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<div id="content">
<h1 class="title">Acoustics Final Report
<br />
<span class="subtitle">Northeastern University MUSC2350, Spring 2020</span>
</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org74d2593">Preface</a></li>
<li><a href="#org8468a5f">1. Simulating Strings</a>
<ul>
<li><a href="#org741323d">1.1. Getting set up with Octave</a></li>
<li><a href="#orgbb936e9">1.2. Pure tones and equal-amplitude harmonics</a></li>
<li><a href="#org0fba2db">1.3. Relative amplitudes of harmonics</a></li>
<li><a href="#orgb918584">1.4. Damping</a></li>
<li><a href="#org60491ab">1.5. Soundboard</a></li>
<li><a href="#orgd070ee4">1.6. Subtractive synthesis</a></li>
</ul>
</li>
<li><a href="#org37804e5">2. Woodwinds: Edgetones</a>
<ul>
<li><a href="#orgbfd0ef2">2.1. Sound source</a></li>
<li><a href="#orgd2174a3">2.2. Sound modifier</a></li>
<li><a href="#org18aa0d3">2.3. Modifying the modifier</a></li>
</ul>
</li>
<li><a href="#orgae6cce0">3. Bibliography</a></li>
<li><a href="#orga882c5f">4. Appendix</a>
<ul>
<li><a href="#org947f5a1">4.1. Program listing</a></li>
</ul>
</li>
</ul>
</div>
</div>
<p>
You can find the pdf version of this report <a href="./matt-dailis-acoustics-report.pdf">here</a>.
</p>
<div id="outline-container-org74d2593" class="outline-2">
<h2 id="org74d2593">Preface</h2>
<div class="outline-text-2" id="text-org74d2593">
<p>
This is Matt Dailis's<sup><a id="fnr.1" class="footref" href="#fn.1">1</a></sup> work for
<i>Acoustics and Psychoacoustics of Music</i> taught by Victor
Zappi<sup><a id="fnr.2" class="footref" href="#fn.2">2</a></sup> at Northeastern University,
Spring 2020.
</p>
<p>
In Section 1, I will describe my attempt at using additive synthesis
to simulate the sound of a guitar. I jumped right into this without
examining much prior work, so my approach was a little naive and
primitive. I do explain concepts along the way, and I reference some
alternative approaches at the end.
</p>
<p>
In Section 2, I will give a more concise description of edgetones and
wind instruments, based largely on what we covered in class.
</p>
<p>
The appendix contains a full program listing of my octave
program. It can also be found at
<a href="https://github.com/mattdailis/simulating-strings">https://github.com/mattdailis/simulating-strings</a>.
</p>
</div>
</div>
<div id="outline-container-org8468a5f" class="outline-2">
<h2 id="org8468a5f"><span class="section-number-2">1</span> Simulating Strings</h2>
<div class="outline-text-2" id="text-1">
<p>
My intent is to attempt to simulate a guitar using purely math. I
wanted to know how big the gap is between the theory I learned in
class and the true complexity of a real musical instrument. I pulled
out my notes from class, fired up my laptop, and decided to give it
a try!
</p>
</div>
<div id="outline-container-org741323d" class="outline-3">
<h3 id="org741323d"><span class="section-number-3">1.1</span> Getting set up with Octave</h3>
<div class="outline-text-3" id="text-1-1">
<p>
For this project, I used <i><a href="https://www.gnu.org/software/octave/">GNU Octave</a></i>, an open source programming
language and environment for mathematical modeling, based off of
<a href="https://www.mathworks.com/products/matlab.html">MATLAB</a>.
</p>
<p>
Octave has an <code>audioplayer</code> function, which when provided with a
vector of floating point numbers between \(-1\) and \(1\), treats them as
a waveform and plays them back.
