Puck

MUSICAL PHYSICS

Abstract

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Introduction

Music is almost as old as humans are. From the beginning of mankind, we have created music: beats to work to, songs to pass down history, religious carols, sea shanties, ditties to amuse children, drinking refrains, the list goes on and on. Music is a common thread that connects every human, no matter how old they are or what language they speak. But how does music actually work? How is it that two instruments created thousands of miles away from each other can produce the exact same notes? This DYO sets out to utilize the theoretical work done by hundreds of physicists over the past several thousand years to create two functional wind instruments thus showcasing the beautiful mathematics behind music.

Background and Theory

The first person to theorize on the science behind acoustics was the Greek philosopher Pythagoras. Pythagoras primarily experimented with the science behind vibrating strings, and his experiments were so fundamental that even now, 2600 years later the pythagorean tuning system is still widely used. 200 years later, Aristotle theorized (correctly) that sound waves spread through the air via the motion of the air. However, he proposed this theory predominantly based on philosophy not actual physics and it wasn't until 300 years later that the Roman architect Vitruvius was able to conduct experiments proving Aristotle right and also proving that high frequencies travel slower than low frequencies.

Pythagoras

Aristotle

Vitruvius

After the work of the ancient philosophers and physicists, the next breakthrough in the physics of music came from Galileo Galilei in the 16th century. Galileo was inspired by his father who was a fairly well known composer, musician and mathematician. Through several experiments, Galileo solidified the study of vibrations as a science as well as establishing the connection between pitch and frequency. Stemming from Galileo's work, a French mathematician Marin Mersenne studied the vibrations of stretched strings and created Mersenne's three laws which govern the construction and operation of string instruments.

After Mersenne's work in the arena of string instruments an English physicist named Robert Hooke entered the scene. He became the first producer of a sound wave of a known frequency. He was able to accomplish this using a rotating cog wheel to measure the sound wave. In the 19th century the French physicist FĂ©lix Savart continued Hooke's work and became known for this device (now known Savart's disk) which is commonly used in physics demonstrations. Another French Physicist named Joseph Sauveur took Mersenne's work even further documenting his detailed studies into the relationship between frequency, pitch, and waves in stretched strings.

Galileo Galilei

Marin Mersenne

Robert Hooke

FĂ©lix Savart

Joseph Sauveur

This work in vibrational strings then transitioned to studies in standing waves in the 1860s when Franz Melde conducted an experiment using vibrating strings and examining their patterns when set oscillating by a tuning fork.

A standing wave pattern that exists in an air column that is open at both ends is a vibrational pattern created when sound waves that are reflected by the vibrational frequency of the source intersect with incident waves from the source. The source in the case of an open air column is the opening at which air (or sound) is entered at. This could be either end since both ends of an open air column are open. The pattern is known as a standing wave pattern because the waves do not appear to move at the location they intersect at. This location is known as the node. In an air column that has both ends open (as is the case with the recorder I built) the two open ends have antinodes (the waves do not intersect here) and the node is located in the center of the column. The fundamental frequency of any standing wave pattern is known as the pattern that has the lowest frequency and the longest wavelength pattern. It is also known as the first harmonic. In the case of an air column that is open at both ends this would look like the diagram depicted below. This is where the equation used in this DYO is derived from.

Franz Melde

Above is the 1st Harmonic for an air column open at both ends

Above is the 1st Harmonic for an air column closed at one end

As with a column of air that is open at both ends, a standing wave pattern in a column that is closed at one end is a vibrational pattern created when sound waves that are reflected by the vibrational frequency of the source intersect with incident waves from the source. The source in the case of an air column that is closed at one end is the open end of the column. However, with an air column that is closed at one end, the air at the end that is closed is not free to move, meaning that it is forced to assume the position of a node. The open end of the column is the only antinode position. The fundamental frequency (aka 1st harmonic) of an air column that is closed at one end can be drawn as shown to the right. This is where the equation that is used in the pan pipes construction is derived from. The equations for the two different types of air columns are different because depending on the type of column, the fundamental frequency will have a different longest wavelength. This is because the length between adjacent antinodes is equal to half of a wavelength. Two antinodes in an open air column exist in the same distance as one antinode in the closed end air column, leading to the difference in fundamental frequency equations. The principles of standing sound waves can also be applied to stringed instruments, and this was where they were first discovered. However for the purpose of this project I chose to focus on wind instruments.

Pan Pipe Methods

Materials Used

  • 1/2 inch diameter PVC Pipe

  • Hot Glue

  • Cardboard

  • Paint

  • PVC Pipe Cutter

  • Hot Glue Gun

  • Scissors

  • Exacto Knife

  • Ruler

Method

First, the length of each piece of pipe was calculated using the fundamental frequency formula of a pipe that is closed at one end. These lengths were marked on the pipes and then cut using a PVC pipe cutter. The eight pipes were then glued standing up onto a piece of cardboard. Once the glue hardened, the cardboard at the base of each pipe was cut so that the ends of each pipe were closed. The eight pipes were then glued in ascending note order to a piece of cardboard in order to create the full pan pipe. The notes produced by the pipe were then tested using two apps: Pitch and Physics Toolbox.

