therefore requires understanding        in air.
the most basic form of vibration; the simple                I will deal later with how these pressure
harmonic motion studied in the previous lecture.        oscillations get set up. For now I will just
                                                        assume that they are set up and that they are
Physics 101A - Physics for the life sciences                                                                2

impinging on your eardrum, causing it to vibrate        thereby increasing its elevation from the floor by
itself and which is the start of the sensory            about one meter, you will have reduced the
processing of sound. For a very good diagram of         pressure on your eardrums by 13 Pa. You can
the ear see fig. 11.30 of Hecht.                        change the pressure on your eardrums by 1 Pa by
     The purest form of sound is, like that of any      nodding your head up and down by 77 mm
oscillation, simple harmonic motion. Your               (about 3 inches). By nodding your head up and
sensory response to such a pressure oscillation is      down over an extent of 144 mm you could then
that it is a "pure tone". Graphically, this has the     cause a pressure oscillation in your ears of
form shown below.                                       amplitude 1 Pa.
                                                            This is obviously a very small change in
                                                        pressure. Another way of imagining a pressure of
 p                                                      1 Pa on your eardrum is that it would be the
                                                        pressure caused on your right ear by a movement
 po                                                     of your head sideways to the right at 2.5 mm per
                                                        second! (At the same time there would be a
                                                        "vacuum" of 1 Pa on your left ear.) Therefore
                                                        you can cause a pressure oscillation of amplitude
                                                        1 Pa in each of your ears by oscillating your head
     Here you should note that the pressure does        sideways at a velocity amplitude of 2.5 mm per
not oscillate between positive and negative values.     second. (The pressure oscillation in your right
Rather it oscillates between pressures that are         and left ears would, of course, be 180 degrees out
above and below quiet atmospheric pressure. The         of phase with each other. It is left as an exercise
pressures involved in sound are therefore               for you to calculate what the extent of this motion
pressure deviations from the quiet level.               would be if you were oscillating your head at 1
     Like all pure oscillations, this is described by   cycle per second.)
an amplitude, a frequency and a phase. I will deal          Thus a Pascal of pressure oscillation is very
with these in turn.                                     small. You would certainly not be able to sense it
The amplitude of sound                                  by normal motions of your head. Yet if you could
     It is fairly obvious that the amplitude of a       oscillate your head, up and down by 144 mm or
pressure oscillation is related to the loudness of      sideways at 2.5 mm/s maximum velocity, at 1000
the sound. The greater the pressure level on the        times per second you would hear a piercing
eardrum the farther it will move and the greater        sound of 1000 Hz at a level about equal to that of
the sensation the movement produces. However,           car horn at one meter from your ear!
what is probably not so obvious is how little the           This shows just how sensitive the human ear
pressure has to oscillate for you to hear it.           actually is to the pressure oscillations that are
     You might object that you are in no position       involved in sound. (It also gives grist to your
to consider this matter since you have not yet          parents for their grinding into your mind the
studied what pressure actually is. However, I will      dangers of listening to too much rock music.) As
try to make it understandable by relating it to an      we all know the human ear is capable of hearing
experience you must have had with the pressure          sound pressure levels which are much lower than
on your ears when you have a sudden change in           that of a car horn at 1 meter. In fact, the limit at
altitude. This can sometimes be experienced in          1000 Hz is a pressure variation of about 60
driving a car through a mountainous region but it       microPa, corresponding to an up and own motion
is more usually felt when you are in an airplane        of the head of less than 10 microns!
which is descending.                                        This incredible sensitivity of the ear to
     Normal atmospheric pressure at sea-level is        pressure variations is achieved by extremely fine
100,000 Pascals (referred to by the International       tuning of the various parts of the inner ear that
Symbol Pa). It weather reports it is often referred     pass the pressure oscillations to the "hair-cells" in
to as 100 kiloPascals (kPa). When you undergo           the basilar membranes of your cochleas. (It is
any change in elevation of your head you will           these hair-cells that fire when stressed by an
experience a change in pressure on your ears of         oscillation so as to pass a signal to the brain that a
about 13 Pa per meter of elevation. That is to say,     sound has been detected. At the lowest level of
if you raise your head out of bed in the morning,       sound that can be heard these haircells oscillate at
                                                        less than the diameter of an atom! )
Lecture 9 - What is sound? What is Music?                                                                  3

