IB Physics HLII Derek Ewald B. 03Mar14 Saxophone Lab Research Question How do different positions of the mouthpiece (changing the length of the neck) of a saxophone affect the frequency of the sound wave produced? Background When two waves of the same speed, wavelength and amplitude that are traveling in opposite directions meet, they create a standing wave due to superposition. The different possibilities of standing waves produce different harmonics. A saxophone works with closed-open standing waves as the mouthpiece will be the closed end (right), while the horn will be the open end (left). The harmonic integers are Source 1 A musical note is a notation representing the pitch or frequency. If a note is of a higher frequency than the in tune note, it means that it is sharp. Conversely, if a note is of a lower frequency than the in tune note, then it means that it is flat. The longer the instrument the more flat it will be (reason why low instruments are longer). On the contrary, the shorter an instrument the more sharp it will be. Hence, if you alter the position of the mouthpiece (change the length of the neck of the saxophone), the frequency will change. This experiment will examine resulting frequencies from different neck lengths. Different musical notes have a different frequency. However, even though these notes are discrete (A, Bb, B, C.), waves have a continuous spectrum of frequencies, meaning that one may not hit the perfect frequency on an instrument (i.e. being in tune). These slight modifications in frequency will make someone sound out of tune, making it harmonically disastrous while playing with a band that is in tune. For this experiment, the tuning of z will be used. Also, the note that will be played will be an Alto Saxophone G, which transposed to a concert key is a Bb. A Bb is of a slightly higher frequency than that of A at.
IB Physics HLII Derek Ewald B. 03Mar14 Variables Independent: The length of the alto saxophone as measured by the Cork Distance Mouthpiece (distance from the end of the cork on the neckpiece to the end of the mouthpiece see picture below). Dependent: Frequency of the sound wave (Hertz) Controlled Variable Temperature Size, material, length of the saxophone (not including the mouthpiece) Size and type of mouthpiece and reed Loudness/Amplitude of the wave Position of the mouth Mouthpiece clip pressure How it will be controlled Unlike light, sounds needs a medium to travel in such as air. If the temperature is higher, then the speed of sound is also higher. Velocity is wavelength x frequency. Therefore, if the velocity is higher, the frequency will also increase, making a note sharp. In order to keep this constant, the experiment was set up in a room with a temperature of. Different materials of the saxophone itself have different natural frequencies of resonance and thus will produce different sounds. The size and length of the saxophone, as mentioned before, will alter the frequency of sound produced. In order to control this, the same musical instrument (i.e. alto saxophone) must be used. Different mouthpieces and reeds have distinct ways that the air flows. This changes the pitch and amplitude of the wave produced. In order to keep this constant, the same Rico Royal C7 alto saxophone mouthpiece and Vandoren 2½ reed will be used throughout the experiment. Sometimes, when a saxophonist is playing too loud or too soft, they play out of tune. Therefore, in order to keep this constant, the note will be played with an amplitude of. Mouth positioning is also an important factor when considering the frequency of a tone. Saxophonists that have good control over the mouthpiece can bend notes just with the positioning of the tongue and bottom lip. If the bottom lip moves up, the air space between the mouthpiece and the reed will decrease, increasing the frequency of the note. The opposite happens if the bottom lip moves down. Thus, this position must be kept constant (although extremely difficult to do so). Therefore the same person will be used in this experiment, reducing this variable as much as possible. There is a metal clip that holds the reed on the mouthpiece (as seen below). This means that the tightness of the reed on the mouthpiece can be adjusted. Hence, this will affect the distance between the tip of the reed with the tip of the mouthpiece, which will affect the pitch of the note (as mentioned above). To keep this constant, the tightness of this clip will not be adjusted throughout the lab.
