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\begin{document}
\centerline{{\Large {\bf {\em From Music to Mathematics: Exploring the Connections}}}}
\vs
\centerline{{\large {\bf Gareth E. Roberts}}}
\vspace{.25in}
\centerline{\Large {\bf The Monochord Lab: Length Versus Pitch}}
\vspace{.25in}
\noindent{ {\bf Names:} \underline{\hspace{6in}}
\vs
\nin {\bf Note for instructors:} Since the point of this project is to derive (and hear) the
Pythagorean scale, this lab should be taught before proceeding to the material
in Chapter~4. The lab requires the use of a monochord (a one-stringed instrument)
containing a movable fret that easily allows different string lengths to be plucked.
Monochords vary in prices depending on quality. The company Artec Educational sells a
kit to construct a monochord for under~\$15. It is also important to have a ruler
running parallel to the string to determine string lengths.
Students can work in small groups, with one monochord
per group. The project has two parts; the second part should not be started (or viewed) until the
first part is completed correctly.
\vs
The goal of this lab project is for you to explore the relationship
between the length of a string and the pitch sounded when the
string is plucked. The device you will use to investigate this
relationship is called a {\bf monochord}. You will recreate a famous and ancient
musical scale known as the {\bf Pythagorean scale}.
You should work in a group of three to four people (one monochord
per group) answering questions and filling in your results as
you complete the lab. Please turn in one lab report per group,
listing the names of all the members at the top of the first page.
After the lab is complete, we will perform some simple pieces of
music as a class. Before beginning the lab,
please read the safety instructions below.
\vspace{-0.15in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Safety}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{itemize}
\item The end of the string is sharp. Be careful when handling the monochord
near the tuner end of the instrument.
\item Over-tightening the string may cause it to break, allowing the two broken
pieces to fly around somewhat. The broken ends will be sharp, so handle any broken string
carefully. Keep your face away from the string so that you do not get hit in the
eye by a breaking string.
\item This instrument is not for ``twanging'' as rock guitarists play their instruments.
This type of playing may generate a louder sound, but the string life is reduced significantly.
Sufficient volume may be generated with just a gentle ``pluck'' of the string.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{The Pythagorean Scale}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As legend goes, the great Greek mathematician and philosopher Pythagoras was walking
by a blacksmith's shop when he noticed that certain sounds of the hammers on the anvils
were more consonant than others. Upon investigation, he discovered that the
nicer sounds came from hammers whose weights were in simple proportions like
$\rat{2}{1}$, $\rat{3}{2}$, and $\rat{4}{3}$. After further musical experiments with strings and lutes, he and
his followers (the Pythagoreans) became convinced that basic musical harmony could be
expressed through simple ratios of whole numbers. Their general belief was that
the lower the numbers in the ratio, the better the notes sounded together.
Thus, the natural building blocks of mathematics (small whole numbers) were aligned
with the natural interval relationships used to create harmonious music. This helps
explain the devotion of the Pythagoreans to rational numbers, numbers that can
be expressed as the ratio of two integers.
Using the monochord, you will investigate this Pythagorean belief and
find the ratios used in the oldest musical scale, the Pythagorean scale.
The goal is to investigate the relationship between the length of the
string vibrating and the change in pitch (musical interval) produced.
\subsection*{Part I: Simple Ratios}
\begin{enumerate}
\item The tension $T$ of the string may be adjusted by tightening the peg at one end of
the monochord. What happens to the pitch as the string is tightened?
\vs
\item Place the sliding fret at exactly half the length of the string. Hold one finger over the fret and pluck the
string gently. What is the relationship between the pitch of this note versus
the note produced without the fret? If you are not sure about the relationship, you can
try finding the notes on a piano, although first you will have to ``tune'' the monochord (on
the open, unfretted string) to a note on the piano.
\ms
\item Place the sliding fret at exactly $1/4$ of the string's length.
Hold the string down over the fret and pluck the smaller portion of the string.