     The highest level of sound that can be              example, suppose you came across two strange
tolerated at the ear without real pain is a pressure     spherical growths in the forest, one of 10 mm
oscillation of about 60 Pa. The range of pressure        diameter and the other of 100 mm diameter, and
levels in sound that we can perceive is therefore        you took them both home to study. You would
about 1 million.                                         regard a subsequent growth of a half-millimeter
     This is an enormous range. How do we cope           by the 10mm diameter sphere with the same
with this range in determining the loudness of a         interest as a growth of 5 mm by the 100 mm
sound?                                                   diameter sphere, even though it was actually 20
     Here you can perhaps already see a problem          times less. This is such a general behaviour of
with the simple linear scale. You say there is no        our perception systems that it is referred to as
sound, i.e. zero sound, when there is actually a         Fechner's Law, after the first scientist in
pressure oscillation of 60 µPa. By increasing the        psychology to formulate it.
sound level to, say, a pressure oscillation of                Similarly with sound. Within the range of
120 µPa you will hear the increase of 60 µPa.            pressure levels that we can hear we seem to be
However, there is no chance whatsoever that you          able to detect changes of about 10%. (This is
will notice a change of 60 µPa at the top end your       only if there are no distracting noises and we are
scale, i.e. from 60.000000 Pa to 60.000060 Pa.           concentrating fully on a pure tone.) Thus the
How then does the central nervous system                 change of a sound that changes from a pressure
respond to changes over such a range?                    level of 1 mPa to 1.1 mPa is detected as easily as
     Here I will try to introduce a little psychology,   a pressure change from 50 mPa to 55 mPa, as is a
in case you have not already come across it. But         change from 120 µPa to 132 µPa.
again you should take into account that I am just             Changes in sound level are therefore referred
a physicist who has never taken a psychology             to in percentages. The unit used for such
course and am therefore speaking as an amateur.          specifications is the "decibel (dB)". For reasons
However, you should also take into account that I        that I hope I can make clear, this is actually
use the word amateur here again in its original          defined to be 12.202…%. One dB is therefore
meaning; as a lover of the subject. I regard             about the smallest change in sound level that can
psychology as one of the most fascinating                be perceived.
subjects for human thought and I have maintained              To see where this weird number comes from,
for some time that every physicist should study          consider the result of 20 successive increases of 1
some psychology before they try to teach                 dB each. Since this is a compounding problem,
physics.                                                 as in compound interest, the result is as shown in
     But also, I have maintained that every              the table on the next page.
psychologist, if they are to understand how the               You can see that the result of 20 dB of
human mind thinks and why it is structured the           increase, where each decibel is about a 12.2%
way it is, should study some physics. The                increase, is just multiplying by 10. If you have
difficulty is in getting many of them to agree, just     learned a little arithmetic of this type, you could
as it is difficult to get many physicists to agree to    see that the actual percentage increase to get
take seriously what, to them, is the totally             exactly 10 after 20 such percentage steps would
unscientific babble of psychology.                       be the 20th root of 10 minus 1.0 times 100, or
     The aspect of psychology that I want to             12.2018454...
introduce here is that of the perception of changes           But along the way to 20 steps there are some
in stimuli. This subject is of extremely                 interesting levels that are reached. The first is at
importance in basic psychology since it seems            3 dB. There the result is very close to the square
that all of human neural activity is devoted to          root of 2. You can check this by noting that at
dealing with change. At the most basic level our         6 dB the increase is about × 2. Thus two
perception systems seem to be based on detecting         consecutive increases of 3 dB, i.e. each of
percentage changes in stimuli rather than the             × 1.413, results in ( × 1.413)2, which is × 1.995,
actual changes in the level of stimuli. Thus if a        i.e. close to × 2. This can be further checked by
ball changes its diameter from 10 cm to 10.1 cm          noting that 12 dB is about × 4 (i.e. two
it is not the 1 mm that we notice b ut rather the 1      consecutive × 2's) while 18 dB is about × 8, or
percent.                                                 three consecutive × 2's.
     The simplest example of this is indeed
noticing a change in the size of an object. For
Physics 101A - Physics for the life sciences                                                                         4

