Physics IA Final Document PDF

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Physics Internal Assessment for Mr. Nguyen's class, THHS IB Physics. Received a 6. Do not plagiarize....

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1

IB Physics 2018/2019 25 February 2019

Guitar String Length Vs. Frequency IA Abstract: This experiment will measure the correlation between the length of a guitar string and the frequency of the pitch it produces by analyzing standing waves. The independent variable is the length of the guitar string (altered by fretting it), and the dependent variable is the frequency (Hz) produced. The experiment is controlled by collecting data from one guitar string (the low E2) on the same guitar. It is hypothesized that as the length of the guitar string becomes shorter, the frequency of the pitch produced will be higher, as the variables have an inverse relationship. After careful data analysis, this relationship was confirmed, with an overall error of 0.09%. Personal Engagement: Digital synthesis of music has become more prevalent in the industry, especially with the development of professional software such as FL Studio and Ableton. In addition, companies such as Apple have developed novice-friendly music interfaces such as GarageBand that allow anyone the opportunity to experiment with composition and production. Even professionals choose to utilize this simple application (or the extension Logic Pro) for their projects, piecing together loops for a unique sound. The undeniable user compatibility and availability of these platforms leads to a reliance on them, especially in the pop industry, and therefore inevitably limits the need for studio musicians. Even in a world where the precision of musical machinery erases the fallibility of human performances, the authentic sound and soul of a real instrument is unreplicatable. As a musician who has supported, promoted, and utilized digital instrumentation for all my music composition, I feel it is my duty to branch out as an artist and embrace the challenge of learning the guitar. I decided to choose this experiment as a way to train myself to play in tune.

2 Introduction: Sound is a wave created by vibrations of objects. The frequency of a wave is the number of cycles of the wave that happen in a given time period. Sound is produced on a guitar because of standing waves, in which the nodes (fixed points) of the strings stay stationary, and the antinodes oscillate back and forth on the strings. The strings vibrate from the bottom of the guitar or the bridge, to the top, called the nut (refer to Fig. 1 1). These strings are tightly stretched, and the amount of tension determines the pitch of the open string. Each pitch, when played in tune by pressing down on the fretboard or plucking an open string, has a particular established frequency. A chart for the pitches included on the low E string (E2-E3) is included below (Fig. 2 2). This also served as the reference for the theoretical frequencies used in this experiment.

Figure 1: Diagram of the parts of an acoustic guitar

Figure 2: Chart of reference pitches

The equation for frequency of a standing wave on a guitar string is as follows 3: √" #/%

f= 2 where T=tension, m=mass, and L=string length. !

1

PinedaI, James. “Parts of An Acoustic Guitar: Head Stock, Tuning Keys and What Are Included?” Best Beginner Guitar, Best Acoustic Guitar, Publisher Name Https://Bestbeginnerguitartoday.com/ Publisher Logo, 8 Nov. 2018, bestbeginnerguitartoday.com/parts-of-an-acoustic-guitar/. 2

“Guitar Fundamentals: Wavelength, Frequency, & Speed | Science Project.” Science Buddies, www.sciencebuddies.org/science-fair-projects/project-ideas/Music_p010/music/guitar-fundamentals-wavelength-frequencyspeed#procedure.

3

Nave, R. “Vibrating String.” Total Internal Reflection, hyperphysics.phy-astr.gsu.edu/hbase/Waves/string.html#c4.

3 Since string tension cannot be calculated on the guitar, it was taken out of the equation. The mass of the string is also negligible in this case. When tension and mass are omitted from the equation since they are controlled, the relationship between frequency and length can be derived: √𝐿

f𝛼 2𝐿→ f𝛼

𝐿 1/2𝐿−1 → 2

f𝛼

𝐿−1/2 → 2

f𝛼

1 2√ 𝐿

As demonstrated by the derivation, length is inversely proportional to frequency. The goal of this experiment is to confirm this relationship.

Materials: -

Acoustic-electric or acoustic guitar in standard tuning (EADGBE)

-

Tuner to measure frequency of pitch (www.alexdemartos.es/wtuner/ used 4 )

-

Meter stick with 1 mm intervals (uncertainty of +/- 1 mm)

Note: the reference chart of established frequencies for given pitches will be helpful during data collection

Figure 3: Diagram of Setup

4

Figure 4: Close up of Fretboard

Martos, Alejandro Pérez González de. Wtuner.com - Online Instrument Tuner, www.alexdemartos.es/wtuner/.