What is the musical relationship between this note and the one sounded
by plucking half the string?
What is the relationship between this note and the one sounded
on the open string?
\ms
\item Place the sliding fret at exactly $1/3$ of the string's length.
Hold the string down over the fret and pluck the two pieces to either side
of the fret. What is the musical interval between these two
pitches? Draw an important conclusion:
\nin Cutting the length of the string in {\bf half}
\underline{\hspace{1.5in}} (raises or lowers) the pitch by \underline{\hspace{1.5in}} (what musical interval?).
\item Next, investigate what happens if the ratio of the string lengths is $\rat{2}{3}$. In other
words, what effect does changing the string length to $2/3$ its original value have on the pitch?
\ms
\item Finally, investigate what happens if the ratio of the string lengths is $\rat{3}{4}$. In other
words, what effect does changing the string length to $3/4$ its original value have on
the pitch? Given that $\frac{3}{4} = \frac{1}{2} \cdot \frac{3}{2}$, how could this result have been
predicted using the answers to the previous two questions?
\vspace{0.5in}
\end{enumerate}
\nin Before proceeding to the next part of the lab,
check with your instructor to make sure that you have answered the questions above correctly.
\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Part 2: Ratios for the Pythagorean Scale}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given the ground work accomplished above, we are now ready to construct the entire
Pythagorean scale. Although it sounds like the usual
major scale, it differs in some important ways from the scale found
on a modern piano. These differences are explored in great detail in Chapter~4.
Thus far, we have uncovered three important facts concerning the relationship
between the ratios of string lengths and the corresponding musical interval.
\begin{enumerate}
\item String lengths in a ratio of $\rat{1}{2}$ are an octave apart. Specifically,
cutting the length of the string in half raises the pitch an octave. Conversely,
doubling the length of the string lowers the pitch an octave.
\item String lengths in a ratio of $\rat{2}{3}$ are a perfect fifth apart. Specifically,
cutting the length of the string by a factor of $2/3$ raises the pitch by a perfect fifth.
\item String lengths in a ratio of $\rat{3}{4}$ are a perfect fourth apart. Specifically,
cutting the length of the string by a factor of $3/4$ raises the pitch by a perfect fourth.
\end{enumerate}
Notice the musical and mathematical simplicity of these facts. Simple ratios lead to the ``perfect" intervals
of an octave, fifth, and fourth. It is no accident that these are the primary intervals of early music
(e.g., Gregorian chant used octaves, while Medieval polyphony used fourths and fifths).
They also underscore the critical tonic-dominant harmonic relationship
and the V--I and I--IV--V chord progressions commonly found in music.
It is also important to recognize that fact 3 can be derived from the first two facts.
Multiplying the length of the string by $1/2$ raises the pitch an octave.
Then, multiplying the length of the new string by a factor of $3/2$ will {\em lower}
the pitch by a perfect fifth (longer strings have lower pitches). Going up an octave
and then down by a perfect fifth is equivalent to going up by a perfect fourth
(e.g., middle C up an octave to the next highest~C and then down a perfect fifth gives F,
and the interval between middle C and this F is a perfect fourth). Since
\begin{equation}
\frac{1}{2} \cdot \frac{3}{2} \; = \; \frac{3}{4},
\label{Eq:MonochordMult}
\end{equation}
the net effect of shortening the string length by a factor of $3/4$ is to raise the pitch by a perfect fourth.
Notice that we multiply the two ratios in Equation~(\ref{Eq:MonochordMult}), as opposed
to adding them. {\em We are always interested in the ratios of string lengths, not their difference!}
\vs
The goal for Part 2 of this lab is to discover the remaining ratios in the Pythagorean scale.
The scale is essentially the same as the major scale, except the string lengths
are slightly different from the modern ones. We will derive this scale using only
facts 1 and~2 above and the circle of fifths (see Section~2.4 of the text).