</p>
<div class="org-src-container">
<pre class="src src-octave" id="org5995aad">audioplayer (vector<span style="color: #006FE0;">,</span> bit_rate<span style="color: #006FE0;">,</span> bit_depth)
</pre>
</div>
<p>
A <i>sample</i> is a discrete measurement of <i>sound pressure level</i> (SPL)
averaged over a predefined duration of time. The <code>bit_rate</code> variable
represents the number of samples to play per second. I chose to set
this to be \(44100\), which is a standard bit rate used for
CDs.<sup><a id="fnr.3" class="footref" href="#fn.3">3</a></sup> The <code>bit_depth</code>
variable has to do with the precision of the floating point numbers
themselves.
</p>
</div>
</div>
<div id="outline-container-orgbb936e9" class="outline-3">
<h3 id="orgbb936e9"><span class="section-number-3">1.2</span> Pure tones and equal-amplitude harmonics</h3>
<div class="outline-text-3" id="text-1-2">
<p>
My approach to simulating strings was to use <i>additive
synthesis</i><sup><a id="fnr.4" class="footref" href="#fn.4">4</a></sup>. This
means that I attempted to simulate a vibrating string by building it
up as a sum of partials. To start, I generated a waveform for the the
fundamental frequency using octave's <code>sinewave</code> function <i>(See Listing
<a href="#orga3802c7">1</a>)</i>. Given a vector size and a period, it returns a set of
values between -1 and 1 in the form of a sine wave with the specified
period.
</p>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 1: </span>Octave provides a convenient sinewave function, which asks for a vector size and a period measured in number of samples</label><pre class="src src-octave" id="orga3802c7">f1 <span style="color: #006FE0;">=</span> sinewave(bitRate <span style="color: #006FE0;">*</span> 4<span style="color: #006FE0;">,</span> bitRate <span style="color: #006FE0;">/</span> 440)
</pre>
</div>
<p>
I defined my own convenience function <code>puretone</code> which would take the
bit rate, duration, and frequency and return the corresponding sine
wave <i>(See Listing <a href="#org20dd99f">2</a>)</i>.
</p>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 2: </span>I defined my own <code>puretone</code> function which allows me to think in terms of frequency instead of period</label><pre class="src src-octave" id="org20dd99f"><span style="color: #0000FF;">function</span> <span style="color: #006699;">puretone</span>(seconds<span style="color: #006FE0;">,</span> frequency)
sinewave(bitRate <span style="color: #006FE0;">*</span> seconds<span style="color: #006FE0;">,</span>
bitRate<span style="color: #006FE0;">/</span>frequency)<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
</pre>
</div>
<p>
Now I had the ability to make pure tones, but I wanted <i>harmonics</i>. A
harmonic is a partial whose frequency is an <i>integer multiple of the
fundamental</i>.<sup><a id="fnr.5" class="footref" href="#fn.5">5</a></sup> We usually
only care about the first six harmonics or so, because after that
they start to get to very high frequencies near the edge of human
hearing. I defined a <code>createharmonics</code> function that returns a sum of
six harmonics <i>(See Listing <a href="#org0e04cd6">3</a>)</i>. Notice that the
returned vector must be divided by six to make sure the whole range
of values is between \(-1\) and \(1\).
