Today, we learn what makes a musical tone. The
University of Houston's College of Engineering
presents this series about the machines that make
our civilization run, and the people whose
ingenuity created them.
Change was afoot by 1600.
For 2000 years the alchemical philosophers had
tried to understand reality through reason. They'd
tried to deduce truth. Now a new breed of
philosopher was inventing the process of
observation that we call the scientific method.
Take the matter of determining what sound is. Ever
since Pythagoras plucked strings around 530 BC,
we'd known that the length of a taut string
determined its pitch. Pythagoras used different
ratios of string length to build musical scales.
Halve the length of a string and you raise its
pitch an octave. Two-thirds the original length
raises the pitch by a fifth (from do
to sol), and so forth. Pythagoras
couldn't tell us what sound was, but he lent a
cosmological significance to different pitches.
By 1600, the last alchemists were still talking
about Pythagoras's harmony of the spheres. But the
new experimental scientists noticed you could vary
pitch without changing a string's length. Galileo's
father Vincenzio was a famous lutenist and music
theorist. In 1590 he wrote what every lute player
knew -- that you changed the pitch of a string by
increasing its tension or its density as well as
its length. Vincenzio's famous son Galileo clearly
explained what that meant. The vibrating string
drives vibrations in the air. A pitch is created by
air vibrating on the eardrum. The rate of vibration
is what determines pitch.
It was Decartes's closest friend and supporter, a
philosopher named Marin Marsenne, who converted
that to a formula. In 1637 Marsenne published a
treatise on music and musical instruments, his
Harmonie Universelle. The title was a
lingering echo of Pythagoras. But in it Marsenne
showed that, while a string's frequency varies
directly with its length, it also varies as the
square root of the tension divided by the string's
Marsenne also recognized overtones, but he was
never able to tell there they came from. No one yet
understood that a string can support shorter waves
along with the one that runs its full length and
produces the fundamental tone.
Another question outlived Marsenne: How do
vibrations travel in air? Newton finally solved
that one in his Principia, but he also
came up short. He thought the air temperature
stayed constant as sound passed through it.
Actually the temperature rises and falls slightly,
making sound travel faster than Newton predicted.
We didn't get that part right until the 19th
An understanding of sound had to be in place before
we could do systematic acoustical design of halls
or instruments -- before we could standardize
pitches (more on that in another episode). It's a
simple fact that once we left the old alchemy, not
only did our understanding of music change: so too
did the very character of the music we listen to
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
Dostrovsky, S., Bell, J. F., Truesdell, C., Physics
of Music. The New Grove Dictionary of Music &
Musicians, (Stanley Sadie, ed.) Vol. 14, pp.
Lindley, M., Pythagorean Intonation. The New
Grove Dictionary of Music & Musicians,
(Stanley Sadie, ed.) Vol. 15, pp. 485-487.
See also the relevant Encyclopaedia
Tyndall, J., Sound: A Course of Eight lectures
Delivered at The Royal Institution of Great
Britain. New York: D. Appleton and Company,
The matter of standardizing pitch is the subject of
Marsenne's formula actually said that the frequency
of a vibrating string varied as the inverse square
root of its cross-sectional area. That would
be correct so long as all strings were made of the
same material. It is more correct to say that it
varies in proportion to the string's density
expressed in grams per meter of length or pounds
per foot of length.
Figure from Tyndall's book on
sound showing how one
might bow a Pythagorean monochord to get overtones
The Engines of Our Ingenuity is
Copyright © 1988-1998 by John H.
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