For example, to find the second note of the major scale, we go up two perfect fifths
from the tonic and then down an octave. The net effect is raising the pitch
by a whole step. If the starting note is middle C, then up a perfect fifth gives G,
and up another perfect fifth gives D$'$ (the first two ``hours" of the circle of fifths).
We then go down an octave to find the note D just above the starting C.
Using the ratios given in facts 1 and~2,
what fraction of the full string do we take to obtain
the second note of the scale? Use a calculator to find the actual length of the
string (round to two decimal places) and play it with the open string to
hear the first two notes of the Pythagorean scale (Do and Re).
The next note to find would be the sixth of the scale since that is a perfect fifth
above the second. From there we can find the third and finally the seventh note of the scale,
the leading tone. Make sure that each note is obtained using only the
ratios from facts 1 and~2. Complete Table~1 and then try playing
the full scale, perhaps as a class. Do you hear any differences
between your scale and the major scale on a modern instrument?
\ms
\vs
\begin{table}[h]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}
\hline
\makebox[1.1in] {{\bf Scale Degree}} & \makebox[1.3in]{{\bf Solf\`{e}ge Syllable}} & \makebox[1.5in]{{\bf Interval}}
& \makebox[1in]{{\bf Ratio}} & \makebox[1in]{{\bf Length}} \\
\hline\hline
& & & & \\
1 & \mbox{Do} & \mbox{Unison} & $\displaystyle{ \frac{1}{1}} \cdot L$ & \\[0.2in]
\hline
& & & & \\
2 & \mbox{Re} & \mbox{Major Second} & & \\[0.2in]
\hline
& & & & \\
3 & \mbox{Mi} & \mbox{Major Third} & & \\[0.2in]
\hline
& & & & \\
4 & \mbox{Fa} & \mbox{Perfect Fourth} & $\displaystyle{ \frac{3}{4}} \cdot L$ & \\[0.2in]
\hline
& & & & \\
5 & \mbox{Sol} & \mbox{Perfect Fifth} & $\displaystyle{ \frac{2}{3}} \cdot L$ & \\[0.2in]
\hline
& & & & \\
6 & \mbox{La} & \mbox{Major Sixth} & & \\[0.2in]
\hline
& & & & \\
7 & \mbox{Ti} & \mbox{Major Seventh} & & \\[0.2in]
\hline
& & & & \\
1 & \mbox{Do} & \mbox{Octave} & $\displaystyle{ \frac{1}{2}} \cdot L$ & \\[0.2in]
\hline\hline
\end{tabular}
\caption{The lengths and ratios of the Pythagorean scale. $L$ represents the length of the full string. Give lengths in the last column
to two decimal places.}
\end{center}
\end{table}
\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Some Concluding Questions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
\item Find the prime factorization of the numerator and denominator in each ratio in the fourth
column of Table~1. For example, $3/4 = 3/2^2$. What do you notice about each factorization?
\vspace{1in}
\item According to Table~1,
what is the ratio of string lengths that are a whole step apart?
Be sure to check {\bf all} five whole steps in the scale (they should be identical).
This ratio, call it~$W$, is the factor used to shorten the string length
in order to raise the pitch by a whole step. Express $W$ as a ratio of
two integers.
\vspace{1in}
\item According to Table~1,
what is the ratio of string lengths that are a half step apart?
Be sure to check {\bf both} half steps in the scale (they should be identical).
This ratio, call it $H$, is the factor used to shorten the string length
in order to raise the pitch by a half step. Express $H$ as a ratio of
two integers.
\vspace{1in}
\item Recall that two half steps are equivalent to one whole step. Hence, it should be the
case that $H \cdot H = H^2 = W$. Is this equation valid?
Compute the quantity $H^2/W$, giving your answer as a ratio
of two integers and in decimal form (to five decimal places). Also give the prime factorization
of the numerator and denominator. This value is important in the theory of tuning and is
known as the {\em Pythagorean comma}. This comma, as well as the
problems it creates, is discussed in Section~4.1 of the text.
\end{enumerate}
\end{document}