Physics IA Andr As Paz Ram A rez

Physics IA 2016 - Investigation into the strings of a tuned guitar
Andrés Paz

Introduction
When studying waves it was very stimulating to think about how music and physics relate. In the past I've done experiments that were related to music and perception, as to test if a certain group of people actually hear the differences between recordings played at different bit rates.

The guitar is a very, if not the most, common musical instrument. This gave me an initial idea to do something that had to do with it and physics in any way. In this paper, I'll try to investigate the tension in the different guitar strings by measuring their frequency and knowing their linear mass densities. The reason behind this is that it seems reasonable to think that the tension in them should be similar otherwise the guitar neck would twist as some strings will need need much more tension than others. Also it would be very hard to play if some strings were very loose and others were very tight.

The purpose of the investigation has more than one relevant relationship, therefore I decided to make a separate experiment to prove the relationship and then I proceeded to study its behaviour in a guitar, where I was interested in investigating how the tensions of each string changed or not, according to the frequency they produced when struck.

Research Question
How is tension related to frequency in the strings of a tuned guitar?

Independent variable: frequency
Dependent variable: tension
Constant variables:
length or wavelength (from the bridge to the fret)
mass per unit length (for each individual string)

Hypothesis
According to the theory and to a basic knowledge of how guitars are built, I expect the tension in all the strings to be roughly the same.

Theory
The vibrations in a guitar string are examples of standing waves. These waves can satisfy the same relationship that originates from the definition of waves:
v=fλ
Where v is the speed of the wave, f is the frequency and λ is the wavelength.
In this investigation, I am concerned in how guitars manage to produce the sounds they do. A direct way of seeing this is by rearranging this equation in terms of frequency (or pitch):
f=vλ
It is then seen that frequency relies upon the speed of the wave and its wavelength. We also know that the speed of a wave depends upon the tension (T) and the mass per unit length (µ), which is given by the formula:
v=Tμ
This all meaning that if we want to change the frequency of the waves in the strings of a guitar, we must either change the strings tension or mass per unit length. The wavelength can't really be changed because all of the strings have the same length: from the bridge of the guitar to the fret that is being pressed (or the first fret if none is pressed)..
The equation defining the speed of a wave tells us that tension is proportional to the speed, meaning that when tension is increased, then the wave's speed will be greater. It also shows that the linear mass density (mass per unit length) is inversely proportional to the speed of the wave, which means that if a string happens to be very heavy then the speed of a wave on it will be lower.
This leads the way into incorporating the next equation, which is the most relevant to this investigation, and which I'll proceed to use in order to investigate the relation between tension and frequency in the strings of a guitar:
f= Tμλ
To show the graphical relation between tension and frequency can be done by doing a small experiment. A sonometer available in the physics lab and the free audio software called Audacity were used to perform this..

Fig. 1 - Sonometer with applied tension (T) (5T in this image)

Tension can be simply adjusted by turning a machine head, but I can not measure it that way. It is possible to work out tension by recording the frequency and knowing the linear mass density of a string, Besides, I had an interest to see the relation of tension in the frequency of the wave, so being able to measure the changes in tension beforehand was a better idea.
The mass used to put tension into the sonometer's string was measured with an uncertainty of 0.1 g, or 0.001 kg. To create different tensions I simply placed the mass on different parts of the lever. The first part represented the original tension, the second represented 2T, the third 3T and subsequently did the rest.
In order to measure the frequency I had to use Audacity. I simply recorded the sounds created for each tension and plotted them in a spectrum analysis, as seen below:

Fig. 2 Spectrum Analysis (in sonometer)

The uncertainties for frequency were set to ± 1 because the cursor in the software did not measure any decimal points.


Raw data for string in sonometer:

Mass/kg ± 0.001
Tension/N ±0.001


Frequency/Hz ± 1


Mean Hz
Unc. In mean Hz
1.367
(1T) 13.405
111
112
114
115
117
113.8
1.8
1.367
(2T) 26.811
167
174
162
172
159
166.8
5
1.367
(3T) 40.216
210
210
209
211
207
209.4
1.2
1.367
(4T) 53.621
236
237
237
237
237
236.8
0.2
1.367
(5T) 67.027
272
271
269
271
270
270.6
0.8

Processed data:
According to the theory, f is proportional to T, therefore the square root(s) for the tension values were calculated and compared to the average frequencies recorded with Audacity.

