Understanding the Relationship between Beat Rate and the Difference in Frequency between Two Notes.

Understanding the Relationship between Beat Rate and the Difference in Frequency between Two Notes. Hrishi Giridhar 1 & Deepak Kumar Choudhary 2 1,2 Podar International School ARTICLE INFO Received 15 November 2017 Accepted 9 December 2017 Published 14 December 2017 ABSTRACT As a guitarist, part of my essential ear training involves me learning to tune my guitar with accuracy. I noticed that while tuning my guitar, at times there was a beating effect- an effect that made the notes coming out of my guitar sound like a motor. I later studied about interference of waves in my IB Physics class in school, and immediately made the connection to the beating sound I heard so often when tuning my guitar. I realized that, if I understood this phenomenon properly, I could use it to identify how far apart two notes are in frequency, which could improve the ease with which I can tune my guitar. 1. Introduction A beat is a phenomena observed in instruments that can create sustained notes. When two notes that differ slightly in frequency are played simultaneously, it creates a beat - a periodic oscillation of volume. The phenomenon observed gives an effect like that of a tremolo. This occurs as the two sound waves alternatively interfere constructively and destructively. This can be better understood in the picture. 2. A beat occurs due to interference where, by the superposition principle, the resulting amplitude of the two waves is equal to the sum of the individual amplitudes. In the picture above, amplitude is on the y-axis and time is on the x-axis. When the two waves are in phase, the amplitude will be maximum, and when they are out of phase, the amplitude will be 0. This is why a beat is a periodic oscillation of volume. I modelled the above sound waves (as sine waves) using the software Audacity 1. If I can understand the relationship between the rate of beating and the difference in frequency between two notes, I will be able to understand how the beating varies as two notes' frequencies become closer / further away from each other, which can effectively help me identify faults in tuning, and help me tune my guitar with greater ease and accuracy. Therefore, I ask: what is the relationship between the difference in frequency between two notes and the 1 Audacity is the name of an open source multilingual audio editor and recorder software that is used to record and edit sounds. beat rate (in beats per second) of the notes when played simultaneously? Research project: The purpose of this experiment is to understand how the rate of beat changes, based on the difference in frequency/pitch between the two notes. One beat is one oscillation of volume. I will conduct this experiment by setting one string on the guitar (Note #1) tuned to a particular frequency (330Hz) and left constant, while adjusting the frequency of another string (Note #2). The difference in frequency between the two notes is my independent variable since I am adjusting it. The beat rate is my dependent variable and it should Imperial Journal of Interdisciplinary Research (IJIR) Page 206

vary based on the difference in frequency between the two notes. I will ensure that the temperature in the room remains constant throughout the experiment. This is because changes in the temperature can affect the tuning of the guitar - therefore, the experiment is best carried out at room temperature. I will not turn on any fans, air conditioners or heaters mid-way the experiment. This will ensure that the frequency of Note#1 remains fixed at 330Hz throughout the experiment. I will also make sure there is no background noise while conducting this experiment, since it can affect the tuner's ability to detect the desired note and its frequency accurately. Hence, I will record all readings in a silent room. In addition, I will use same tuner is used to measure the frequencies throughout the experiment, since different tuners may have different sensitivities, and may measure notes slightly differently. The strings must also be struck with the same force for each reading. I will do this by attempting to pluck the strings with the same amount of force each time. This is necessary since, the force with which the strings are struck can affect the frequency emitted by small amounts. Lastly, the experiment needs to be conducted in a stable magnetic field. An external magnetic field can cause interference and buzzing with the sound emitted from an electric guitar, affecting the frequency emitted as well as the tuner's ability to recognize the notes. I will attempt to maintain a stable magnetic field by limiting the number of electronic gadgets placed nearby, since they can interfere with the magnetic field of the electric guitar's pickups. An electric guitar is an instrument that converts the vibrations