           dB                           Result of                        ( −6)               6
                                    12.202% increase     p final                        −
           1                          × 1.12202                    =   10 20     = 10       20   = 10 −0.3 = 0.501
           2                          × 1.259
           3                          × 1.413                (Try this on your calculator and see.)
           4                          × 1.585               Thus a 6 dB decrease in sound level is about
           5                          × 1.778           a half, just as an increase in 6 dB was about × 2.
           6                          × 1.995               So you can easily calculate the actual pressure
           7                          × 2.239           ratio of two sound levels from their difference in
           8                          × 2.512
           9                          × 2.918                                           dB
                                                                       p final
          10                          × 3.162                                   = 10        20

          11                          × 3.548                          pinitial
          12                          × 3.981               But what about the case when you know the
          13                          × 4.467           ratio of the final to the initial pressure level and
          14                          × 5.012           you want to find the corresponding dB change?
          15                          × 5.624               Here you must understand logarithms. You
          16                          × 6.310           take the logarithm of both sides of the equation
          17                          × 7.080           above to get
          18                          × 7.943
                                                                                          dB 
          19                          × 8.913                        p final                  dB
          20                          × 10.000                log10            = log10 10 20  =
                                                                     pinitial                 20
                                                                                               
     Thus for any amplitude of an oscillation,
including AC voltages in electrical circuits, it is        Rearranging this equation to get the unknown
common to refer to times 2 in amplitude as a            dB on the left hand side gives
“6 dB increase”.
     Just in case there are some of you who are                                      p 
not yet completely bored I will now raise a final                      dB = 20 log10  final 
point; one that I introduce to show the connection                                    pinitial 
with some mathematics you may have learned, or
at least have been exposed to.                              The remarkable fact about this equation is
     Since the ratio of the final pressure level to     that, since it is dB that you perceive in a sound
the initial pressure level for an increase of 1 dB is   level change, you naturally process sound levels
the twentieth root of 10, i.e.                          logarithmically. And since all your senses are
                                                        based on detection of percentage changes, you are
                                                        essentially a logarithmic processor. (Bet you
            ratio for 1 dB =         10 20              didn't know that. Bet you didn't necessarily want
                                                        to know it.)
                                                            What I have been telling you to now is the
then for n dB the ratio is                              use of decibels to express changes in a sound
                               n                        level from one value to another. This is the most
             p final                                    common use of decibels. However, in sound
                        =   10 20                       there is another usage in which 0 dB is given a
                                                        very specific meaning. That is a sound for which
    Thus for any number of decibels, n, including       the amplitude of the pressure oscillation is
fractions of a decibel, you can calculate the ratio     28 µPa, or for which the extent of the pressure
of final to initial pressures for any sound level       oscillation is 56 µPa. Earlier I had taken this to be
change. Notice that this formula even applies to        the human threshold level for the perception of
negative decibels, where the sound level actually       sound. The reason for the seemingly arbitrary
decreases. Thus if the sound level goes down by         value is that it is the pressure level at which sound
6 dB, then the formula gives                            will transmit exactly 10-16 Watts into one square
Lecture 9 - What is sound? What is Music?                                                                                            5

