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Project Synopsis

The Final Synopsis should be at least 5 pages in length double spaced, and size 12

font. The synopsis is the final report that has all the conclusions you made during

your research over the semester and should include the following:

Title:

This should be well thought so that it is clear to the audience what the

whole project is about or encapsulates the process you used to organize your

project.

Who was assigned what tasks in your group?

How did your project address the methods learned in this course and how does

your project apply to the real world?

Include all the formulas you used for our project. Any conclusions and no more

than 1-2 pages of calculations and mathematical models that helped you arrive at

your conclusions.

What problems did you face during your research?

Was there any information that stumbled you and prevented you from making a

solid conclusion for your project?

Did everyone work on the project why or why not?

or

Did some of the members put in more time than others or was the workload evenly

split up between all the group members?

What did you learn about working together as a group?

The 6th page should include your bibliography and any cited works or web

addresses used for your synopsis and project.

1

Application of Calculus in Music and Harmonics

The project was divided into two parts. That is, types of electronic sound files and their

associated equations, and application of derivatives on music. The members tackled the parts as

follows:

Sub-topic

Types of electronic sound files and the

associated equations

Application of derivatives

Members

Mustafa Khaleel

Omar Al Hatem

Noor Al Hatem

Zahra Yaghubi

There are different electronic sound files, each represented differently mathematically.

Some of the files considered on this project include WAV and MP3 (Kande & Divandari, 2019).

The WAV, has a function expressed as:

f(t − T) + g(t) where:

T: Time taken by sound to travel from the speaker to the receiver

g(t): The background noise function

The MP3 had a function expressed as:

f(t) = ∑ f(n)sinc(t + n)

Derivatives in Music

2

Sound wave is defined as a pattern of disturbance as a result of movement of energy through a

medium as it propagate away from the source. The disturbance pattern can be represented as sine

function.

f(t) = sin(2πft)

Where f is the frequency.

The derivative of the sound function is:

f ′ (t) = 2πf cos(2πft) applying the derivatives of trignometric function and chain rule

The derivative function shows the fluctuations and dynamic changes of the sound and how well

the volume can be balanced. In addition, the derivative function is used to determine the speed

with which the volume changes as well as the intensity of the sound.

Sample Calculation

Assuming the function of the music to be:

f(t) = tsin(t)

The graph of the function looks like:

The upper wave form function is:

3

𝑓 (𝑡 ) = 𝑡

Its derivative becomes

𝑓 ′ (𝑡 ) = 1

The lower wave form function is:

𝑓 (𝑡) = −𝑡

Its derivative becomes

𝑓 ′ (𝑡) = −1

The derivative of the lower and upper wave form means that the volume (amplitude) of the

music increases by 2 for every progression and 1 across the time axis. However, in real life a

steady crescendo like this would be less rapid. In such a case the equation may be 0.0001tsin (t),

where the volume changes with 1/5000 for each progression of a unit of time. The amplitude,

therefore, gives information to the speaker regarding how loud one section of the music should

be compared to the rest. The derivative of the functions tells the speaker how much to crescendo

by or diminuendo by over time (Vines, Nuzzo, & Levitin, 2005).

Formulas Used.

f(t) = sin(2πft)

f(t − T) + g(t)

f(t) = ∑ f(n)sinc(t + n)

Conclusion

Music is considered to represent the dynamics of human emotion. When listening to the music,

human emotions do as the dynamics and tension created by music fluctuates. In conclusion,

calculus is a powerful tool in showing how the fluctuations and dynamics changes, as well as

how well the volume is balanced.

4

Mathematical models

f(t) = sin(2πft)

let u = 2πft

du

= 2πf

dt

𝑦 = sin (𝑢)

𝑓 ′ (𝑡 ) =

dy

= cos (u)

du

𝑓 ′ (𝑡 ) =

dy dy du

=

×

dt du dt

𝑓 ′ (𝑡 ) =

dy

= cos(𝑢) × 2πf = 2πf cos(2πft)

dt

f ′ (t) = 2πf cos(2πft)

Problem faced:

Analysis of music require a combination of mathematical concepts, of which some are

not covered in our course. For instance, Fourier series that transform function from time domain

to frequency domain and vice versa, play a major role in the analysis of sound wave. As a group

we had little idea about Fourier series since it is a topic to be covered later.

About the group members

Every member of the group participated and the time allocation was uniformly distributed.

Working as a group was an awesome experience mostly when brainstorming on ideas. It was

interesting to learn from each other and examine the topic from the perspective of other

members.

