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linear mathematical music.nb
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linear mathematical music.nb
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(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
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(* Beginning of Notebook Content *)
Notebook[{
Cell[CellGroupData[{
Cell["Mathematical Music", "Section",
CellChangeTimes->{{3.881164249977367*^9,
3.881164252468671*^9}},ExpressionUUID->"f47f6a60-ab4b-480d-86b3-\
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Cell["\<\
I first considered the connection between math and music when I was listening \
to the song \[OpenCurlyDoubleQuote]No Money\[CloseCurlyDoubleQuote] by \
Galantis. It\[CloseCurlyQuote]s a fascinating song with many different \
instruments and a touch of something electronic as well, and I was curious if \
I could somehow represent the song in a different way. Being a mathematician, \
I naturally tried to use equations and expressions to represent the song, and \
so began my journey in the realm of mathematical music.\
\>", "Text",
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Cell["\<\
Mathematical music is a broad term and has varying meanings to people. It\
\[CloseCurlyQuote]s most commonly viewed as the math behind frequency, pitch, \
rhythm, and other facets of music, such as how chords of prime numbers (2nd, \
3rd, 5th, and 7th) sound instinctively right for musicians. However, I was \
interested in something different: what mathematical equation represents the \
melody of a song? If I had an equation, could I turn that into a piece of \
music?\
\>", "Text",
CellChangeTimes->{{3.881081531952589*^9, 3.881081594365638*^9}, {
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Cell[CellGroupData[{
Cell["The General Plan", "Subsection",
CellChangeTimes->{{3.881164264433848*^9,
3.8811642675148277`*^9}},ExpressionUUID->"94b41c84-ae45-40da-ab6c-\
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Cell["\<\
Before I could start coming up with a solution, I had to determine precisely \
what I was trying to find. While I was trying to find the connection between \
math and music, it is important to note that I was pursuing this connection \
only one way\[LongDash] turning a given equation into a song. The \
corresponding connection would be to find an equation given a song, but that \
was not my goal in this part of the project. \
\>", "Text",
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"My general plan was to find the values of an equation at certain points and \
then convert these values into notes. For example, if I was looking at the \
function ",
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FontSlant->"Italic"],
", then at every integer value of ",
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" from 1 to 10, the melody would be comprised of notes with number values \
from 1 to 10. However, this presented a problem immediately: the range of \
functions. "
}], "Text",
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Cell["\<\
There were only so many notes that I could use, but some equations went on to \
infinity (such as quadratic equations) while others were periodic with ranges \
beyond possible notes (such as sin functions with large amplitudes). And so, \
I split my solution into two cases and came up with slightly separate ways of \
dealing with each type of function.\
\>", "Text",
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"My first case included all functions that had an infinite range. It didn\
\[CloseCurlyQuote]t matter if it went to positive or negative infinity\
\[LongDash] if the function increased/decreased without limit, then it was \
considered in this case. To determine whether a function had an infinite \
range, I used the functions MaxValue and MinValue. I used these functions \
with an if statement to determine if a function had an infinite range, and \
put it all inside a custom function, which I named ",
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"."
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"Determining if a function had an infinite range with my ",
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":"
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Cell["\<\
Testing it out on some functions, we see it works as intended:\
\>", "Text",
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"Once I could determine if a function was infinite, I needed to find a way \
to express these infinite notes as values. My solution was to place a cap on \
the function\[CloseCurlyQuote]s values. After experimenting with number \
values for notes in the SoundNote function, I determined that a reasonable \
range was between -30 and +30, which meant I had to limit the values of these \
functions to be between negative and positive 30. And so, I created the ",
StyleBox["value",
FontSlant->"Italic"],
" function to find the values of the function for notes. Essentially, if the \
function\[CloseCurlyQuote]s value was below -30, I set it to -30, and if it \
was above +30, I set it to +30. If it was between -30 and +30, I set the note \
value to be the floor of the function\[CloseCurlyQuote]s value (since \
SoundNote only accepted integers)."
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Cell["\<\
What exactly is the note value? To explain it best, think of the notes on a \
piano. Reduce the 88 keys to a reasonable range (which I took as 60 notes). \
Imagine middle C (C4) as 0, and for every half-step up, increase the note \
value by 1. In that sense, the note value for D# is 3. For every half-step \
down, decrease the note value by 1. In that sense, the note value for B is \
-1. With a range of -30 to +30, the notes go from F#1 to F#6, a range of 5 \
octaves.\
\>", "Text",
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Cell[TextData[{
"Using the ",
StyleBox["value",
FontSlant->"Italic"],
" function to change a function\[CloseCurlyQuote]s values into usable note \
values:"
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3.881083546301227*^9}},ExpressionUUID->"96d740ac-a3a9-442d-87bd-\
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CellLabel->"In[77]:=",ExpressionUUID->"9d1d85ed-d647-4f15-8e2d-d6801294f579"],
Cell["\<\
We can see this function working as intended after testing it out on a \
quadratic function.\
\>", "Text",
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"I noticed that the shape of the notes were very similar to the quadratic \
function, which was exactly what I wanted. However, this didn\
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I experimented with many different things in order to reach the right timing \
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\[CloseCurlyQuote]s tempo or making the tempo a factor of the note values. I\
\[CloseCurlyQuote]ve included the some of my experiments below.\
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