meter of surface area. (More about Watts etc.                                           As the frequency passes this range our
later.)                                                                            sensitivity again levels off until at some higher
    With 0 dB as the threshold for sound                                           frequency it drops very quickly. The frequency
perception, the pressure levels for any sound that                                 at which this drop occurs depends a great deal on
can be heard have positive dBs.                                                    your age and whether or not you have had any
    Returning to the definition of zero decibels                                   child-hood diseases that effected your hearing
for the threshold of the perception of sound being                                 and, of course, whether you have been listening to
a pressure oscillation of amplitude 28 µPa, the                                    too much rock music. For old people it is
absolute level of loudness for a pressure                                          typically about 8000 Hz while for young adults it
oscillation of amplitude p is given by                                             is about 15 000 Hz. For babies it can be as high
                                                                                   as 20 000 Hz.
                                                                                        Devotees of heavy rock music will claim that
                                                                                   frequencies well below 100 Hz are very important
                                                 p2                              in their music. (They could be because that is
                              db = 20 log10          −5 
                                             2.8 × 10                            about all it seems that they can hear.) However, at
                                                                                   these frequencies you are not really hearing the
    Some representative levels for various human                                   sound through your ears but feeling it through
activities are given in Hecht Chapter 11 Table                                     vibrations of your body. You only start hearing
11.4 (page 406 of 2nd edition).                                                    the sound, i.e. perceiving it through your ears,
    It should be noted that humans can detect                                      when its frequency is about 120 Hz or greater.
sounds that are at negative decibels when they are                                      The bump in sensitivity at about 3000 Hz is
in the frequency range of 2000 to 4000 Hz. This                                    not an accident of development. It is directly
is because of the enhanced sensitivity to sounds                                   related to your ability to perceive the direction of
in this frequency range. People with good                                          a sharp click. This is because you need response
hearing can detect sounds at a loudness of −10                                     in this frequency range to follow sharp clicks that
dB in this frequency range.                                                        last about one-third of a millisecond. This
    Overall, the sensitivity of humans to sound as                                 enhancement of the response is therefore the
a function of frequency is somewhat as shown in                                    result of natural selection of the way the ear is
the graph below. This graph shows that the                                         constructed. Looking at the diagram in Hecht
ability to hear sounds falls drastically for                                       mentioned earlier, it results from the external
frequencies below 400 Hz. Between 400 to                                           auditory meatus (ear canal) which reaches from
2000 Hz it is fairly constant until the bump in                                    the "earhole" in the outer ear to the tympanic
sensitivity I already mentioned between 2000 and                                   membrane (eardrum). This is just of the right
4000 Hz. This bump is the explanation of the                                       length to resonate, somewhat like a small pipe will
unpleasant "screech", such as that of chalk                                        resonate by blowing across it, at 3000 Hz. Thus
rubbed backward on a blackboard, associated                                        this canal is not just to allow the sensitive active
with sounds in this frequency range                                                components of the ear to be protected deep within
                                                                                   the hardest bone in the body (the mastoid) but
                                                                                   also to enhance the chances that this protection
                                                                                   will not be needed.

                                                                                        Of course, the frequency of sound is not
                                                                                   limited by our ability to hear it. It is just that
                                                                                   nature seems to have picked out just the range we
                                                                                   need to hear for our survival. Smaller mammals,
                                                                                   for example, have their frequency range adjusted
                                                                                   upward to the higher frequencies, the frequency
                                                                                   being roughly inversely proportional to the
                                                                                   animal's dimensions. Conversely, elephants have
                                                                                   their frequency range adjusted downward, again
                                                                                   roughly in inverse proportion to the animal's
                            100   200   400   1000   2000 4000   10 000   20 000   dimensions. In particular, the frequency of
                                              FREQUENCY - Hz
                                                                                   enhanced perception in the mid-range is the
                                                                                   resonant frequency of the ear canal of the animal,
                                                                                   and this is inversely proportional to the length of
Physics 101A - Physics for the life sciences                                                             6