5

References

Kande, R. R., & Divandari, J. (2019). Mathematics, Music, and Architecture.

Lostanlen, V., Andén, J., & Lagrange, M. (2019). Fourier at the heart of computer music: From

harmonic sounds to texture. Comptes Rendus Physique, 20(5), 461-473.

Vines, B. W., Nuzzo, R. L., & Levitin, D. J. (2005). Analyzing Temporal Dynamics in Music;

Differential Calculus, Physics, and Functional Data Analysis Techniques. Music

Perception, 23(2), 137-152.

Presentation

prepared by video14-15 minutes. This project has taken your group all semester to create

so anything less than14minutes may suggest you did not do much in research for the

project. Must have all members active for the presentation Clearly explain to the class the

title and what your group researched.

How was your project related to this course?

Any Mathematical models, analytics, and calculations you used to arrive at

your group’s conclusions.

I will be looking at overall depth of analysis used. So, show this in the presentation,

but even more so in the synopsis.

Make the presentation with a high level of energy. You should be excited about sharing the

presentation to the class. In saying that, selling to the class like you were competing for

American Idol.

Application of Calculus in Music and

Harmonics

Introduction

Interestingly, there is the math behind every beautiful sound of music that

we listen to. When we hear a flutist, a signal is usually sent from their fingers

to the ears. As the flute is played, it makes vibrations that travel through the

air to the listeners’ eardrums. Sound is the transfer of energy that produces

air pressure waves. The vibrations are oscillations in air pressure which are

translated as sound. The waves’ frequency determines the sound’s pitch,

and the volume is influenced by the strength and size of the waves.

Objectives

The purpose of our project is to present a detailed analysis of how

calculus could be applied in harmonics and music

Electronic Sound files

•

WAV FILE

𝑓 𝑡 − 𝑇 + 𝑔(𝑡)

Where:

T: Time taken by sound to travel from the speaker to the receiver

g t : The background noise function

•

THE MP3 FILE

𝑓 𝑡 = σ 𝑓 𝑛 𝑠𝑖𝑛𝑐(𝑡 + 𝑛)

Application of Derivative

f t = sin 2πft

Where f is the frequency.

The derivative of the sound function is:

f ′ t = 2πf cos 2πft applying the derivatives of trignometric function

The derivative function shows the fluctuations and dynamic changes

of the sound and how well the volume can be balanced. In addition,

the derivative function is used to determine the speed with which the

volume changes as well as the intensity of the sound.

Example

Assuming the function of the music to be:

f t = tsin(t)

The graph of the function:

Example continued

The upper wave form function is:

𝑓 𝑡 =𝑡

Its derivative becomes

𝑓′ 𝑡 = 1

The lower wave form function is:

𝑓 𝑡 = −𝑡

Its derivative becomes

𝑓 ′ 𝑡 = −1

Example continued

The derivative of the lower and upper wave form means that the

volume (amplitude) of the music increases by 2 for every progression

and 1 across the time axis. However, in real life a steady crescendo like

this would be less rapid. In such a case the equation may be 0.0001tsin

(t), where the volume changes with 1/5000 for each progression of a

unit of time. The amplitude, therefore, gives information to the speaker

regarding how loud one section of the music should be compared to

the rest. The derivative of the functions tells the speaker how much to

crescendo by or diminuendo over time.

Conclusion

Music is considered to represent the dynamics of human emotion.

When listening to the music, human emotions do as the dynamics and

tension created by music fluctuates. In conclusion, calculus is a

powerful tool in showing how the fluctuations and dynamics changes,

as well as how well the volume is balanced.

Challenges Faced

Analysis of music require a combination of mathematical concepts, of

which some are not covered in our course. For instance, Fourier series

that transform function from time domain to frequency domain and

vice versa, play a major role in the analysis of sound wave. As a group

we had little idea about Fourier series since it is a topic to be covered

later.

References

Kande, R. R., & Divandari, J. (2019). Mathematics, Music, and

Architecture.

Lostanlen, V., Andén, J., & Lagrange, M. (2019). Fourier at the heart

of computer music: From harmonic sounds to texture. Comptes

Rendus Physique, 20(5), 461-473.

Vines, B. W., Nuzzo, R. L., & Levitin, D. J. (2005). Analyzing Temporal

Dynamics in Music; Differential Calculus, Physics, and Functional

Data Analysis Techniques. Music Perception, 23(2), 137-152.

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