IB Physics HLII Derek Ewald B. 03Mar14 Diagram / Materials Saxophone a. Alto saxophone (Yamaha) b. A. Sax Mouthpiece (Rico Royal C7) and fastening clip c. Reed (Vandoren 2 1/2) Computer with Logger Pro software installed Microphone (frequency sensor) Vernier Caliper Procedure / Method 1. Gather materials, assemble the saxophone, and start up the computer with Logger Pro with the microphone frequency sensor set for 10,000 samples/second for 0.05 seconds. Zero the amplitude (arbitrary units) 2. Connect the mouthpiece of the neck of the saxophone to a Cork Distance Distance (see diagram above) of 0.028 m. 3. Play a Alto Saxophone G (Concert Bb) at (amplitude - loudness), start collection of the data and record the frequency. 4. Repeat Step 3 for a total of 5 trials 5. Once completed with the first set of trials with the mouthpiece set at 0.028 m, change the position of the the mouthpiece to 0.023 m, 0.018 m, 0.013 m, 0.008 m Cork Distance Difference, respectively and repeat Steps 3-
Data Collection and Processing Table 1 shows the raw data collected during the experiment. Total Distance refers to the total length of the cork (use the picture above for reference). Total Distance (±0.0005) Table 1: Raw Data Cork Distance Difference (±0.0005) Time taken to finish the highest integer number of periods (±0.0001) Number of Periods 0.034 0.0280 0.0496 23 0.034 0.0280 0.0499 23 0.034 0.0280 0.0497 23 0.034 0.0280 0.0498 23 0.034 0.0280 0.0499 23 0.034 0.0230 0.0495 23 0.034 0.0230 0.0497 23 0.034 0.0230 0.0497 23 0.034 0.0230 0.0493 23 0.034 0.0230 0.0496 23 0.034 0.0180 0.0491 23 0.034 0.0180 0.0492 23 0.034 0.0180 0.0489 23 0.034 0.0180 0.0493 23 0.034 0.0180 0.0490 23 0.034 0.0130 0.0497 24 0.034 0.0130 0.0498 24 0.034 0.0130 0.0480 23 0.034 0.0130 0.0480 23 0.034 0.0130 0.0481 23 0.034 0.0080 0.0493 24 0.034 0.0080 0.0494 24 0.034 0.0080 0.0493 24 0.034 0.0080 0.0494 24 0.034 0.0080 0.0496 24
Diagram 1 is a sample graph of the sound wave produced by the saxophone. This is considered as raw data. Diagram 1 Table 2 is the processed data and the sample calculations show how these values were calculated. Table 2: Processed Data Cork Distance Mouthpiece (±0.001) Time Taken %Unct Length of 1 Period Period Abs.Unct Frequency Frequency Abs.Unct 0.006 0.20% 0.002157 0.000004 463.7 0.9 0.006 0.20% 0.002170 0.000004 460.9 0.9 0.006 0.20% 0.002161 0.000004 462.8 0.9 0.006 0.20% 0.002165 0.000004 461.8 0.9 0.006 0.20% 0.002170 0.000004 460.9 0.9 0.011 0.20% 0.002152 0.000004 464.6 0.9 0.011 0.20% 0.002161 0.000004 462.8 0.9 0.011 0.20% 0.002161 0.000004 462.8 0.9 0.011 0.20% 0.002143 0.000004 466.5 0.9 0.011 0.20% 0.002157 0.000004 463.7 0.9 0.016 0.20% 0.002135 0.000004 468.4 1.0 0.016 0.20% 0.002139 0.000004 467.5 1.0 0.016 0.20% 0.002126 0.000004 470.3 1.0 0.016 0.20% 0.002143 0.000004 466.5 0.9 0.016 0.20% 0.002130 0.000004 469.4 1.0 0.021 0.20% 0.002071 0.000004 482.9 1.0 0.021 0.20% 0.002075 0.000004 481.9 1.0 0.021 0.21% 0.002087 0.000004 479.2 1.0 0.021 0.21% 0.002087 0.000004 479.2 1.0 0.021 0.21% 0.002091 0.000004 478.2 1.0 0.026 0.20% 0.002054 0.000004 486.8 1.0 0.026 0.20% 0.002058 0.000004 485.8 1.0 0.026 0.20% 0.002054 0.000004 486.8 1.0 0.026 0.20% 0.002058 0.000004 485.8 1.0 0.026 0.20% 0.002067 0.000004 483.9 1.0
Sample Calculations Raw data Cork Distance Mouthpiece: When two or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the absolute uncertainties. In order to find the frequency, you must find the length of 1 period. The data that is found in the sample sound wave Diagram 1 shows how many periods there are within 0.05 seconds. However, in order to make the data more accurate, the amount of time taken to finish the highest integer number of periods is recorded (this will be close to 0.05 seconds). This section is highlighted in the Graph 1. Also, the number of integer periods is also recorded. Hence, with this raw data, the length of one period can be found Cork Distance Difference: Total Distance: Time taken to finish the highest integer number of periods: (0.2% uncertainty) Number of integer periods: This value carries on the percent uncertainty (0.2%). Hence, the absolute uncertainty of the period is Therefore, the length of one period is Knowing the length of one period, the frequency can now be found. The frequency is defined as the inverse of the period. Periods per second is also known as hertz. The value continues to have a.uncertainty. So the absolute uncertainty is Therefore, the frequency of the sound wave is
Table 3 shows the average values for the recorded frequencies for each mouthpiece position Table 3 - Processed Data: Averages Mouthpiece Cork Distance Average Frequency Frequency Abs.Unct 0.006 462.0 1.7 0.011 464.1 2.4 0.016 468.4 1.9 0.021 480.3 2.6 0.026 485.8 2.0 Sample Calculations: Greatest Residual on the Average Raw Values of frequency of a Cork Mouthpiece Distance of 0.006. Average Greatest residual. The following numbers are the difference between the raw values and the average. The greatest one of these is 1.7. Therefore, this is the greatest residual. Hence the average value and uncertainty is: 463.7 460.9 462.8 461.8 460.9 1.7 1.1 0.7 0.2 1.1
