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\title{Math, Mind, and Music}
\author{Quiz 2}
\date{March 18, 2005}
\maketitle

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Solve 3 out of four questions for a 100\% score.  Extra credit is
possible.

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\begin{enumerate}

\item Psychophysics is interested in determining the relationship
between the magnitude of a sensation experienced in the mind and the
magnitude of the physical stimulus that gave rise to it (e.g.,
frequency of a tone and pitch).
\begin{enumerate}
\item Briefly (!) describe the mel scale. 
\item Why is the mel scale not an appropriate musical scale?
\item Briefly (!) describe the probe-tone technique, and what it is
used for.
\end{enumerate}
\hfill (5)

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\item Gestalt heuristics are powerful grouping mechanisms in auditory
scene analysis.
\begin{enumerate}
\item List at least four heuristics that play part in auditory stream
segregation.
\item Describe an example of competing heuristics (and the winner and
why).
\item Speculate about the relationship between (left/right)handedness
and pitch perception.
\end{enumerate}
\hfill(5)

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\item The list on the next page contains an intricate description of
the subjective characteristics of different key signatures from 1784.

Comment on the music-historical context that provided the basis for
such judgments.  (You are not required to analyze the descriptions of
single keys in any detail!)

Would you find similar descriptions in our times?
\hfill(5)

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\item The following statement can be found on the internet.
Find arguments in support of, or against this thesis.

``If the electromagnetic zone around the earth vibrates, it does so
with a frequency of 7.5 Hz because the speed of light is 300,000 km/s
and the circumference of the earth is 40,000 km. Therefore the
predicted strong harmonics of this vibration should have frequencies
of 7.5 Hz times the various harmonics numbers. Interestingly, the
frequencies resulting exactly match those used in Indian music.

\begin{verbatim}
Harmonic  h          48    54    60    64    72    80    90    96
Frequency h*7.5 Hz  360   405   450   480   540   600   675   720
Indian note          pa   dha    ni    sa    ri    ma    ga    pa
Western scale        F     G     A     Bb    C     D     E     F
\end{verbatim}

The modern standardised scale has A=440 Hz and the others adjusted
according to the equitempered scale which does not quite fit this
table. However the trend has been for A to increase with time and it
had got to 450 Hz before the standard was set. Based on Indian music,
the earth's natural resonance, a study of the rhythm speed for great
composers and on other evidence, I believe that 450 Hz is the true and
correct A.''
\hfill(5)



\end{enumerate}

\end{document}

\section*{Solutions}

\begin{enumerate}
\item
\begin{enumerate}
\item Steps on the mel scale are defined by ``equal perceived pitch
difference''.
\item The perception of (relative) pitch differences is not uniform
across the auditory range.  Thus, in particular, octaves don't have a
special role on the mel scale although they are a universal interval
of consonance.
\item TBW
\end{enumerate}

\item
\begin{enumerate}
\item Similarity of timbre, similarity of pitch, temporal proximity,
good continuation, location, onset and offset, experience
\item For example: timbre vs.\ smooth continuation.  Timbre (and
experience) usually wins---even if the flute line could be smoothly
continued by the piano, we will not have difficulty to segregate the
streams.
\item The right-ear preference for following the high-pitched stream
appears to be an asymmetry not unlike handedness.  Speculation:
Left-handed people also have a preference to locate the high-pitch
stream on their side.
\end{enumerate}

\item The list was written at a time where mean-tone temperaments
were common and well-tempered tunings started to catch on.  Thus, keys
that are only a few translation by fifths away from C major sound
pure; the further away from C a key is, the more out-of-tune the keys
get.  (Cf.\ descriptions for F\# major or G\# major.)

\item This statement appears to be complete nonsense for a number of
reasons.

\begin{itemize}
\item While it is true that the ionosphere of the earth can act as a
standing electromagnetic wave guide with a fundamental frequence at
about the claimed 7.5 Hz (often referred to as ELF vibrations), it is
neither a perfect wave guide, nor is the earth a perfect sphere.
Thus, the reasonance will not be very sharp, and scaled up to the 60th
harmonic, the error in frequency is likely huge, certainly more than
$\pm 10$ Hz.

\item The earth is not a string, but more like a drum.  I.e., the
natural harmonics of the earth will not be simple integer multiples of
the fundamental.  In particular, the higher harmonics will cluster
much closer together than for a string/pipe.

\item The text talks about electromagnitic waves, not acoustic ones.
Thus, the earliest stage the earths ELF field could couple into the
human body is the auditory nerve transmittion from the inner ear to
the brain.  However, since the human ear has a place coding for
frequency,  this signal has already nothing to do with the original
frequency of the acoustic stimulus.
\end{itemize}
If you are interested, an example (with explanation) of measured ELF
spectra of the earth can be found at
\verb+http://www.bath.ac.uk/~eesmf/eicrspe.html+.
\end{enumerate}
\end{document}
