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\title{Math, Mind, and Music}
\author{Quiz 1}
\date{February 25, 2005}
\maketitle

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\begin{enumerate}

\item Match the  descriptions of spectra 
\begin{enumerate}
\item Fundamental and all harmonics have approximately equal strength
\item Fundamental is strong, higher harmonics present but decreasing
in strength
\item Only harmonics 1,2,3 and 10,11,12 are present
\item Only harmonics 1,2,3 and 6,7,8 are present
\item All frequencies above the fundamental (not only harmonics) are
strong
\end{enumerate}
with the following corresponding subjective impressions of the sound:
\begin{itemize}
\item[(1)] Percussive/noisy
\item[(2)] Dissonant
\item[(3)] Full
\item[(4)] Sharp
\item[(5)] Two clearly distinct tones
\end{itemize}
\hfill(5)



\item Answer either (a) or (b), not both.

\begin{enumerate}
\item Recall that the open end of a pipe is always a region of maximal
wave amplitude, while the closed end of a pipe is always a region of
zero motion.  Does a pipe which is closed at both ends produce a full
set of harmonics?  Explain! 

\item Why do the sounds of two instruments always reinforce, and never
cancel out?
\end{enumerate}
\hfill (5)


\item Consider the following question:

Is the human auditory system's ability to discriminate frequencies
different when presented with \emph{purely sinusoidal tones} as
opposed to \emph{natural tones} having a full set of harmonics?


\begin{enumerate}
\item State and explain your expectation with regard to the \emph{just
noticeable difference}---the smallest perceivable difference in
frequency between two consecutive single tones of equal loudness.

\item State and explain your expectation with regard to the
\emph{limit of frequency discrimination}---the smallest difference in
frequency at which two tones played simultaneously are perceived as
separate.  

\item Describe an experiment to test your expectation for either (a)
or (b).
\end{enumerate}
\hfill (3+3+4)

\end{enumerate}


\section*{Solutions}

\begin{enumerate}

\item a4, b3, c5, d2, e1

\item
\begin{enumerate}
\item Yes.  The fundamental mode clearly has zero displacement at both
ends, and a single region of maximal amplitude at the center.  If you
divide the pipe in two equal halfs, then each of the halfs supports a
pattern of vibration that looks like the fundamental of the entire
pipe.  The same goes for a division in three, four, etc.  As this
corresponds to a doubling, tripling, etc.\ of the frequency, all
harmonics can be realized.  (In fact, this is exactly the same
situation as for a string.)
\item Quote from the \emph{FAQ in music acoustics}, the University of
New South Wales, visited March 3, 2005:

``The answer to this involves several different effects that
complicate the sound of musical instruments. To hear the effect of
destructive interference, you have first to eliminate each of these
effects, and it is rare that they are all eliminated together, which
is why you don't normally hear destructive interference in
practice. Neverthe less, when two instruments are nearly but not
exactly in tune, you do hear the phenomenon of beats (listen to the
sound files of beats). This is an example of constructive and
destructive interference: the slight difference in frequency causes
the phase relationship to change slowly. When the two waves are in
phase it sounds loud but, when they are out of phase it is soft. 

Beats between real musical instruments do have variation in loudness,
but the loudness doesn't usually go to zero. There are several reasons
for this: 

\begin{itemize}
\item First, real musical instruments rarely have exactly equal
amplitude, so even when they are exactly out of phase, their waves
don't cancel. (And they often have vibrato.) Even if they were of
equal amplitude, it is unlikely that you would be equally distant from
both. Further, this effect is enhanced by the logarithmic response of
your ears (see \emph{what is a decibel?} for details). If two waves
cancel to 99\%, the resulting sound is 40 dB softer than one
instrument alone. Listeners often judge that 40dB softer sounds about
1/16 as loud. So even if you achieve cancellation with this precision
(hard to do), the effect is not as spectacular as you might expect.
\item Second, real musical instruments don't play sine waves: they
have several harmonics. Even when the conditions are such that two
harmonics cancel, other harmonics may not. 
\item Third: unless you are in an anechoic chamber, the points of
cancellation are not where you calculate them to be, because of
multiple reflections off walls. 
\item Fourth: you have two ears. Even if one ear is at the point of
cancellation, the other is not. 
\item Fifth: your ears are most sensitive in the range 1--4 kHz, which
means wavelengths of ${\sim\!300}$ mm to ${\sim\!100}$ mm. Nearly
all instruments radiate fairly strongly in this range, so you must get
cancellation in this frequency range to get cancellation. However,
because of these short wavelengths the region of cancellation (if
there is one, in spite of the complications mentioned above) is very
small, and the chance of having even one ear at the point is small.
\item Sixth: instruments and peoples ears move on a scale of at least
10s of mm (often more) during performance.
\end{itemize}

Despite all of the above, it is possible to set up conditions under
which you can experience the interference effects. Simply set up a
sine wave source (eg.\ an electronic tuner) in a room. There will be
reflections off walls and other objects that cause the amplitude to
vary strongly from place to place. Cover one ear, put the other near a
wall, and move your head towards and away from it. If you were to
drive two identical speakers with the same signal but reversed in
phase, and if you did it in an anechoic chamber, then you should get
cancellation on the plane of symmetry between the speakers. If you put
one ear on this plane, and neglecting the reflections from your body,
you'd expect to get pretty good cancellation. (I've never tried this
experiment in an anechoic chamber, but I've certainly noticed the
effect of reflections from walls.) 

In Berio's \emph{Sequenza VII}, a sine wave is played throughout on
the note B4. It creates an eery ambiance: one doesn't know where it is
coming from and it seems to get louder and softer, for several of the
reasons discussed above, including the motion of performer and
audience.''

\verb+http://www.phys.unsw.edu.au/~jw/musFAQ.html#interference+
\end{enumerate}

\item 
\begin{enumerate}
\item The JNR curve grows much less than by a factor of two per octave
over a large portion of the audible range (see powerpoint slides).
Therefore, even in cases where the difference of the fundamentals is
below the JNR, the difference of one of the higher harmonics may
exceed it.  Moreover, the JNR actually increases at very low
frequencies.  Thus, comparing natural tones should typically be easier
than comparing pure sinusoidal tones.  Note, however, that a very
strong fundamental may mask this effect.

\item The limit for frequency discrimination is roughly 30 times
larger than the JND, so the same argument applies.

\item We'll describe an experiment for the JNR; an experiment for the
corresponding limit of frequency discrimination would be similar.

First, for a given test subject and fundamental frequency, determine
the JNR for pure sinusoidal tones by a separate experiment, or use a
reasonable value from the literature.  Next, choose a difference of
the fundamental frequencies which is just below the JNR.  We can then
generate a sequence of pairs of tones in which an increasing portion
of the energy is contained in higher harmonics.  There are obviously
many ways to do this; an elegant way is to have the energy per mode,
i.e.\ the square of the amplitude of a particular mode,
Poisson-distributed.  If our conjecture from (a) holds, there should
be a point at which the subject is able to reliably compare the pitch
of the two tones in a pair.  Randomization of the order in which the
samples are presented will help reduce adaptation effects and
interference of expectation.


\end{enumerate}
\end{enumerate}

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