Tone Versus Pitch in the Violin Middle b Using Fourier Series

1. Tone Versus Pitch in the Violin Middle b Using Fourier Series Jenna KrieschelDr. Jeff SeldenDepartment of Mathematics PROPOSED REASEARCH PROJECT Introduction Wave Formation of a Saw tooth formation after a Fourier Transform Purpose Statement 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. 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 Picture from: http://mathworld.wolfram.com/FourierSeriesSawtoothWave.html 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. 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. Saw toothformationof Bowing ExpectedResults 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. 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. 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 Background from: http://hattieh29.edublogs.org/files/2011/05/violin-1bijw9x.jpg