</p>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 3: </span><code>createharmonics</code> generates the first six harmonics and adds them together</label><pre class="src src-octave" id="org0e04cd6">createharmonics(duration<span style="color: #006FE0;">,</span> fundamental)<span style="color: #006FE0;">:</span>
f1 <span style="color: #006FE0;">=</span> puretone(duration<span style="color: #006FE0;">,</span> fundamental)<span style="color: #006FE0;">;</span>
f2 <span style="color: #006FE0;">=</span> puretone(duration<span style="color: #006FE0;">,</span> fundamental <span style="color: #006FE0;">*</span> 2)<span style="color: #006FE0;">;</span>
f3 <span style="color: #006FE0;">=</span> puretone(duration<span style="color: #006FE0;">,</span> fundamental <span style="color: #006FE0;">*</span> 3)<span style="color: #006FE0;">;</span>
f4 <span style="color: #006FE0;">=</span> puretone(duration<span style="color: #006FE0;">,</span> fundamental <span style="color: #006FE0;">*</span> 4)<span style="color: #006FE0;">;</span>
f5 <span style="color: #006FE0;">=</span> puretone(duration<span style="color: #006FE0;">,</span> fundamental <span style="color: #006FE0;">*</span> 5)<span style="color: #006FE0;">;</span>
f6 <span style="color: #006FE0;">=</span> puretone(duration<span style="color: #006FE0;">,</span> fundamental <span style="color: #006FE0;">*</span> 6)<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">return</span> (f1 <span style="color: #006FE0;">+</span> f2 <span style="color: #006FE0;">+</span> f3 <span style="color: #006FE0;">+</span> f4 <span style="color: #006FE0;">+</span> f5 <span style="color: #006FE0;">+</span> f6) <span style="color: #006FE0;">/</span> 6<span style="color: #006FE0;">;</span>
</pre>
</div>
<p>
I was so excited about the fact that my equations were producing the
pitches that I wanted that I created a sample song using this function.
</p>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 4: </span>A sample song using the functions created so far - it sort of sounds like music!</label><pre class="src src-octave" id="org874ed42">A3 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 220)<span style="color: #006FE0;">;</span>
A4 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 440)<span style="color: #006FE0;">;</span>
A5 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 880)<span style="color: #006FE0;">;</span>
B4 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 495)<span style="color: #006FE0;">;</span>
C5 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 523.26)<span style="color: #006FE0;">;</span>
D4 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 293.33)<span style="color: #006FE0;">;</span>
D5 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 293.33 <span style="color: #006FE0;">*</span> 2)<span style="color: #006FE0;">;</span>
E4 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 330)<span style="color: #006FE0;">;</span>
E5 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 660)<span style="color: #006FE0;">;</span>
F5 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 348.84 <span style="color: #006FE0;">*</span> 2)<span style="color: #006FE0;">;</span>
GS4 <span style="color: #006FE0;">=</span> createharmonics(0.5<span style="color: #006FE0;">,</span> 415.305)<span style="color: #006FE0;">;</span>
aMinor <span style="color: #006FE0;">=</span> [A4<span style="color: #006FE0;">,</span> (C5 <span style="color: #006FE0;">+</span> E5) <span style="color: #006FE0;">/</span> 2<span style="color: #006FE0;">,</span>
E4<span style="color: #006FE0;">,</span> (C5 <span style="color: #006FE0;">+</span> E5) <span style="color: #006FE0;">/</span> 2]<span style="color: #006FE0;">;</span>
eMajor <span style="color: #006FE0;">=</span> [B4<span style="color: #006FE0;">,</span> (E5 <span style="color: #006FE0;">+</span> GS4) <span style="color: #006FE0;">/</span> 2<span style="color: #006FE0;">,</span>
E4<span style="color: #006FE0;">,</span> (D5 <span style="color: #006FE0;">+</span> GS4) <span style="color: #006FE0;">/</span> 2]<span style="color: #006FE0;">;</span>
dMinor <span style="color: #006FE0;">=</span> [A4<span style="color: #006FE0;">,</span> (D5 <span style="color: #006FE0;">+</span> F5) <span style="color: #006FE0;">/</span> 2<span style="color: #006FE0;">,</span>
D4<span style="color: #006FE0;">,</span> (D5 <span style="color: #006FE0;">+</span> F5) <span style="color: #006FE0;">/</span> 2]<span style="color: #006FE0;">;</span>
song <span style="color: #006FE0;">=</span> [aMinor<span style="color: #006FE0;">,</span> eMajor<span style="color: #006FE0;">,</span> aMinor<span style="color: #006FE0;">,</span> eMajor<span style="color: #006FE0;">,</span>
dMinor<span style="color: #006FE0;">,</span> aMinor<span style="color: #006FE0;">,</span> eMajor<span style="color: #006FE0;">,</span>
A4<span style="color: #006FE0;">,</span> E4<span style="color: #006FE0;">,</span> A3]<span style="color: #006FE0;">;</span>
playSound(song<span style="color: #006FE0;">,</span> bitRate)
</pre>
</div>
<p>
You can hear the result here:
</p>
<p>
<audio controls src="audio/string-simulation-0.wav"></audio>
</p>
<p>
After listening to the result, I could recognize this as music, but
it sounded nothing like a guitar. What's missing?