of its strings into electrical signals. It does this by the use of a pickup, which is a magnet coiled by a thin wire. When the metal strings of the guitar are strummed, they vibrate and they cut the magnetic field of the pickup, thereby inducing a current. A plectrum is a small tool that is held by the guitarist and is usually used to strum the strings of a guitar. An ideal plectrum A guitar amplifier is an electronic device that converts electric signals received from the pickup of an electric guitar into sound. Quite simply, it amplifies electric signals. The wire that connects the electric guitar to the amplifier is called a lead cable. A guitar can be tuned by twisting the tuning knobs at the top of the guitar. There are several materials/apparatus required to carry out this experiment. These include: an electric guitar, a guitar amplifier, a lead cable, a tuner, a plectrum and a laptop. A background of the electric guitar and its parts and peripherals Lastly, a tuner is a device that can identify the notes coming out a guitar amplifier, and their respective frequencies. Conducting the experiment For this experiment, the amplifier must be connected to a power source and switched on. The guitar will then be connected to the amplifier using a lead cable. The master volume of the amplifier will be turned up to at least 50% to ensure the notes are loud enough to be detected by the tuner, which will be placed right in front of the amplifier. I will place a laptop near the amplifier as well. Imperial Journal of Interdisciplinary Research (IJIR) Page 207

A screenshot of the software 'Amazing slow downer' Then, I will tune the first (and thinnest string) of the guitar to 330Hz, using the tuner. This is Note #1 and will be kept constant throughout the experiment. I will then tune the second string (second thinnest string) to 320Hz- this is Note #2. The difference in frequency between the two notes is 10Hz. S.R No. Note #1/Hz Note #2/Hz Difference in frequency/hz Time required for 10 beats ± 0.1Hz ± 0.1 (0.1Hz+0.1Hz Hz ) = ± 0.2Hz t 1 / s t 2 / s t 3 / s 1 330.0 328.0 2.0 5.16 5.17 4.96 2 330.0 327.0 3.0 3.43 3.13 3.24 3 330.0 326.0 4.0 2.40 2.46 2.61 4 330.0 325.0 5.0 1.93 1.98 2.07 5 330.0 324.0 6.0 1.62 1.55 1.72 6 330.0 323.0 7.0 1.35 1.38 1.44 7 330.0 322.0 8.0 1.24 1.31 1.19 8 330.0 321.0 9.0 1.09 1.16 1.02 9 330.0 320.0 10.0 0.96 1.04 0.92 Next, both strings will be plucked simultaneously, using a plectrum. I will listen closely for a beat - for a periodic oscillation of volume. I will use the software Amazing Slow Downer 2 that will be installed on my laptop. This software allows me to record the sound, and play back the recording at a slower speed. I will play back the recording at 50% of its original speed and I will use the software's inbuilt clock to record the amount of time required for 10 beats to occur. Next, both strings will be plucked simultaneously, using a plectrum. I will listen closely for a beat - for a periodic oscillation of volume. I will use the software Amazing Slow Downer 3 that will be installed on my laptop. This software allows me to record the sound, and play back the recording at a slower speed. I will play back the recording at 50% of its original speed 2 Amazing Slow Downer is a software that can change the speed of music - from 20% (one fifth speed) to 200% (double speed) without changing the pitch. 3 Amazing Slow Downer is a software that can change the speed of music - from 20% (one fifth speed) to 200% (double speed) without changing the pitch. and I will use the software's inbuilt clock to record the amount of time required for 10 beats to occur. I will then repeat the experiment two more times, to get 3 readings in total. An average of these readings will be taken to get the average time. After taking the first set of readings, Note #2 should be re-tuned, to 321Hz. The difference in frequency between the two notes is now 9Hz. The same experiment should be conducted again, and the average time should be calculated. This whole process should be repeated, using the following frequencies for Note #2: 322Hz, 323Hz, 324Hz, 325Hz, 326Hz, 327Hz and 328Hz. From the data, the beat rate (in beats per second) can be calculated, seen below: After conducting the experiment for all the values of Note#2, my results were as follows: At this point, I noticed how, as the difference in frequency, the time taken for 10 beats reduced. Therefore the rate of beating increased along with the difference in frequency. Since a tuner is a digital device, I've taken the uncertainty of the frequencies for Note #1 and Note #2 as the least count of the tuner, which is ±0.1Hz. To find the uncertainty of the beat rate, I took the maximum minus the minimum time required for each trial. I did not divide this value by two since it was resulting in extremely small uncertainties, and I used the highest uncertainty value (0.30s) for all data points. I have graphed my data below: a graph of difference in frequency Vs beat rate, including the horizontal and vertical error bars. I notice a strong linear relationship between the two variables. S. R N o. Time required for 10 beats Average time t 1 / s t 2 / s t 3 / s t A / s Uncerta inty in 't' Maximu m - minimu m time Beat rate (Beats per second) = 1 5.16 5.17 4.96 5.10 0.21 1.96 2 3.43 3.13 3.24 3.27 0.30 3.06 3 2.40 2.46 2.61 2.49 0.21 4.02 Imperial Journal of Interdisciplinary Research (IJIR) Page 208