(Tension)/ (N) ±0.001
Frequency/Hz
3.661
113.8 ± 1.8
5.178
166.8 ± 5
6.342
209.4 ± 1.2
7.323
236.8 ± 0.2
8.187
270.6 ± 0.8


Fig. 3 Graph of Frequency (Hz) vs. Tension/ (N)

Plotting the values of the square root of tension against frequency, makes it easier to notice that their relation is proportional. As shown by Fig. 2, a LoggerPro graph of frequency against tension, the variables have a linear relation.

Changing the length of the string have an effect in the frequency of the wave as well. This relation is inversely proportional.

One can conclude saying that the vibrations on a string indeed behave as standing waves, according to the theory.

Application of theory in a guitar:

Method (frequency and mass per unit length were measured to work out tension in each string)

A guitar's strings all have the same length, which would be from the bridge to the fret that is being pressed (or simply the last fret in this case), so that means that to change the values for frequency (or pitches) one must then change the speed of the wave. As shown before in the theory, we can do this either by changing the tension in the string or by having different values for μ. In the case of the guitar, both the frequency and μ actually have to be changed for each string so that the resultant tension is similar in each string is similar. This is what I wanted to investigate. Synthesizing, to measure the tension in the six strings of a guitar can be done by knowing frequency, μ and length.

Measurement of frequency (f):
To measure the frequency I had to record the sound produced when each string in the guitar was stroked with Audacity. To ensure that I had accurate values, I made sure the recordings took place in a silent room where little or no significant noise was present. For the recording of each string, I stroked the string three times.


Fig. 4 Audacity recording process

To show the frequencies in each stroke, I used the "analyse spectrum" option. Although the software is accurate, I repeated this procedure two times for each string. The cursor for the spectrum was very sensible but not enough to give decimal places, so I decided to give the frequencies an uncertainty of ±1. Nevertheless, I think it is valuable to say that the program automatically gives a value for the highest peak in a selected area, as seen in the image below:



Fig. 5 Linear frequency analysis

Range of independent variable:

String


Frequency/Hz ± 1



Mean Freq.
E (E4)
327
327
327
327
327
327
327
B (B3)
247
248
248
249
248
248
248
G (G3)
194
194
196
196
194
196
195
D (D3)
146
146
146
146
147
145
146
A (A2)
110
111
110
110
109
110
110
E (E2)
82
82
82
82
82
82
82


Measurement and control of the controlled variables [L, and μ or m/L]:
To measure the length of the strings, a ruler with an uncertainty of ±0.1 cm was used. A measurement from the bridge of the guitar to the end of the first fret was made, as this was the length where the strings would be vibrating. The measurement was made two times to ensure it was correct. The length was set as 65.4 ±0.1cm.

Measuring the mass per unit length of each string could not be made directly because taking each string out to measure its mass and putting it correctly again in the guitar would be too complicated. Therefore, it was required to know the volume in order to get the mass, m=d v.
Firstly, to get values for volume I used the equation v=πr2l, where r is the radius of the string and l is the length of the (vibrating) string. Each string's diameter was measured with a micrometer caliper that had an uncertainty of ±0.01mm. This measurement was made several times for each string to ensure the reliability of the values. After this, the values were halved in order to have values for each radius. Next, to measure the density of each string, an internet database with different material densities was used. The first two strings were made out of plain steel, while the rest of the strings (third to sixth) were all bronze wound strings. After this, each string's mass could be calculated. Thus, the individual values for m/L or μ could finally be calculated as well.

String
Material
Density (kg/m3)
Diameter/±.01mm
radius/±0.00001m
Length/±.001m
Volume (m3)
Mass (kg)±0.000005
µ=m/L (kg±0.000005/±.001m)
E (E4)
Plain steel
7850
0.27
0.000135
0.654
3.74E-08
0.000294
0.000449
B (B3)
Plain steel
7850
0.03
0.000185
0.654
7.03E-08
0.000552
0.000844
G (G3)
Bronze wound
8700
0.05
0.000285
0.654
1.67E-07
0.001452
0.002220
D (D3)
Bronze wound
8700
0.08
0.000405
0.654
3.37E-07
0.002932
0.004483
A (A2)
Bronze wound
8700
1.05
0.000525
0.654
5.66E-07
0.004927
0.007533
E (E2)
Bronze wound
8700
1.32
0.00066
0.654
8.95E-07
0.007786
0.011906

To control these variables didn't require me to do something special. To ensure the length was the same for all strings I played all of them without pressing any fret. To ensure the mass per unit length was constant for each string just required me to use the same string for each recording. I used the same six strings along the experiment.