Graph 1 illustrates the effect of the Cork Distance Mouthpiece (dependent variable) on the frequency of the note played (independent variable). Sample Calculations: Graph 1 In order to determine the Minimum Uncertainty Slope (orange), the top left of the uncertainty box of the smallest (x) point and the bottom right of the largest (x) point. To accomplish this, the value must be added or subtracted to find the edge of the uncertainty box. As this is addition and subtraction, the place value, not significant figures, are important on determining the amount of values required for the sum or difference. Smallest point (top left of uncertainty box) X: Y: Largest point (Bottom right of uncertainty box) X: Y: Equation of line: In order to determine the Maximum Uncertainty Smallest point (bottom right of uncertainty box)
Slope (green), the bottom right of the uncertainty X: box of the smallest (x) point and the top left of the largest (x) point. To accomplish this, the value Y: must be added or subtracted to find the edge of the uncertainty box. As this is addition and subtraction, the place value, not significant figures, are important on determining the amount of values required for the sum or difference. Largest point (top left of uncertainty box) X: Y: Equation of line: Table 4 demonstrates the process data of the slope and the Y-intercept with their respective uncertainties. Table 4 - Processed Data: Slope + Y Int Slope Mean 1275 Max-Mean 249 Mean-Min 359 Abs. Uncertainty Greatest Residual 359 Y intercept Mean 451.7 Max-Mean 7.4 Mean-Min 2.1 Sample Calculations: Table 4 To find the uncertainty of the slope, the greatest residual must be found: the maximum value was subtracted with the mean value and the minimum was subtracted from the mean. The larger of the two values is the greatest residual or uncertainty for the average distance (they are the same). Therefore the greatest residual is. The slope is To find the uncertainty of the y intercept, the greatest residual must be found: the maximum value was subtracted with the mean value and the minimum was subtracted from the mean. The larger of the two values is the greatest residual. Therefore the greatest residual is The Y intercept is then Conclusion and Evaluation
As shown by the gathered results, there is a linear relationship between the length of the saxophone (found by the Cork Distance Mouthpiece) and the frequency of the note played. In Graph 3, the trend line is represented by the equation where y is the frequency in hertz and x is the Cork Distance Mouthpiece in meters. This is a positive regression, meaning that the greater Cork Distance Mouthpiece (less Cork Distance Difference), the higher the frequency. This is due to the fact that the musical instrument would be slightly shorter. As seen by the formula,the smaller L is (proportional to Cork Distance Difference), the higher the frequency. The slope,, is the rate at which the frequency changes with regard to the length of the instrument (as measured by the Cork Distance Mouthpiece). It can be used to find the value of the ideal place to position the mouthpiece on the saxophone. Graph 1 is not proportional as it does not pass through the origin. The reason for this is because although the mouthpiece might be at the edge of the cork on the saxophone neckpiece, it will still produce a sound (of a lower frequency), but will not even reach close to zero. Additionally, the lowest frequency of a musical instrument is the lowest note of an organ, a C-1 (8.18 hertz). An alto saxophone would never even reach that note as its lowest note is its Bb (concert C#) of 138.59 hertz. For this reason, there were very few systematic errors. The y-intercept is and is within the uncertainty limits. One potential and very important systematic error would be temperature. This would be a systematic error as it would shift the data either completely up or down, affecting only the y intercept and not the slope. This is due to the fact that unlike light, sounds needs a medium to travel in such as air. If the temperature is higher, then the speed of sound is also higher. Velocity is the product of wavelength and frequency. Therefore, if the velocity is higher, the frequency will also increase, making the tone of a note sharp and shifting the data up. There were more random errors than systematic errors as depicted in Graph 1. The trend line (blue), minimum and maximum lines do not pass through all the uncertainty bars. This is caused by one data point s y value (frequency) being too low. Additionally, the vertical uncertainty bars were too small as the uncertainty of the raw data in relation to the time taken was too small. Although the uncertainty was that of the precision of the digital apparatus, more human errors should have been accounted in the experiment. The horizontal uncertainty bars were directly related to the precision of the Verneir Caliper and were appropriate size in relation to the graph.