4 1.93 1.98 2.07 1.99 0.14 5.02 5 1.62 1.55 1.72 1.63 0.17 6.13 6 1.35 1.38 1.44 1.39 0.09 7.19 7 1.24 1.31 1.19 1.25 0.12 8.02 8 1.09 1.16 1.02 1.09 0.14 9.17 9 0.96 1.04 0.92 0.97 0.12 10.27 Maxima S.R Beat rate (beats No per second) 1 1.96 2 10.27 Minima S.R Beat rate (beats No per second) 1 2 1.96 0.30 10.27 0.30 Uncertai nty 0.30 0.30 Uncertainty Maxima 1.96-0.30 = 1.66 10.27+0.30 = 10.57 Minima 1.96+0.30 = 2.26 10.27-0.30 = 9.97 - LN R = LN K + n. LN This is nothing but a standard straight line equation, where LN R = y LN K = c (y-intercept) n = m (gradient) LN = x I then plot a graph of LN R Vs LN below:, as seen Below is a graph of the original line, the maxima line and the minima line, and their respective equations: From the 3 trends, the 3 gradients obtained are: m : 1.029 m MAX : 1.113 m MIN : 0.963 Average gradient: m MAX + m MIN = 1.113 + 0.963 = 1.04 2 2 Error: m MAX - m MIN = 1.113-0.963 = 0.075 rounded off to 0.08 2 2 Therefore, the final value for the gradient that I have obtained is: 1.04 ± 0.08 (Notice that the gradient of the line 1) From the graph, I can evaluate that: - R(beat rate) (difference in frequency) - Therefore: R = K. (where K = constant) - If I add log on both sides: The equation of the above line is: y=1.019x-0.020, where 0.020 is the y-intercept, which in this case, represents LN K. LN K = -0.020 Therefore = K K (constant) = 0.98 As seen in the above graphs, the trend observed has a strong positive linear relationship between the rate of beating and the difference in frequency between two notes - the greater the difference in frequency, the greater the rate of beating. As seen in my calculations above, the gradient value I calculated is 1.04 ± 0.08. This is a value that is very close to 1, indicating that the beat rate increases with the same amount as the difference in frequency. In the graph of LN ( frequency) Vs LN (beat rate), I calculated K = 0.98. This is also a value very close to 1. The constant tells us the relationship between the two variables; the value of the constant that I obtained is approximately equal to 1. This ties up with the fact that the gradient of the slope is also approximately 1. The value of K and the value of the gradient being 1 implies that the beat rate is not only proportional to the difference in frequency, but is equal to it. For example, if their difference in frequency is 2Hz, the beat rate is 2 beats per second. The same trend is observed for all values. Therefore I evaluate that: Beat rate (beats per second) = difference in frequency between two notes Imperial Journal of Interdisciplinary Research (IJIR) Page 209

i.e. Frequency of beat = Frequency of Note #1 - Frequency of Note #2 F B = F 1 - F 2 When two notes of slightly different frequencies are played together, they start off in phase. The individual amplitudes of the two notes are added, and due to constructive interference and the resulting tone produced had maximum amplitude. However since these two notes have slightly different frequencies, they eventually go out of phase. The amplitudes are added, and due to destructive interference, the resulting tone produced has zero amplitude. Since the amplitude of a wave corresponds to the volume/loudness of the sound, the beat produced is simply a tone that fluctuates/oscillates periodically between a high volume and zero volume. Tuning the guitar Therefore, this can indeed help me tune my guitar! Let s assume that I want to tune two notes to the same frequency. If I play two notes simultaneously and I hear a fast rate of beating, I can automatically assume that the notes are off-tune by several hertz. As I adjust my tuning knobs, I will hear the rate of beating slow down. In order to tune most accurately, I need to tune to a point where the frequency of the beat is so slow, it is unrecognizable to my ears. I now know that, if the beat rate begins to increase again, I have overshot the note, and the difference in frequency between the notes is increasing. If the rate of beating is reducing, I know that the two notes' frequencies are coming closer together and are becoming more in-tune. Errors and solutions The uncertainties get smaller as the rate of beating increases. Since some uncertainties very small, I took the largest uncertainty value (0.30) for my vertical error bars. The trend line goes through all the error bars, touching all the