4 Procedure: 1. Make a chart for one octave of the theoretical perfect frequencies for the pitches of the given string. For reference, an octave is the interval between one pitch and another with double its frequency. In this case, the lowest string (E2) was used. In order to do this, you must know which notes are in one octave from the starting pitch and match that to the fret. For example, G2 is played on the 3rd fret. A complete chart of these values is included in the following section. 2. Measure the length from the bridge (bottom) of the guitar to the nut (top), then measure the length of each fret, beginning at the 1st and ending at the 12th (one octave). (Note: In figures 1 and 2, the meter stick is not positioned for measurement. It is there for the purposes of scale. In order to measure the fret lengths, the meter stick would be flipped around.) 3. Use the tuner to ensure the guitar is in standard tuning (EADGBE). It will automatically indicate when the note is in tune. If the tuner is an online application, ensure the microphone on the computer is on the highest sensitivity. This will give the most accurate frequency reading. 4. Beginning on the open string (no fret), pluck the string and record the frequency displayed. There may be very slight fluctuations as the note resonates, but try to be as precise as possible by recording data right after the string is plucked. Place your finger on the fingerboard as close to the first fret as possible and document the frequency. Continue this method for all 12 frets. In this case, you should be ending on E3. Repeat for 5 trials.

5 Data Collection and Processing: This is a table of the theoretical frequency values for one octave of pitches on the low E string. Included is the corresponding fret and string length (based on the measurements for my guitar specifically).

Table 1: Theoretical Frequency Values of E2-E3 and Corresponding String Length Fret and Length (cm, +/- 1 mm)

Pitch

Theoretical Frequency (Hz)

Open String (66.4)

E2

82.407

1 (62.8)

F2

87.307

2 (59.6)

F2#

92.499

3 (56.2)

G2

97.999

4 (53.1)

G2#

103.83

5 (50.2)

A2

110.00

6 (47.8)

A2#

116.54

7 (44.9)

B2

123.47

8 (42.6)

C3

130.81

9 (40.3)

C3#

138.59

10 (38.2)

D3

146.83

11 (36.2)

D3#

155.56

12 (34.2)

E3

164.81

These theoretical values were taken from Fig. 2.

6 Table 2: Raw Data: Measured Frequency Values of E2-E3 and Corresponding String Length Below are the measured values for 5 trials of the experiment. This is raw data, and was later altered in a separate table (Table 3) using Excel to form a graph based on the derived equation.

Fret and Length (cm, uncertainty +/- 1 mm)

Pitch

Measured Frequency (Hz) Trial 1

Trial 2

Trial 3 Trial 4

Trial 5

Open String (66.4)

E2

82.4

82.5

82.3

82.4

82.4

1 (62.8)

F2

87.8

87.7

87.1

87.2

87.5

2 (59.6)

F2#

92.7

92.9

92.7

92.6

92.7

3 (56.2)

G2

98.1

98.2

98.1

98.2

98.1

4 (53.1)

G2#

103.9

103.9

103.8

103.8

103.9

5 (50.2)

A2

109.7

109.9

110

109.9

110

6 (47.8)

A2#

116.6

116.3

116.4

116.5

116.4

7 (44.9)

B2

123.5

123.8

123.6

123.4

123.5

8 (42.6)

C3

131

131.1

131.1

131.1

131.1

9 (40.3)

C3#

138.8

138.5

138.4

138.6

138.7

10 (38.2)

D3

146.7

146.8

147

146.6

146.8

11 (36.2)

D3#

155.5

155.2

155.2

155.4

155.3

12 (34.2)

E3

164.5

164.6

164.8

164.6

164.5

7 Table 3: Derived Data Included in this table are the inverse length measurement; the average, maximum, and minimum frequencies per pitch from all 5 trials; and the uncertainty of the frequency measurements.

1/2√L (cm)

Average Frequency (+/- 0.01 Hz)

Maximum Frequency Per Pitch (Hz)

Minimum Frequency Per Pitch (Hz)

Uncertainty of Frequency (maxmin)/2

0.061

82.40

82.50

82.30

0.10

0.063

87.46

87.80

87.10

0.35

0.065

92.72

92.90

92.60

0.15

0.067

98.14

98.20

98.10

0.05

0.069

103.86

103.90

103.80

0.05

0.071

109.90

110.00

109.70

0.15

0.072

116.44

116.60

116.30

0.15

0.075

123.56

123.80

123.40

0.20

0.077

131.08

131.10

131.00

0.05

0.079

138.60

138.80

138.40

0.20

0.081

146.78

147.00

146.60

0.20

0.083

155.32

155.50

155.20

0.15

0.085

164.60

164.80

164.50

0.15

8 Figure 5: Graph of Theoretical Frequency Values (E2-E3) vs. Inverse Length of String

This graph was linearized by plotting the theoretical frequency values for E2-E3 vs. the inverse length of the string (1/2√L), since frequency and length are inversely proportional. This resulted in an R^2 value of 0.9919, meaning the linear trendline fit the data points well. The reason the plots seem so far to the right is because this is a graph of one octave of pitches. Therefore, the lower octaves and frequencies are not pictured on this graph. To reiterate, this graph only depicts the perfect frequencies for the low E string (open string) to E3 (the 12th fret, one octave higher from the starting pitch). It is also important to note the inverse string length was given 3 places past the decimal point, while the string length was measured one place past the decimal point. The reason the inverse was graphed in this manner was for the most accurate linearization. If different pitches/frequencies were plotted against the same inverse string length values for the sake of rounding, the graph would be less linear and therefore less accurate, as it would not be showing the actual relationship between the values. This graph does not need error bars, as it graphs the perfect frequencies.

