# Application of Mathematics in Culinary Arts Paper and Presentation

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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.
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
What problems did you face during your research?
Was there any information that stumbled you and prevented you from making a
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
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.
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
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.