points. If I had not chosen the maximum uncertainty for the vertical error bars, the line may not have passed through all the vertical error bars. The error for the difference in frequency between Note #1 and Note #2 is ±0.2Hz. This is because, the individual uncertainties of the two notes were ±0.1Hz. When the difference was calculated, by propagation of uncertainties, the errors were added to give ±0.2Hz. The line of best fit passes through all horizontal error bars. I found it very difficult to record the time taken for 10 beats, especially for beat rates above 6 beats/second. I could not start and stop a stopwatch accurately: there was far too large a human error in doing so. To overcome this, I used the software Amazing Slow Downer which allowed me to record the beat and play it back at a slower speed. The software had an in-built stopwatch that slowed down along with the beat. This allowed me to significantly reduce errors in recording the data. An error may have occurred when plucking the strings of the guitar - the frequency of the note emitted is affected by the force with which the string is plucked. As a human being, it is impossible for me to pluck the strings with the exact same amount of force for each reading (random, human error). Plucking a string with a high amount of force will cause the frequency emitted to be slightly higher, than if the string is plucked with less force. In order to minimize this error, I attempted to control and maintain the force with which I plucked the strings. I conducted the experiment in a room. However this room was not soundproofed - sounds from outside were audible inside the room (instrumental error). These sounds, I noticed, interfered with the tuner's ability to identify notes. I closed all entrances, doors and windows in an attempt to minimize the amount of external noise in the room. Although I attempted to carry out this experiment in a stable magnetic field, a laptop needed to be placed near a guitar and a tuner (which was an application on my mobile phone) needed to be placed in front of the amplifier when conducting the experiment. The existence of these electronic gadgets may have caused interference with the note/frequency of the note emitted (random error). It may have also affected the tuner's ability to recognize the note(s) emitted from the guitar. The placement of the mobile phone (tuner) and laptop was inevitable; however I tried to reduce further errors by removing other unnecessary electronic gadgets from the room itself. I also noticed that, after each set of readings, Note #1 would slightly reduce in frequency (by 0.1Hz or so). This is because repeated plucking of the string can slowly cause the string to de-tune (random error). Therefore, I made sure to check the tuning of Note#1 after each set of readings, ensuring that it remains at 330Hz. I chose to carry out this experiment by playing notes on the guitar, since the ultimate aim of my exploration was to help me tune my guitar with greater ease and accuracy. However, as stated above, there were several errors that existed (force of striking the strings affects the frequency, the fact that a guitar does not remain in tune and needs to be checked and re-checked to make sure it is still in tune, etc.). Therefore, perhaps I could have conducted the same experiment using tuning forks, which have fixed frequencies. However, it would have been difficult for me to obtain a flexible range of values using tuning forks. In a nutshell, Although there were errors in this experiment, I attempted to reduce these errors as far as possible to get accurate and precise readings both. To conclude, through my experiment, I observed that when two notes of close frequencies are played together, the Imperial Journal of Interdisciplinary Research (IJIR) Page 210

frequency of the beat observed is equal to the difference in frequency between the two notes. References: [1] Briand, L. C., Daly, J., and Wüst, J., "A unified framework for coupling measurement in objectoriented systems", IEEE Transactions on Software Engineering, 25, 1, January 1999, pp. 91-121. [2] Maletic, J. I., Collard, M. L., and Marcus, A., "Source Code Files as Structured Documents", in Proceedings 10th IEEE International Workshop on Program Comprehension (IWPC'02), Paris, France, June 27-29 2002, pp. 289-292. [3] Marcus, A., Semantic Driven Program Analysis, Kent State University, Kent, OH, USA, Doctoral Thesis, 2003. [4] Marcus, A. and Maletic, J. I., "Recovering Documentation-to-Source-Code Traceability Links using Latent Semantic Indexing", in Proceedings 25th IEEE/ACM International Conference on Software Engineering (ICSE'03), Portland, OR, May 3-10 2003, pp. 125-137. [5] Salton, G., Automatic Text Processing: The Transformation, Analysis and Retrieval of Information by Computer, Addison-Wesley, 1989. Imperial Journal of Interdisciplinary Research (IJIR) Page 211