Jenna Krieschel
Dr. Jeff Selden
Department of
Mathematics
PROPOSED REASEARCH PROJECT
Introduction
There are many notes on the violin that can be played
on more than one string. When bowing the violin, it is
a common practice to use the fourth finger position in
playing the note on the previous string instead of
playing that same note on the open string itself. There
is a common belief among players that the sound of
the open string note is somehow different if it is
played using the fourth finger. This difference, usually
called the tone of the note by players, is believed to be
from the body of the instrument changing the energy
as the sound is created. This ‘tone’ is a complement to
the idea of pitch, which is what is notated in the
written music and is actually the particular
wavelength that makes the fundamental frequency, or
sound of the note. The bowing of these notes creates
what is called a saw tooth formation from the bow
slipping and sticking to the string as the instrument is
played, as seen below.
Saw tooth formation of Bowing
Image from: http://waynestegall.com/politics/images/earthmusic/sawtooth.jpg
Importance of Study
This research may help define the cause of
differences in tone of the violin and provide a
mathematical insight into the sound of the violin.
This may also give physical mechanisms that
quantify the sound difference that violin players hear
from these notes played on different strings.
Background from: http://hattieh29.edublogs.org/files/2011/05/violin-1bijw9x.jpg
Wave Formation of a Saw tooth formation after a
Fourier Transform
Picture from: http://mathworld.wolfram.com/FourierSeriesSawtoothWave.html
Methods and Proposed Data Analysis
The saw tooth formations from bowing the instrument at the note regarded as the
middle b will be established from both the D and A strings either from past
mathematical data, if available, or will be estimated using a violin and a computer
program that will estimate this saw tooth formation from the sticking and slipping
of the bow on the string as played on a single instrument. The saw tooth formation
will be run through a Fourier Transform, a form of mathematical data analysis, that
should output something like the graph above, or that of the sound wave produced
by the instrument. The wave with the least nodes (the one with the fewest ‘bumps’)
is the one that will be regarded as the wavelength of the sound and with the
harmonics that are expected with that frequency of sound will be regarded as our
pitch. This wave and its harmonics should show up exactly identical in the two
wave formations of middle b. The rest of the formation should not be the same and
will give us the tone of the note. These differences in the wave formation will be
broken down and compared in hopes of finding the physical trait of the violin that is
creating that addition to the sound of the note.
Acknowledgements
Dr. Jeff Selden and Rachel Farris
The University of Wyoming Department of Mathematics
Susan Stoddard, Zackie Salmon, Pilar Flores and Talysa Stockert
The University of Wyoming McNair Scholars Program
Purpose Statement
This study will examine the differences between the
tone and pitch of the violin middle b as played on
both the D and A strings of the violin. In terms of
this study pitch will be regarded as the ‘note’ on the
treble cleft, and tone will be regarded as the clarity of
the sound. To clarify further, the note will be
regarded as the wavelength of sound that is expected
from that placement on the treble cleft. The tone will
then be anything that shows up that is not the
particular wavelength of sound that is expected in the
note.
Hypothesis
It is hypothesized that there will be a distinct
mathematical difference between the tone and the
pitch of the middle b when played on different
strings. When the middle b is played on each string,
as a violinist, I know that I will be creating the same
pitch because despite them being played on different
strings, they are the same in the music as the note on
the treble cleft. I expect to find that each different
way of producing the pitch will give a unique
measurably different tone.
Expected Results
Once these different tones are found, I expect to be
able to identify which physical trait of the violin is
creating the difference in tone. It is likely that these
traits will be that of string diameter, and string
tension.