Solving for tension (T):
Rearranging f= Tμ2L for T would give the equation:

f 2L=Tμ T=f 2L μ T=4f2L2μ

With this equation I was able to work out the tensions in each string and then use this to see how the frequencies and µ in each string affect the tension. As I said before, I expected the tensions to be roughly the same, but this was not really the case.

String
µ=m/L (kg/m)
Frequency/Hz ±1
Tension/N
E (E4)
0.000449
327
79.895
B (B3)
0.000844
248
86.38
G (G3)
0.00222
195
140.48
D (D3)
0.004483
146
158.9
A (A2)
0.007533
110
151.62
E (E2)
0.011906
82
133.15


Summary of data (include units and uncertainties, justify uncertainties):
Data: length, linear mass density, frequency and tension for each string

String
Diameter (±.01mm)
radius (±.001m)
Length (±.001m)
µ=m/L (kg±0.000005/±.001m)
Frequency/Hz ±1
Tension/ N
E (E4)
0.27
0.000135
0.654
0.000449
327
79.895
B (B3)
0.37
0.000185
0.654
0.000844
248
86.38
G (G3)
0.57
0.000285
0.654
0.00222
195
140.48
D (D3)
0.81
0.000405
0.654
0.004483
146
158.9
A (A2)
1.05
0.000525
0.654
0.007533
110
151.62
E (E2)
1.32
0.00066
0.654
0.011906
82
133.15


Processing Data:
It is important to point out again that in the case of the guitar, six different strings are being compared, therefore the only value which remains constant is the length. In order to have a more complete approach to the relation between frequency and tension, it is also good to know that frequency is related to both tension and linear mass density. For this reason, both variables are valuable to compare in order to see the relation between the tensions in a guitar.


μ (±0.000007kgm/L±.001m)
Tension/ (kgs)
Frequency/Hz ±1
0.02119
8.938
327
0.02905
9.294
248
0.04712
11.852
195
0.06695
12.606
146
0.08679
12.313
110
0.10911
11.539
82


Graph:


Fig. 6 Tension ( N) vs. Frequency (Hz) for the strings of a guitar


Conclusion:
The strings have what I would consider strongly differing values for tension. This does not support my initial guess, which I expected to be right because of my knowledge of how guitar strings are built.

Firstly, I explained this by mentioning some things: I can notice that between the strings that are made from the same material, hence those that have the same density, they have a small range of difference between them. This may say various things. Either there's an initial mistake in their calculated densities and therefore their tensions differ, or, the strings were not from the same manufacturer, thus they were not built in order to require similar tensions.

Nevertheless, after evaluating my results again and looking for more information about tensions in the the strings of a guitar, I found a free web application that calculated tensions and was able to compare string sets (http://stringtensionpro.com/). Information or values for string, tuning, material, gauge, (no need for density, as you choose the materials) were needed to get tension. I placed my calculated values and the result for tension was very similar to my original results.

This leaves me with space for an interesting conclusion. Even though I considered my calculated tensions to not be in an acceptable range, and I thought that the reason behind it could be fairly justified, an external calculator confirmed them. So that means that actually, the tensions in a guitar's string are made to have 'roughly' the same tension, which is regulated through the changing of density and mass per unit length.


Values for tension using D'addario's "String Tension Pro" web application.


Evaluation:
Improvements:
I spent a lot for my method.
I could have measured m/L differently. I think much more consistent results would have been collected if I had measured the masses in each string by taking them off the guitar, measuring and then putting the string back.
My research question was easily answerable by the experiment in the sonometer, but to say I got a clear relationship between tension and frequency in the strings in a guitar is greatly arguable. The guitar is affected by many other factors that compliment or influence each other.
My hypothesis was correct, but I thought it wasn't. The reason I considered the possibility of having incorrect results could be explained by considering that the strings came from different manufacturers, provided that I got the guitar from my college's music room, and probably its strings have been changed constantly. Another reason was simply that the strings were maybe too old, and for that reasons their tensions were different. A third reason was that maybe the way I calculated m/L was incorrect.
The results were confirmed by an external calculator. This meant that actually my results were inside what can be considered "similar tensions". It is interesting to point out that despite all my reasons to consider my results incorrect, they were proven by something else.

Resources:
http://www.student.thinkib.net/physics
http://www.bsharp.org/physics/guitar
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html#c2
http://www.engineeringtoolbox.com/metal-alloys-densities-d_50.html
http://hypertextbook.com/facts/2004/KarenSutherland.shtml