that canal. Of particular interest are bats, which     2.273 ms but rather as a pure tone of the A above
navigate completely by sound. These creatures          middle C.
have frequency responses typically up to                    How do we know that the brain does not do
200,000 Hz and, for small bats, even higher. This      this by counting the cross-overs for one second
high frequency response is very important to           and adding them up to 440? Firstly, we know no
them because of the way they catch prey. They          mechanism by which the brain can count this fast,
emit high-frequency sounds as chirps and detect        whereas we do know that the brain has
the sound that reflects from the prey.                 mechanisms for timing intervals down to 10 µs or
     In modern medicine very high frequency            less. Secondly, a good musician can tell that the
sound is used to image the fetus in pregnant           tone is A440 in about one tenth of a second, i.e.
women. Typically such sounds can go to 2 MHz           in only 44 oscillations. And he/she can tell that it
(2 million cycles per second). In physics              is 440 Hz to within an accuracy of about 1 Hz!
research sound waves of up to several GHz              This cannot be done by counting cycles. It can
(billion of cycles per second) have been used.         only be achieved by observing the time interval
However, these very high frequency sound waves         between cross-overs.
are not propagated in air but rather, in the case of        The timing accuracy that we can accomplish
medicine in the liquids of the human body or, in       is incredible. For the perception of a 440 Hz tone
the case of physics in solids.                         to within 1 cycle it is 1/440, or about 0.23%, of
                                                       2.27 ms. That is about 5 µs!
Perception of the frequency of sound                        How we do this timing is not yet fully
     So far I have dealt with the amplitude of         understood. However, it is very likely related to
sound and how it relates to our perception of          the "flame-speed" at which neural pulses reach
loudness over various frequencies of sound.            the synaptic connections of the various dendrites
Turning now to our perception of frequency.            connected to an axon. This flame speed can be as
How do we do that?                                     high as 200 m/s. At this speed a time of 5 µs
      The perception of the frequency of a tone,       corresponds to a distance along the dendrite of
commonly referred to at the perception of its          1 mm. Thus length differences of 1 mm can
"pitch", is one of the great puzzles of science.       result in a timing accuracy of 5 µs. This distance
Many researchers believe that the secret of how        is a thousand times larger than the usual cellular
we do it will lead to a greater understanding of the   dimension of one micron.
basic processes within the human brain.
     Whatever, what we do know is that the             The Perception of Harmony
perception of the frequency of a tone is by the             In has been known from antiquity that certain
detection of the cross-over points in the pressure     tones will appear to be in harmony. This
oscillation graph when the pressure is going           harmony is the basis of music. The first to
positive. This comes about because it is at these      realize that there is a mathematical basis for
instants that the shear forces in the hair-cells of    harmony were the Early Greeks. Since music
the cochlea are the greatest and the hair-cells        was regarded as having been invented by the
consequently most likely to fire.                      Gods this confirmed in the minds of the Ancient
     To see what this means, consider a graph of       Greeks that mathematics was the language of the
the pressure oscillation for a pure 440 Hz note        Gods.
(the A above middle C).                                     The person who first reported this
                                                       relationship was Pythagoras (yes the man of the
                                                       right-angled triangle). He noticed that tones made
       p         A pure 440 Hz tone                    by plucking a stretched string held down at
                                                       certain points were in harmony with the tone from
      po                                    t          plucking the full string if the points left half the
           2.273 ms      2.273 ms                      string free, one-third of the string free, one-
                                                       quarter of the string free, etc. (This series of
                                                       fractions are therefore referred to as "The
   The cross-over points are 2.273 milliseconds        Harmonic Series".)
apart. The brain detects the cross-over points and          We now know that harmony is related to the
senses that they are indeed 2.273 ms apart. Of         actual frequencies of the sources. Thus a
course, it does not interpret this result as           frequency of 880Hz is in harmony with a
Lecture 9 - What is sound? What is Music?                                                                   7

frequency of 440 Hz, and a frequency of
1320 Hz is in harmony with a frequency of
660 Hz. When Pythagoras shortened the
vibrating portion of a string to one-third he was
increasing the frequency of the vibration 3 times,
and correspondingly for the other fractions. In                 B C D E F          G A   B C D
terms of timing intervals, the relationships
between these frequencies can be seen in the               Thus middle C and the C above middle C
diagram below.                                        form an octave because counting both C's they
                                                      span eight white keys. Similarly middle C and
               440 Hz
                                                      the G above middle C, counting both the C and
                                                      the G, span five white keys and therefore form a
                                                      fifth. Middle C and the F above it form a fourth
      2.273 ms          2.273 ms
         660 Hz
                                                      by spanning four white keys while middle C and
                                                      the E above it form a third by spanning three
                                                      white keys. The minor third spans middle C and
      880 Hz                                          the E-flat just above it. (It takes real musical
                                                      ability to see the harmony between middle C and
   1320 Hz                                                 The significance of these intervals to music in
                                                      general is that if tones having these relationships
                                                      are played in sequence they will form a "tune".
                                                      If they are played simultaneously they will form
    Thus a frequency of 880 Hz produces timing        "chords".
intervals between every second cross-over that are    Pitch is a Logarithmic scale too
equal to the interval between the cross-overs for          By now you may be having a sneaking
the 440 Hz tone. Similarly, the interval between      suspicion that there are the dreaded logarithms
every third cross-over of the 1320Hz tone is the      lurking here too. This feeling can be probably be
same as that between cross-overs of the 440 Hz        enhanced to a real fear by me pointing out that a
tone. Also, the interval between every third cross-   fifth plus a fourth is an octave.
over of the 660Hz tone is the same as the interval
between every second cross-over for the 440 Hz
    This matching up of intervals is what puts
different tones into harmony. It seems that the                     B   C D E   F G A B C D
simultaneous firings of the hair-cells at these
coincidental cross-overs gives a buzz that we call                         Fifth    Fourth
    In the terms of the music of the "Western"
civilization (i.e of Europe and North America),
two tones that have a frequency ratio of 2 form an        The mathematics of this confirms the fear.
"octave". Two tones that have a frequency ratio                  Fifth + fourth = Octave
of 3/2 form a "fifth". Two tones that have a
frequency ratio of 4/3 form what is called a                       3         4
                                                                        ×         = 2
"fourth", two that have a frequency ratio of 5/4                   2         3
form a "third" and two that form a ratio of 6/5           Simply put, musical intervals add by
form a "minor third".                                 multiplying. Also simply put, the multiplying of
    This seemingly weird system of notation is        ratios results in the addition of their logarithms.
explained by the layout of the horizontal lines on    To see this more fully, consider the layout of the
music sheets, which is perhaps more easily            piano keyboard over a range of several octaves:
understood by non-musicians in the layout of the
piano keyboard.
Physics 101A - Physics for the life sciences                                                                                            8

