The guitar is an instrument that encompasses many of the ideas that were presented in math It functions as a string instrument combined with a Helmholtz resonator that can vary its timbre



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The guitar is an instrument that encompasses many of the ideas that were presented in math 5. It functions as a string instrument combined with a Helmholtz resonator that can vary its timbre. This project involves the study of how my guitar, a Taylor 710ce, and how other guitars function and produce the characteristic sounds they have.

On its most elementary level, a guitar is a six stringed instrument that produces sound by plucking or picking the strings and pressing them against what is called a fret board. These strings have varying amounts of mass to help them oscillate at their desired frequencies at ideal tensions. A guitar usually has its lowest open string (its heaviest) tuned to an E4 and the highest (its lightest) tuned to an E6, some guitars however can play almost up to an E8 using the fret board alone. The frequency that each string plays is determined by the equation F1 = Velocity of the string / 2(length). The speed of sound on the string is determined by the equation V = (Tension of the string / Mass per unit length)^(1/2). Changing any one of these variables can produce a wide range of notes on all of the strings.



Shortening the length of the string is the primary way to play different notes on a single string. This is done by placing the finger on a string and pressing it against the fret board. A fret board is a piece of wood on the neck of the guitar under the strings with many rows of metal bars called frets emerging from the surface. Pressing the string down one fret closer to the bridge of the guitar corresponds to raising the fundamental frequency of the string by one semi-tone; this is basically shortening the length part of the equation. The pitch can also be changed by bending the string with the finger while holding it down; this corresponds with increasing the tension on the string which in turn raises the frequency. The final way to alter the pitch of the string is to force it to resonate at certain modes. This is done by placing the finger on a node of the lowest desired mode.

An acoustic guitar has to generate these tones using just the energy input from the player. However this is an extraordinarily small amount of energy used to produce a sound that is both audible pleasant. It is a constant battle between the guitars sustain and its volume for a guitar to produce music. When the strings vibrate, they actually displace very little air and therefore their vibrations in the air do not contribute much to the pressure changes that you interpret as sound. Most of the energy instead is dissipated into the guitar through what is called the soundboard, found underneath the bridge. This causes all of the wood in the guitar to vibrate along with the air inside of the body. The wood and air form very complex vibration modes, the following picture is an analysis of the vibrating body of wood and air of a model guitar, displaying varying amplitudes and modes that resonate at different frequencies. The sound hole under the string projects this sound well and also functions as a Helmholtz resonator which will be discussed later3. This way of producing a loud sound comes at a cost mainly that the strings vibrate for a much shorter time. Since all of the energy in the system remains the same (including the guitar and the air) decay time is drastically decreased in order to produce a higher intensity of sound. If the string was suspended across two poles nailed into concrete, it would vibrate for a very long time but would not produce much sound.2

All guitars have the capability of using these techniques to play a wide assortment of notes at an audible volume, however perhaps what is more interesting is the vast assortment of timbres that every different guitar is capable of producing. Each individual guitar has its own general timbre because every part of the guitar is unique and has an effect on the timbre of the instrument. The most influential parts are the types of strings that are used, types of wood, and general body shape. Instruments made of lighter wood tend to have a brighter sound as the attenuate to higher frequencies better. Instruments made of heavier denser wood usually have a smooth sound with a mellow bass. Inertia plays a big role in wood timbre. Heavier wood is harder to move and has the most inertia so it vibrates more slowly like lower frequencies. Lighter wood vibrates more quickly because it has less inertia and presents faster frequencies. There are different types of strings that guitars can use as well; steel and nylon are two popular types that are used on two different types of guitars. Steel strings tend to have a brighter sound and nylon strings have a smooth sound with more bass. The guitar bodies that these two different strings are placed on are also structurally different which contributes to their different timbres.

In order to demonstrate a general display of timbre, I decided to record my guitar while picking the low E string on four different spots. Each area would produce a note that sounded much different than all of the others. The positions picked were right next to the bridge, between the sound hole and the bridge, right over the sound hole, and the middle of the string. When anything that produces a periodic signal and does not produce a pure tone is played, the fundamental frequency will resonate but integer multiples of this frequency called partials will also resonate. This is the case for an open ended pipe or string instrument.

Bridge: Middle:



These partials are excited based on whether or not the resonating mode has a node at the particular strike location. Partials that have nodes closer to the played area will resonate more strongly than others. This concept was best demonstrated in the recordings and spectrograms from the bridge pick and the middle of the string. When the bridge was played, it sounded very “buzzy” and it had a lot of “twang” like you might hear in old country music. This is because many of the higher partials of the note were greatly excited. The spectrogram demonstrated this with many evenly spaced bold lines high in the spectrum. The spectrogram of the string being plucked in the middle also demonstrated an important concept. When compared to the other spectrograms, it was apparent that every even partial appeared to be missing or was much fainter than the odd ones. This clearly demonstrated that the string was very excited in the middle and therefore even modes would have a difficult time vibrating because they all would have an anti-node in the center of the string. Since this area was vibrating the most, these modes were almost non-existent in the spectrogram. It is also important to note that the fundamental frequency was very prominent in this recording.

Finally I decided to do a recording of the Helmholtz frequency of my guitar. A Helmholtz resonator functions by vibrating a small column of air over a chamber of larger air and produces a low pitch. The resonant frequency of an ideal Helmholtz resonator can be determined by the equation:

F = (speed of sound/2(Pi)) X (area of the hole / (length of the column of air X volume))^(1/2)



However a guitar body is not an ideal Helmholtz resonator so I decided to stimulate the body of my guitar using white noise which contains all frequencies at equal amplitudes. This would cause the guitar to resonate at its natural frequency.

The guitar had a fundamental frequency of 94 Hz, an F#2, within the range that most steel string guitar bodies are tuned to, with other smaller peaks present at 160 Hz (E3) and 211 Hz (G#3), 94 Hz however appeared to be the Helmholtz frequency.

References:

1. Taylor Guitars Official Website. http://www.taylorguitars.com/



2. Wolf, Joe. Guitar Acoustics University of New South Wales.

http://www.phys.unsw.edu.au/music/guitaracoustics/construction.html

3. M. J. Elejabarrieta et al. Air Cavity Modes in the Resonance Box of the Guitar: The Effect of the Sound Hole. Journal of Sound and Vibration. Volume 252, Issue 3. pp 584-590. 2002


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