9 Figure 6: Graph of Recorded Frequency Values (E2-E3) vs. Inverse Length of String

This graph was linearized in the same manner as the graph of theoretical values, and the inverse string length measurements were also carried out to 3 places past the decimal point. The frequency data for all 5 trials is depicted on the graph in addition to the average measured Hz value per pitch. The average data points served as the basis for the linear trendline, which had an R^2 value of 0.9923, meaning the data fit the line almost perfectly. Figure 5 (the theoretical graph) illustrates that the relationship between the frequency of each pitch and the length of the string is inversely proportional. As the length of the string decreases, the frequency increases. The same is true for Figure 6 (the experimental graph). It is interesting to note that the R^2 value for Figure 6 is slightly higher than that of Figure 5. This may be because there is less precision in the values, as the theoretical frequencies were listed to 3 places at most past the decimal point, and average frequencies were listed to 2.

10 Error Analysis: The experimental data almost perfectly matched the theoretical data for frequencies. Included is a table of percent error calculated for each pitch.

Table 4: Percent Error Values for Each Pitch (E2-E3) Pitch

Theoretical Frequency (Hz)

Average Frequency (+/- 0.01 Hz)

Percent Error (|Experimental Value -Theoretical Value| / Theoretical)

E2

82.407

82.40

0.01%

F2

87.307

87.46

0.11%

F2#

92.499

92.72

0.24%

G2

97.999

98.14

0.14%

G2#

103.83

103.86

0.03%

A2

110.00

109.90

0.09%

A2#

116.54

116.44

0.09%

B2

123.47

123.56

0.07%

C3

130.81

131.08

0.21%

C3#

138.59

138.60

0.01%

D3

146.83

146.78

0.03%

D3#

155.56

155.32

0.15%

E3

164.81

164.60

0.13%

Averaging the percent error values gives an overall 0.09% error, which is very low, signifying accurate data. Data was collected in a consistent manner to eliminate error. When the meter stick was used to measure the length of each fret from the bridge, I leaned directly over the fretboard to eliminate any sort of parallax error. Measurements were rounded by a factor of 1 mm due to the uncertainty of the meter stick. The experiment was well controlled. There were hardly any variations between the measured frequencies for each pitch; the largest uncertainty calculation was 0.35, and the data was precise. This experiment is easily repeatable due to low error, and the

11 frequency data would be similar to mine should it be reproduced. The largest amount of variability in this would come from the length of the frets, as every guitar is different. However, the relationship would still be indirectly proportional, resulting in another linear graph. There were no real systematic errors, though it is important to note the online tuner only listed the Hz value 1 place past the decimal point, as opposed to the 3 listed on the chart of theoretical values. Human error was limited as well. The guitar was tuned immediately before data collection. Frequency data was taken in one short period, not allowing an opportunity for the guitar to fall out of tune. Length measurements were recorded as accurately possible within the bounds of the uncertainty of the meter stick (+/- 1 mm). Though the data for each length was rounded to the nearest millimeter, it did not negatively impact the results of the study. The only instance where human error could have minutely impacted the experiment was upon the reading of frequency data due to the slight fluctuations displayed on the tuner as the note resonated. Conclusion: Based on analysis of the data, as the length of the string decreases, the frequency increases. Each successive pitch in the octave (chromatic progression) has a higher frequency. This means frequency and string length are indeed indirectly proportional, confirming the derivation of the listed frequency equation and resulting in two linear gradients with R^2 values of almost 1. The experimental data closely matched the theoretical data, with an overall percent error of 0.09%, meaning it is reliable. For future experiments, I would consider using a string instrument with no frets to collect data. It would inevitably be more challenging to measure the string length per note, but it would be another way to prove the inverse relationship between it and frequency.

12 Work Cited “Guitar Fundamentals: Wavelength, Frequency, & Speed | Science Project.” Science Buddies, www.sciencebuddies.org/science-fair-projects/project-ideas/Music_p010/music/guitarfundamentals-wavelength-frequency-speed#procedure. Martos, Alejandro Pérez González de. Wtuner.com - Online Instrument Tuner, www.alexdemartos.es/wtuner/. Nave, R. “Vibrating String.” Total Internal Reflection, hyperphysics.phyastr.gsu.edu/hbase/Waves/string.html#c4. PinedaI, James. “Parts of An Acoustic Guitar: Head Stock, Tuning Keys and What Are Included?” Best Beginner Guitar, Best Acoustic Guitar, Publisher Name Https://Bestbeginnerguitartoday.com/ Publisher Logo, 8 Nov. 2018, bestbeginnerguitartoday.com/parts-of-an-acoustic-guitar/....