How does atmospheric temperature ranging between 10℃ to 55℃ with a regular interval of 5℃ affect the frequency of E (1st string) and E (6th string) of an acoustic guitar?

Do the parents serve a major influence in choosing their children’s career? The statement is a poser which triggers manifold opinion. However, I personally believe that parental opinion do act as an anchorage in children’s decision making. For instance, my father is a sound engineer and spends almost entire working hour in listening and recording tracks and my mother is a professional singer. I was brought up in an environment encircled with musical instrument and other forms of art. Consciously or Unconsciously, I inherited their passion and dedication towards the art, culture and music.

 

Since childhood, I was mesmerised by the tone of the acoustic guitar which my father used to play at night. String instruments have always influenced my interest in the cultural world. Finally, it was in my third standard when I started acquiring a formal training in playing acoustic guitar. After playing guitar since an age of nine years the only problem while playing guitar is again its most mesmerising factor – the tone. Guitars have to be tuned properly almost every time it requires to be played. Due to lack of formal education on working mechanism of a guitar as well as string theory, till now, I thought the reason of detuning being loosening of keys of guitar. However, currently, after learning oscillation and harmonic motion, I learnt that there exist a relation between frequency and temperature. This theory has given born to a curiosity in my mind.

 

After contacting with one of our family friend settled in Boston, United States, I came across the fact that guitar strings are tightened with different intensity during the winters than that of summers in Boston. Establishing with the fact I learnt from Boston, I wanted to dive deeper into the effect of atmospheric temperature in the tune of guitar. I studied a few journals and articles on temperature dependency of frequency, and found a few equations where temperature affects the frequency of any oscillating object; but failed to find any direct information about the effect on temperature on specifically guitar strings. To find the answer to my question, I have decided to explore and find the answer to my question through this assessment.

Acoustic guitar is one of the most popular used musical instrument and also one of the most widely used musical instrument available in today’s world. This is because of its mesmerising soothing tone, portability, light weighted structure and also financial affordability.

 

Structurally, an acoustic guitar is divided into several parts as shown in Figure 1.

In an acoustic guitar six strings are present, which is tied by the keys at one end and by the pins placed at Bridge on the other side. The keys help to loosen or strengthen the intensity of tightness of the individual string. When the intensity of strain increases, every string pitch also increases and vice versa. The keys are there to tune the strings to it’s fundamental tone accurately.

 

Each string in guitar comprise a fundamental tone. From Figure 2, normal six string guitar corresponds to the fundamental tones of E, B, G, D, A, E from 1^st to 6^th string respectively.

Each string has its fundamental frequency which determines the tone of each string. The fundamental frequency of each string of an acoustic guitar is shown below in a tabular form.

As per Vincenzo Galilei , propagation velocity of a wave produced by a vibrating string is directly proportional to the square root of tension applied on the string. It is also inversely proportional to the square root of linear density of the string.

 

The mathematical expression of the above mentioned relation is stated below -

 

\(v ∝ \sqrt{\frac{T}{\mu}}\)

 

Here, v = the propagation velocity of wave,

T = string tension,

µ = linear density of string.

Experimentally, it has been found that,

\(v \sqrt{\frac{T}{\mu}}\)

 

Linear density = the mass of any body of unit length.

Hence, the previous equation can be written as follows -

 

\(v = \sqrt{\frac{T}{\frac{m}L}}\)  .........(1)

 

Here, m =string mass

L = string length.

Frequency is inversely proportional to the wavelength of the wave, considering that the wave velocity stays constant.

 

The mathematical expression,

\(f =\frac{v}{λ}\) .........(2)

 

Where, f = wave frequency,

λ = the wavelength of the wave,

v = wave velocity.

From equation (1) and (2 -

\(f = \frac{1}{\lambda}\sqrt{\frac{T}{\frac{m}L}}\) ........(3)

The law states that the fundamental harmonic nodes produced by a vibration lies at both ends of the string. So, if the length of string be L, then the wavelength of the vibration will be 2L. Thus, from equation (3), we concluded:

 

\(f = \frac{1}{2L}\sqrt{\frac{T}{\frac{m}L}}\) .........(4)

Length of any solid objects increases with an increase in temperature. This occurs because the kinetic energy of the object molecules increased. The mathematical expression for increase in length -

∆L = L₀ × α × ∆T ... ... ... (5)

 

Here, ∆L = the change in length of string,
α = co-efficient of thermal expansion,
∆T = change in temperature.

 

Co-efficient of thermal expansion is defined as the increase in length of a solid object of unit length for an increase in temperature of unity.

Let, the initial length of a string be L₀, the coefficient of thermal expansion of the material of string be α, the fundamental frequency of the string be f₀, mass of the string be m, tension in the string be  T_e, therefore, the expression of the frequency can be expressed as:

\(f_{o}=\frac{1}{2L_{o}}\sqrt{\frac{T _{e}}{\frac{m}{L_o}}}\)

 

Let, the string is heated and the change in string temperature to be ∆T. Therefore, the final length (L_f) of the string can be expressed as:

L_f  = L₀ + L₀ × α × ∆T

 

=> L_f  = L₀(1 + α∆T)

 

Therefore, the expression of final frequency (f_f) of the string can be expressed as -

 

 \(f_{f}=\frac{1}{2L_{f}}\sqrt{\frac{T _{e}}{\frac{m}{L_f}}}\)

 

 \(=> f_f  =\frac{1}{2L_{o}(1\,+\, α∆T)}\sqrt{\frac{T _{e}}{\frac{m}{L_o(1\,+\,α∆T)}}}\)           (6)

 

In the above equation (6), physical parameters such as initial length of the string (L₀), coefficient of linear expansion of string (α), mass of string (m), initial fundamental frequency of string (f₀) are constant parameters and do not depend on temperature or change in temperature.