                                                                                           From your calculator you can quickly get this
                                                                                      to be 1.05946..., which is equivalent to an
 B C D E F G A B C D E F G A B C D E F G A B C D E F G A B C D
                                                                                      increase of 5.946..%.
       O CTAVE              OC TAVE                   O CTAVE                  A
                                                                           OC T V E        Thus, taking a value that is of sufficient
                 125.6 Hz               MIDDLE C                502.6 Hz              accuracy for music, a semitone is a 5.95%
                                        = 251. 3 Hz
                                                                                      increase in frequency.
                            125. 6 Hz             251.3 Hz                            The Western Musical Scale
     Thus it is equal octaves that have equal                                              Like the logarithmic scale for pressure where
distances along the keyboard, not equal frequency                                     there is need for an absolute value to be set, there
intervals. The frequency of middle C is 261.3 Hz                                      is a need in music for some defining frequency to
(see later). Thus the frequency of the C below                                        set the scale. In the case of pressure this was
middle C is 125.6 Hz while the frequency of the                                       28 µPa as the threshold of hearing. In the case of
C above middle C is 502.6 Hz. The frequency                                           modern Western music the frequency reference is
interval for the lower octave is thus 125.6 Hz                                        a tone of 440 Hz and is referred to as a form of
while the frequency interval of the next octave                                       the tone "A”. This particular form of the tone is
above is 261.3 Hz                                                                     therefore referred to as the "A440" tone.
     You can now perhaps see that this scale is the                                        All other musical notes are then derived from
same sort of scale as the decibel scale for                                           this by using the frequency ratio for a semitone.
loudness. The only difference is that now the                                         The result is the table below.
natural unit is × 2 rather than × 10.                                                      The A above middle C -            = 440 Hz
     Why not use the × 10 unit for frequency like                                          A#(BFlat) - One semitone up       = 466.2
we do for pressure levels?                                                                 B - One more semitone up          = 493.9 Hz
     The × 10 unit for pressure levels was an                                              C - One semi-tone up from B = 523.3 Hz
arbitrary unit based on the fact that we have ten                                          C#(DFlat)                         = 554.4 Hz
fingers. A civilization of beings that had 12                                              D                                 = 587.3 Hz
fingers would have used × 12 as the basic unit                                             D#(EFlat)                         = 622.3 Hz
for pressure levels. However, the unit for                                                 E                                 = 659.3 Hz
frequency changes is not arbitrary. It can only be                                         F                                 = 698.5 Hz
× 2. This is because of the way we process tones                                           F#(GFlat)                         = 740.0 Hz
to determine pitch. Frequencies changes that are                                           G                                 = 784.0 Hz
× 2 result in harmony, apparently for birds and                                            G#(AFlat)                         = 830.6 Hz
howling wolves as for us.                                                                  A - One octave up from A440 = 880 Hz
     As with the pressure levels being divided into                                        Musicphiles will know that this was not
smaller units called decibels of about 12. 2                                          always the definition of the musical scale in
percent increase, for music we would like to have                                     Western music. Early definitions had a much
smaller units for the frequency scale (often called                                   lower frequency for the A above middle C,
the "tone scale"). Here the number of sub-units is                                    sometimes even lower than 400 Hz. This has
arbitrary, and can depend on the culture that does                                    resulted in catastrophic destruction of precious
it. In Western music the × 2 unit is divided into                                     Stradivarius violins when their strings were
12 sub-units, which for historical reasons are                                        tightened up to produce the higher frequencies
called "semitones". This is why there are 12 keys                                     demanded by the modern scale.
on the piano keyboard in one octave, 5 black and                                           Now consider how well this musical scale
7 white. (The 8th white is the start of the next                                      produces the harmonic musical intervals. The
octave.)                                                                              octave is produced automatically, since the scale
     Thus a sequence of 12 semitones steps                                            is based fundamentally on that interval. This is
results in one octave. Since each semitone results                                    very important in that the octave is the most basic
in the same percentage increase the ratio that is                                     musical interval; one that even the most tone-deaf
involved in a semitone increase in pitch is given                                     philistine can appreciate. However, consider the
               fupper       1                                                         fifth. In terms of "tones" this is 7 semi-tones, in
                      = (2)12                                                         the above list spanning from C to G. This
               flower                                                                 interval is 1.059463094 raised to the 7th power,
                                                                                      which is actually 1.4983. While this is not
Lecture 9 - What is sound? What is Music?                                                                                           9