However, the trend and value of Graph 1 is very high (0.9367). The value is extremely close to 1 which signifies that the data lies close to the trend line and that there were few evident random errors and is very reliable. The percent error could not be found as there was no accepted value found for this experiment from previous research. However, this data can be used to find the ideal distance for the mouthpiece to be on the cork to be in tune at room temperature by using the equation of Graph 1. As the in tune Bb note is, that can be substituted into the value of y:. Therefore,. At room temperature the alto saxophone must be played with a Cork Distance Mouthpiece of. The controlled variables were attempted to be held constant in this lab. However, some mentioned controls (such as temperature and position of the mouth) were more difficult to be consistent and will be explained in the Weaknesses and Improvements section of this document. The equipment used was reasonable and were used properly for the purposes of this experiment. For instance, the precision of equipment such as the Verneir Caliper ( ) and a digital microphone ( ) helped keep the final uncertainty bars to low values. Additionally, time was effectively used to maximize the efficiency of the lab. The range of values was very precise and judging by the conclusion, very accurate. Accuracy and precision were ensured for each of the five length settings, by conducting five repetition trials, which were made and then later averaged. Due to this, the collected data is very reliable. Also, when graphing and analyzing the results, the trend line goes through most of the error bars. As for anomalous points, there were none because all the evident outliers when releasing when playing a note were due to a human inconsistency and thus were not recorded. Weakness and Improvements As mentioned earlier, the control variables that was very difficult to keep consistent was the position of the mouth on the mouthpiece. It is possible to bend notes just with the positioning of the tongue and bottom lip. If the bottom lip moves up, the air space between the mouthpiece and the reed will decrease, increasing the frequency of the note. The opposite happens if the bottom lip moves down. For example, the third data point at Cork Distance Mouthpiece did not follow the trend line and had too low of a frequency for its corresponding length. This was most likely caused by this human error of the positioning of the bottom lip. It was not as tight as it had to be. To improve upon this error, a total of 10 trials could be taken (and average it later),
while concentrating in keeping the lower lip in the same position for every trial. Apart from conducting more trials, there are no quantitative way to control this on a saxophone. Although all the trials were collected one after the other during the same time of the morning, throughout the experiment the temperature steadily raised. This meant that the speed of sound was higher as it needs a medium to travel through. As velocity is proportional to the frequency, as the velocity is higher the frequency will increase. This would produce a trend line that has many quadratic characteristics because the frequency would not only increase due to a shorter instrument but also a rising temperature. To improve upon this error, the experiment must be conducted in a room that is not subject to temperature changes. Additionally, while playing the saxophone saliva leaves the mouth into the mouthpiece. This not only adds humidity but also partially blocks the sound waves. Humidity will increase the speed of sound and thus the frequency will also increase. Hence, as the trials went on (just like the temperature), the humidity within the instrument increased and thus the frequency increased. When the saliva blocks part of the mouthpiece, the sound waves create a more smooth sine/cosine curve. Some trials with jagged graphs, were immediately discarded and not considered as an individual trial. In order to improve upon this error, more time has to be taken between individual trials to let the humidity leave the instrument. Additionally, the saxophone player must clear (by inhaling) the mouthpiece of any saliva blocking it. In order to have a greater accuracy of the results, more independent variable values could be measured. There is a limit to the domain as that is the length of the whole cork. Therefore, these values for the independent variable would not be able to surpass this domain, but more values within it would be found in order to have a more accurate and reliable regression. Additionally, it would be interesting to investigate the effect musical instrument length on other notes apart from G (Concert Bb), and see if the slope is comparable with the one found in this experiment.
Bibliography 1. Kamehameha Schools. Standing Waves. Digital image. Kamehameha Schools. N.p., n.d. Web. 3 Mar. 2014.