 

Thus, the above equation (7) can be expressed as -

 

\(f_f  ∝ \frac{1}{∆T}\sqrt{T_{e}∆T}\)           (7)

 

Moreover, tension would be varied with temperature. This is because, as the temperature increases (increase in the magnitude of change of temperature), the string length would increase and hence the tension in the guitar string would decrease.

 \(Te ∝\frac{1}{∆T}\)          (8)

 

Combining equation (7) and (8):

 \(f_f  ∝\frac{1}{∆T} \sqrt\frac{{1}}{∆T}×∆T\)

 

\(∴ f_f  ∝ \frac{1}{∆T}\)           (9)

 

From equation (9), it can be stated that with an increase in the magnitude of change in temperature due to increase in temperature, the final frequency of the string will decrease.

 

Therefore, the change in frequency (∆f) of string due to increase in temperature can be expressed as -

 

∆f  =  f_f   −  f₀

 

As the final frequency (f_f) decrease with an increase in temperature, the difference in frequency would increase with an increase in temperature.

 

Therefore, mathematically, it could be expressed as -

 

∆f ∝ − ∆T

 

There is a negative sign before the notation of change in temperature because with an increase in temperature, the frequency would decrease.

In this exploration, two acoustic guitar strings, i.e., 3^rd string (G string) and 4^th string (D string) determines the temperature effect on change of guitar string frequency. These two strings are considered for the exploration as the E(1^st) and B strings are too thin, so they might melt during the experiment as temperature increased to 60°C. On the other hand, the E(6^th) and A strings are too thick, so often they gather rust layer and other impurities on them that cannot be separated without an experimental process. A different reason of not taking E(6^th) and A string was the resistance of the string. As these strings are thick they have higher cross sections. Thus their resistance decreases. Thus, it will take a lot of current to reach the required heat to make the temperature more than 35°C. 3^rd and 4^th strings, as they are placed at the centre of the guitar, their observed nature in these two strings would have been carried out in all the other strings.

 

A power supply (DC battery) was used and the principle of Joule’s Law was followed to heat to increase the guitar string temperature because this is the easiest way to increase the guitar string temperature without taking out the strings from the guitar. If the guitar string temperature is increased by any other method that will also increase the temperature of metallic parts and the body of the guitar. This would have created higher errors in experimental observations.

 

Thereafter, for increase in every 5°C in temperature of each of the two guitar strings, the guitar string was plucked with same intensity using a plectrum and the same position on the string and the frequency was measured using a mobile application.

In the research journal titled as – ‘Effect of temperature changes on the function of the electric guitar’ by Baškarić, Tomislav, et al. in the Proceedings of TEAM 2014 (2014): 472, it was concluded that guitar strings should be manufactured by material having less coefficient of thermal expansion to reduce the extent of effective increase or decrease in length of string controlled by the atmospheric temperature.

It is assumed that with an increase in temperature, the frequency of each string will decrease. This is because due to linear expansion of solid string, the length of string will increase with an increase in its temperature initiated by the increase in atmospheric temperature as shown in equation no. (5). Consequently, it will be loosened without even altering the keys of guitar. In guitar, if the strings are loosened, the pitch of the tone of string decreases. As pitch is directly proportional to frequency, the string frequency will decrease with increase in its temperature. Furthermore, with an increase in length of the string, the fundamental frequency of the string will decrease as discussed in equation no (6).

Temperature of string

The temperature of the string was considered to be the independent variable in this exploration. The temperature was increased from 25°C to 55°C with a regular interval of 5°C using an electric circuit. The temperature was varied over such a range because temperature usually varies over this range in India and other subtropical countries. The temperature of the string was increased by passing electric current through the string. When current passed through any string or wire, according to Joule’s Law of Heating, heat is dissipated by the string. As a result, the temperature of the string was increased.

Frequency of string

The frequency of tone produced by the string at each trial of temperature was the dependent variable in this exploration. Itwas because, frequency is the only driving parameter which was responsible for detuning of any string of guitar. The frequency was measured using a functional microphone in Google Science Journal Application on an android.

• It was made sure that the G string and D string of the guitar by adjusting the turning pegs until the frequency is reached.
• It was made sure that gloves have been worn.
• It was made sure that the aforementioned control variables are being controlled as well as possible and the surroundings are clear of any noise.
• Infrared radiation thermometer was used to measure the initial temperature of the guitar string.

• An acoustic guitar with brass strings connected in the guitar was taken.
• The strings were rubbed using a clean cloth to remove dirt and impurities from it.
• A DC Voltage Supply (battery) of 12 V is obtained and the two ends were connected with two different crocodile to crocodile probes.
• The free end of both the crocodiles probes were connected with two ends of the G string.
• It was verified that the metal parts of the crocodile probe was not in touch with any other string.
• The battery was turned on and the temperature of the string was being constantly observed using the infrared thermometer.
• The temperature was increased from 30°C to 50°C. At every interval of 5°C, the battery has been turned off and the G string was played once using a plectrum.
• The frequency of the string was measured using Google Science Journal Application on a smartphone with a fully functional microphone and noted.
• Once the procedure was completed, the battery is turned off and the guitar was left for some time so that the temperature of the strings fall back to the room temperature.
• After that, the same procedure has been repeated two times.
• Finally, the entire process is repeated for D string.