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C-Major Scale Played on the Pan Pipes

Recorder Methods

Materials Used

  • 1/2 inch diameter PVC Pipe

  • 5/8 inch diameter PVC Pipe

  • 1/2 inch diameter wooden dowel

  • Paint

  • PVC Pipe Cutter

  • Drill

  • Hacksaw

Method

First, the length of the body was calculated by using the fundamental frequency theorem of a pipe that is open at both ends. Then, a piece of 1/2 inch PVC pipe was cut to that length. Then, a 3 cm piece of 5/8 inch was cut, along with a 3 cm piece of 1/2 inch wooden dowel. All three of these pieces then had a 45 degree angle cut into them at one end. After this, the piece of 1/2 inch PVC Pipe (the body of the recorder) had a 3 cm long, 1 cm wide notch cut into the longer side of the end with the 45 degree angle. The piece of 5/8 inch PVC pipe then had a 1 cm wide piece cut out of the shorter side of it (this is the mouthpiece of the recorder). Next, the wooden dowel was inserted into the body of the recorder and the angles of the body of the recorder and the wooden mouthpiece insert were aligned. Next, the mouthpiece of the recorder was slotted over the head of the body of the recorder. The angles of the body of the recorder and the head of the recorder were aligned. At this point the recorder could play one note (v). Next, the locations of the holes were marked on the body of the recorder with a sharpie according to the calculations that had been done to find the length of pipe necessary to make each note. Finally, the holes were drilled out and the notes were tested.

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C5-Scale Played on the Recorder

Theoretical Work

Fundamental Frequency Formula for a Pipe that is Closed at One End

Fundamental Frequency Formula for a Pipe that is Open at Both Ends

In order to determine the length of the pipes for the pan pipe, the fundamental frequency for a pipe closed at one end is used. This formula states that the frequency of a note in Hertz is equal to the speed of sound in meters per second divided by 4 times the length of the pipe in meters. Using algebra one can rearrange this formula to solve for the length of the pipe for any frequency. I performed this algebra (see above) and then substituted the frequency of the eight notes I wanted to create into the equation. These problems are performed below for the frequencies listed to the right.

C4 - 256 Hz

D4 - 294 Hz

E4 - 330 Hz

F4 - 349 Hz

G4 - 392 Hz

A4 - 440 Hz

B4 - 494 Hz

C5 - 523 Hz

In order to determine the length of the pipes for the recorder, the fundamental frequency for a pipe open at both ends is used. This formula states that the frequency of a note in Hertz is equal to the speed of sound in meters per second divided by 2 times the length of the pipe in meters. Using algebra one can rearrange this formula to solve for the length of the pipe for any frequency. I performed this algebra (see above) and then substituted the frequency of the eight notes I wanted to create into the equation. These problems are performed below for the frequencies listed to the right.

C5 - 523 Hz

D5 - 587 Hz

E5 - 659 Hz

F5 - 698 Hz

G5 - 783 Hz

A5 - 880 Hz

B5 - 988 Hz

C6 - 1047 Hz

Pan Pipe Calculations

C4

F4

B4

D4

G4

C5

E4

A4

Recorder Calculations

C5

F5

B5

D5

G5

C6

E5

A5

Results

With a little practice, both the pan pipe and the recorder are easily playable. Mr. Copeland provided four songs to be played. One for the pan pipes and three for the recorder. The music is below in both sheet music format and letter notation, as well as videos of me playing each song.

"Hot Cross Buns" Played on the Pan Pipe

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"Hot Cross Buns" Sheet Music

"Hot Cross Buns" Notes

G4A4B4 G4A4B4 B4B4B4B4A4A4A4A4 G4A4B4

"Twinkle Twinkle Little Star"Played on the Recorder

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"Twinkle Twinkle Little Star" Sheet Music

"Twinkle Twinkle Little Star" Notes

C5C5G5G5 A5A5G5 F5F5E5E5 D5D5C5 G5G5F5F5 E5E5D5 G5G5F5F5 E5E5D5 C5C5G5G5 A5A5G5 F5F5E5E5 D5D5C5

"Oh Susanna"Played on the Recorder

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"Oh Susanna" Sheet Music

"Oh Susanna" Notes

C5D5 E5G5G5A5 G5E5C5D5 E5E5D5C5 D5C5D5 E5G5G5A5 G5E5C5D5 E5E5D5D5 C5C5D5 E5G5G5A5 G5E5C5D5 E5E5D5C5 D5C5D5 E5G5G5A5 G5E5C5D5 E5E5D5D5 C5 F5F5 A5A5A5 G5G5E5C5 D5C5D5 E5G5G5A5 G5E5C5D5 E5E5D5D5 C5