exactly 1.5 (it is about 0.1% off) it is close                   This is indeed the form of graph I showed
enough for the human perception system to say it            earlier of the perceived loudness for various
is a fifth. Now consider the fourth. This spans 5           frequencies of tones.
semi-tones, which is 1.059463094 raised to the                   Such a graph can now be used to explain the
5th power. This is 1.3348, which is again about             Sheppard staircase with which I began the lecture.
0.1% off the ideal of 1.3333, this time a little            The tones in this set were computer generated
high. However, again this is considered close               using only octave intervals. For example, the first
enough for human beings to hear it as a fourth.             tone you may have heard was an "A" made up of
     However, now consider the third. This is 4             all the A's in the audible range, from the subaural
semi-tones, which turns out to 1.260 and is                 A55 to the superaural, for most people, A14080.
almost 1% off the ideal value of 1.25. This
difference can be easily detected by a good                                                                      TO NE COMPONENT
musician. The situation is even worse for the
minor third, which is 3 semi-tones. The actual

                                                            Loudness - dB
value of the ratio is 1.189, which is more than 1%
off the ideal of 1.2.
     I will not go into the detail of why musicians
would adopt a scale that does not exactly fit the
requirements of the musical intervals, except to
say that it has to do with transposing music into
different "keys". For particular pieces of music
some serious musicians will adopt a definition of
the keys that does follow the requirements of                               55    110   220   440    880 1760 3520 7040     14080
harmony exactly, and claim that the music sounds
much better when played in these keys. However,                 Your brain would then tell you "This is an A".
for most musical purposes the deviations of the             However, but what "A"?
thirds and minor thirds from true harmony                       Here the subtlety of the “staircase” comes
doesn't seem to matter since only talented                  into play. The amplitudes of the various
musicians can see much musical harmony in                   components were adjusted so that the loudest was
them anyway. (Try them on a piano and see if                the A440. You then concluded that it was an
you can detect the harmony.)                                A440 tone, though a complex one.
     The progression of the musical tones of                    If the A tone was the first you heard then the
5.95% per semitone is yet another example of the            very next was an Aflat tone in which the
human processing of stimuli on the basis of                 frequencies of all the tones were decreased by
percentage changes. That is to say another                  5.95%. However, the amplitudes were now
example of Fechner's Law.                                   readjusted so that they still fitted a curve that
     Thus the perception of both the loudness and           peaked at 440 Hz, even though there was actually
the pitch of musical tones are on the basis of the          no tone at this frequency. The result is the
logarithms. The natural graph of the loudness of            distribution shown below.
a tone as a function of frequency is therefore a
                                                                                 NEW TONE                          PREVIO US TONE
graph of dB versus octaves.                                                      COMPONENTS                        COMPONENTS
      Loudness - dB