</p>
<p>
First off, in a string, the relative amplitudes of the harmonics are
not all the same.<sup><a id="fnr.6" class="footref" href="#fn.6">6</a></sup>
Secondly, for a plucked instrument, the amplitudes of all of the
harmonics change over time, eventually diminishing to
silence. Lastly, the soundboard of the instrument will act as a
filter affecting the output of the instrument.<sup><a id="fnr.7" class="footref" href="#fn.7">7</a></sup> Let's tackle these issues one by one.
</p>
</div>
</div>
<div id="outline-container-org0fba2db" class="outline-3">
<h3 id="org0fba2db"><span class="section-number-3">1.3</span> Relative amplitudes of harmonics</h3>
<div class="outline-text-3" id="text-1-3">
<p>
First off, the fundamental frequency of a plucked string will
always be the most prevalent harmonic.<sup><a id="fnr.8" class="footref" href="#fn.8">8</a></sup> The
relative amplitudes of harmonics of a plucked string depend on the
pluck location.
</p>
<p>
We model a pluck as a "kink" in the string.<sup><a id="fnr.9" class="footref" href="#fn.9">9</a></sup> The
prevalence of each harmonic depends on whether the initial kink
location is at one of that harmonic's nodes or antinodes. Put
another way, it depends on the <i>similarity</i> of the string shape at
the moment of the pluck to the shape of the resonant mode.
</p>
<p>
Similarity, in linear algebra, is defined as the dot product
between two vectors. The more "aligned" those two vectors are, the
higher their dot product.
</p>
<p>
If we take the <i>fourier transform</i> of the string shape, we should
get an idea for which frequencies are represented. Let's first
define the shape of our string.
</p>
<p>
Let's define a kink in terms of a piecewise function.
</p>
<p>
Let \(k\) be the kink location whose value is between \(0\) and \(1\),
and \(L\) be the length of the string.
</p>
<p>
\[y_1={\frac x kL}, x \leq kL\]
</p>
<p>
\[y_2 = {\frac {1 - {\frac x L}} {1 - k}}, x > kL\]
</p>
<p>
The following pairs of graphs show the kink function on the left,
and its FFT on the right. The only axis worth looking at is the x
axis of the FFTs - each number corresponds to the harmonic index.
</p>
<p>
These images were generated using <i>octave-online</i><sup><a id="fnr.10" class="footref" href="#fn.10">10</a></sup> with the following call:
</p>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 5: </span>This line of code generated the graphs below</label><pre class="src src-octave" id="org994d9ca">v <span style="color: #006FE0;">=</span> kink(1000<span style="color: #006FE0;">,</span> 0.1)
bar(abs(fft(v<span style="color: #006FE0;">-</span>mean(v)))(1<span style="color: #006FE0;">:</span>10)(2<span style="color: #006FE0;">:</span>end))
</pre>
</div>
<div id="org8da4f82" class="figure">
<p><img src="./images/kinkfft50.png" alt="kinkfft50.png" />
</p>
<p><span class="figure-number">Figure 1: </span><code>kink(0.5)</code> and its FFT</p>
</div>
<p>
Notice that the fundamental is always the most prominent, but the
behavior of the rest of the harmonics varies. Observe <i>Figure
<a href="#org8da4f82">1</a></i> - the pluck location is in the center of the string,
which emphasizes odd harmonics, and has no even harmonics because
all even harmonics have a node in the center.