"Mary Had A Little Lamb"Played on the Recorder

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"Mary Had A Little Lamb" Sheet Music

"Mary Had A Little Lamb" Notes

E5D5C5D5 E5E5E5 D5D5D5 E5G5G5 E5D5C5D5 E5E5E5E5 D5D5E5D5 C5 E5D5C5D5 E5E5E5 D5D5D5 E5G5G5 E5D5C5D5 E5E5E5E5 D5D5E5D5 C5

Discussion

As can be seen in the previous videos, two different scales were achieved using the mathematics presented above. It may have taken several thousand years, but physicists have unlocked the key to the creation of music. By following the documented theory I was able to create these two instruments. With enough study, a physicist could create an entire orchestra based purely on mathematical ideals.

Conclusion

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Project Journal Log

April 4

Rube Goldberg Machine Designing

Today I conducted research on the internet to determine what topic I want to conduct my DYO on. After much thought, I have decided to build a Rube Goldberg Machine that utilizes the concepts that we studied in our various lab practicums throughout the year, as well as the recent study we undertook on musical instruments and the physics behind music. I spent the second half of class sketching out several ideas (documented to the left) for this project.

April 6

Today I intended to begin collecting materials and performing calculations to build my Rube Goldberg Machine. However, after a lot of thought I realized that there were two major issues with building a Rube Goldberg Machine as my DYO. The first was the time constraint. Rube Goldberg Machines take a while to construct and the time for this project is very limited (only three weeks). The second is space. This Rube Goldberg Machine would be quite large which would make it difficult to find a location to build it. It's size would also make it impossible to transport meaning it would be difficult to work on outside of the hours that Pritzker is open. Because of these reasons, I decided to switch my topic. The music project that we recently completed really intrigued me so I have decided to build two wind instruments. These are a pan pipe and a recorder. Tomorrow I will begin my construction and calculations. I wasted a couple days by not settling on a topic earlier, but I should still have enough time to complete my DYO.

April 7

Current State of the Pan Pipes

Today I did the calculations to figure out the length of each piece of PVC for the pan pipe. I then measured the lengths out, marked the PVC pipes with sharpie and then cut the pipes to the proper length. I also briefly tested that they were able to make noise by covering the bottom of each pipe with my hand and blowing across it as I would if the instrument were fully constructed.

April 12

Current State of the Pan Pipes

Today I took the pipes that I had cut yesterday and assembled them into the full set of pan pipes. I did this by first using cardboard to cover the ends of each pipe. I then glued my eight pipes in ascending note order to another piece of cardboard. Once the glue had dried, I cut out the excess cardboard so that the pan pipes had the same shape when looked at from both directions. Finally, I painted the pipes to make them look less DIY and more polished.

April 13

Today I worked on this website. I arranged all of the different sections, wrote out my equations digitally, and began to write out my background information section. I also made sure that I had something to share for the progress report that is due in class tomorrow.

April 14

Current State of the Recorder

Today I worked on creating the recorder out of PVC pipe. First, I followed the tutorial posted by Rick Osgood on YouTube. However, when I put the pieces together, I couldn't get it to make noise, and I quickly realized that this is because I am using the same size PVC pipe for both the body and mouthpiece of the recorder. I either need to use 1/2" CPVC and PVC pipe or two different sizes of PVC pipe. Once I realized this, I attempted to instead follow the tutorial posted by MNda TV on YouTube. However, the precision of angles required for this build was extremely difficult to achieve. Tomorrow I will attempt to find either 1/2" CPVC and PVC pipe, or two slightly different sizes of PVC pipe to reattempt the original tutorial. If this fails, I will either return to the second tutorial or attempt to find a new one.

April 15

Current State of the Recorder

Today I continued to work on making the recorder. I got a piece of 2/8" PVC Pipe to act as the mouthpiece. With this new piece, I was able to put cut a new head and body piece for the recorder. I need to cut a new piece for the mouth insert, but once that is completed, the recorder should play a single note. I also completed the calculations to determine the locations for each finger hole on the body.

April 18

Today I fleshed out the background, theory and intro sections of the website. I also transcribed my sheet music and began planning what my abstract and conclusion videos will look like. Additionally, I made an optimistic plan for the rest of the week before the Symposium.

Tuesday: finish recorder build, begin recording videos

Wednesday: finish recording videos

Thursday: finalize website, make poster for Symposium

April 20

Recorder and Poster

Today I finished building my recorder. I had some trouble with getting it to make noise but after some assistance from Mr. Copeland and Ars Lee I was able to get it working. Once I finished the recorder, I recorded the songs that I am playing.

April 21

Poster

Today I recorded my abstract and conclusion videos. Then I created a poster. It contains a bit of background, the calculations I did, the process I followed, and the sheet music I used. I will have my recorder and pan pipes at my booth so that people can play them when they come by. I also finalized my website and made sure that everything is all set for the Symposium tomorrow.

Sources