                                                            Loudness - dB

                      1   2   3   4   5   6   7    8   9
                                                  Octaves                   55    110   220   440   880 17 60 3520 7040   14080
Physics 101A - Physics for the life sciences                                                           10

     You now heard clearly that the pitch had gone     definition that the intensity definition. The
down by one semi-tone to A-flat. This continued        intensity definition is more commonly used in
for the next semitone until after 12 semitones you     physics and engineering while the pressure
are back, of course, to A.                             definition is more commonly used in audiology
     Throughout this whole sequence the level of       and music, and music recording. If you are
all of the components of each tone fitted the curve    confused by the two definitions just follow the
that peaked at 440 Hz. By doing so the barely          definition given here and you will be safe.
audible 14080 Hz has moved down to the quite               However, if you consult other sources on this
audible 7040 Hz while the 55 Hz tone has moved         definition you will find different numbers for the
down to the inaudible 27.5 Hz. The 27.5 Hz tone        pressure at the threshold of sound (i,e. Po)
can then be safely discarded and the 14080 Hz          Usually it will be 20 µPa, instead of the 28 µPa,
tone reintroduced without anybody noticing.            given here. Authours using 20 Pa are using the
This is, of course, just the original tone set and     "effective pressure" of the oscillation, which is
the cycle can be repeated.                             1/√2 of the amplitude and which is derived from
     Incidentally, the lengthening of the duration     the effectiveness of the pressure oscillation in
of the notes was a red herring. It was designed to     delivering energy to the ear.
fool you into thinking that this lengthening was           But that is no doubt more than you wanted to
the cause of the illusion.                             know at this time.
     Again, this illusion is not based on a fault of
hearing. It is only people who can tell pitch that     Example problems:
will experience the illusion. Thus pitch detection
by humans is much more than just a simple              1. A sound level goes up by 30 dB. By how
detection of frequency. It involves a complex             much does the pressure level increase?
processing of time-intervals in the pressure
oscillations on your ear-drums for possible
decoding of significant messages. This system
was not designed to cope with the very unnatural
sounds made by a computer to form the
Sheppard staircase.

Relevant material in Hecht
Chapter 11- Sections 11.7 and 11.8. There are
no solved problems relevant to the course.

    You will note that Hecht uses an alternate
definition of the decibel based on sound intensity
rather than sound pressure. They are both
equivalent in giving the same loudness for the
same decibel level. The intensity definition has       2.   A sound pressure changes from an
the minor advantages of having 10 in front of the           amplitude of 0.025 Pa to an amplitude of
logarithmic definition instead of the 20. For               0.01 Pa. What is the dB change?
those with an interest in mathematics this comes
about because the intensity of a sound wave is
related to the square of its pressure variation
    There is also a minor advantage in the
denominator of the ratio being 10 rather than
the 2.8 × 10 that appears in the pressure
definition. This comes about because a pressure
oscillation of amplitude 28 µPa gives an intensity
of 10-12 Watts per square meter (the SI unit for
    However, I have found that students have
fewer problems understanding the pressure
Lecture 9 - What is sound? What is Music?                                                             11

3.   You decide to invent an even-tempered           4.   But will you have anything close to a ”fifth”
     musical scale that actually has 8 equally            on your piano?
     spaced tones. (That is what "even-tempered"
     means.) After all, an "octave" should have 8         The tone intervals will be:
     tone intervals shouldn't it? What will be the
     percentage change in frequency between               1 tone = 1.0905
     tones?                                               2 tones = 1.0905 x 1.09905 = 1.189
                                                          3 tones = 1.2968
                                                          4 tones = 1.4142
                                                          5 tones = 1.5422
                                                          6 tones = 1.682
                                                          7 tones = 1.834
                                                          8 tones = 2.0

                                                          The closest interval you would have to a fifth
                                                      would be the 5-tone interval 1.5422, which would
                                                      be about 3% off from what it should be, i.e. 1.5.
                                                      Musically, this would be intolerable.
Physics 101A - Physics for the life sciences   12

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