</p>
<div id="orge89f285" class="figure">
<p><img src="./images/kinkfft25.png" alt="kinkfft25.png" />
</p>
<p><span class="figure-number">Figure 2: </span><code>kink(0.25)</code> and its FFT</p>
</div>
<p>
Moving the pluck location to the quarter point of the string (<i>Figure <a href="#orge89f285">2</a></i>), we
see more harmonics pop up, but the fourth and eighth (and all multiples
of four) are still silent, because the kink location is at the node
of the fourth harmonic.
</p>
<div id="orgeb66e4c" class="figure">
<p><img src="./images/kinkfft10.png" alt="kinkfft10.png" />
</p>
<p><span class="figure-number">Figure 3: </span><code>kink(0.1)</code> and its FFT</p>
</div>
<p>
In <i>Figure <a href="#orgeb66e4c">3</a></i>, all nine of the first harmonics are
present. The tenth is not pictured, but it would be zero, because
it has a node at the pluck location.
</p>
<p>
This is the result of scaling the harmonics using the weights from the FFT:
</p>
<p>
<audio controls src="audio/string-simulation-1.wav"></audio>
</p>
<p>
After listening to this result, I found that it sounded a little
better - the fundamental was more prominent than before. It still
did not sound like a physical string though.
</p>
</div>
</div>
<div id="outline-container-orgb918584" class="outline-3">
<h3 id="orgb918584"><span class="section-number-3">1.4</span> Damping</h3>
<div class="outline-text-3" id="text-1-4">
<p>
When one plucks a string, it does not sustain the sound for very
long. Immediately, it starts to lose energy to friction at the
imperfect boundaries of the string, as well as friction with the
fluid (air) in which it is
vibrating.<sup><a id="fnr.11" class="footref" href="#fn.11">11</a></sup> I hoped that
adding damping will at least make it sound plausible that the
strings are being plucked.
</p>
<p>
Let's focus on the kinetic energy lost due to the motion of the
bridge, since that is more significant than the energy lost to the
air.<sup><a id="fnr.12" class="footref" href="#fn.12">12</a></sup> The way we take into
account the bridge motion is by modeling it as an impedance
mismatch, similar to how we would model a tube open on one
end. This results in an exponential decay.
</p>
<div class="org-src-container">
<label class="org-src-name"><span class="listing-number">Listing 6: </span>I found that a decay halflife of about 0.3 seconds sounded good to me</label><pre class="src src-octave"><span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">damping</span>(x<span style="color: #006FE0;">,</span> dampingTime<span style="color: #006FE0;">,</span> bitRate)
y <span style="color: #006FE0;">=</span> 0.5 <span style="color: #006FE0;">^</span> (x <span style="color: #006FE0;">/</span> (dampingTime <span style="color: #006FE0;">*</span> bitRate))<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
</pre>
</div>
<p>
In this model, all of the frequencies decay at the same rate, which
isn't necessarily accurate, although looking at a the spectrum of
plucking my guitar string, I think this is a reasonable
approximation.<sup><a id="fnr.13" class="footref" href="#fn.13">13</a></sup>
</p>
<p>
<audio controls src="audio/string-simulation-2.wav"></audio>
</p>
</div>
</div>
<div id="outline-container-org60491ab" class="outline-3">
<h3 id="org60491ab"><span class="section-number-3">1.5</span> Soundboard</h3>
<div class="outline-text-3" id="text-1-5">
<p>
Okay, we've now made a generic plucked string instrument, but what
makes a guitar a guitar? One of the aspects that has the biggest
contribution to the timbre of a stringed instrument is its
<i>soundboard</i>. A soundboard is a resonance chamber that takes the
input vibration from a string and transforms its frequency
spectrum, behaving as an acoustic filter. In a guitar, the string
transfers its vibration through the bridge and into the top of the
guitar. The top of the guitar is an <i>idiophone</i><sup><a id="fnr.14" class="footref" href="#fn.14">14</a></sup> that creates a pressure wave inside the body as it
vibrates. It is the modes of this piece of wood plus the sound
propagation inside of the body that together create this acoustic
filter.<sup><a id="fnr.15" class="footref" href="#fn.15">15</a></sup>
</p>
<p>
To implement a filter in octave, I intended to use the <code>signal</code>
library. While I did eventually manage to install it, I did not
have enough time to implement this part before the project
deadline. However, I read some papers about soundboard
design. Luthiers install <i>braces</i>, which are strips of wood glued
to the soundboard to create areas of greater stiffness, which
encourages modes that have nodes in those locations.<sup><a id="fnr.16" class="footref" href="#fn.16">16</a></sup>
That same paper included Figure <a href="#org0d868b5">4</a>, which shows the
frequency responses of "good" versus "bad" quality guitars. They
both show peaks around 110 and 220 hertz, though the good guitars
have higher amplitude peaks.
</p>
<div id="org0d868b5" class="figure">
<p><img src="./images/soundboard-frf.png" alt="soundboard-frf.png" />
</p>
<p><span class="figure-number">Figure 4: </span>This diagram was taken from "Frequency Response Function Of A Guitar - A Significant Peak" By Samo Šali</p>
</div>
</div>
</div>
<div id="outline-container-orgd070ee4" class="outline-3">
<h3 id="orgd070ee4"><span class="section-number-3">1.6</span> Subtractive synthesis</h3>
<div class="outline-text-3" id="text-1-6">
<p>
When I got this far in the project, for the first time I actually
searched for "synthesizing guitar sound" on the internet.<sup><a id="fnr.17" class="footref" href="#fn.17">17</a></sup> I found that the most commonly used
algorithm for generating guitar sounds does <i>not</i> use additive
synthesis! Instead, it uses subtractive synthesis, which means it
starts with all possible frequencies (i.e. white noise), and
filters them down to the frequencies of a guitar.
</p>
</div>
<div id="outline-container-org4f18787" class="outline-4">
<h4 id="org4f18787"><span class="section-number-4">1.6.1</span> The Karplus-Strong Algorithm</h4>
<div class="outline-text-4" id="text-1-6-1">
<p>
The Karplus-Strong algorithm<sup><a id="fnr.18" class="footref" href="#fn.18">18</a></sup> is a way of
cheaply synthesizing guitar-like sounds using one or two sine wave
oscillators. It can be summarized by four steps (see Figure
<a href="#org0d01a72">5</a>)
</p>
<ol class="org-ol">
<li>Generate a short burst of white noise</li>
<li>Apply delay</li>
<li>Pass it through a filter</li>
<li>Loop</li>
</ol>
<div id="org0d01a72" class="figure">
<p><img src="./images/karplusstrong.png" alt="karplusstrong.png" />
</p>
<p><span class="figure-number">Figure 5: </span>Karplus Strong</p>
</div>
<p>
The most important part of this algorithm is the interaction of the
delay in step 2 with the filter in step 3. The delay helps select
the frequency, while the filter creates the timbre. The loop step
allows the sound to change over time.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org37804e5" class="outline-2">
<h2 id="org37804e5"><span class="section-number-2">2</span> Woodwinds: Edgetones</h2>
<div class="outline-text-2" id="text-2">
<p>
This section will give a shallow overview of wind instruments, with
a deep dive in the middle on <i>edgetones</i>.
</p>
<p>
A <i>wind instrument</i> is similar to a stringed instrument in that it
has a <i>sound source</i> and a <i>sound modifier</i>. However, instead of
having a vibrating soundboard, wind instruments typically have a
tube that contains a one-dimensional air column through which sound
propagates as a <i>longitudinal wave</i>.<sup><a id="fnr.19" class="footref" href="#fn.19">19</a></sup>
</p>
</div>
<div id="outline-container-orgbfd0ef2" class="outline-3">
<h3 id="orgbfd0ef2"><span class="section-number-3">2.1</span> Sound source</h3>
<div class="outline-text-3" id="text-2-1">
<p>
The sound source is responsible for generating a stream of
vibrating air. We can categorize this generation into three
phenomena: <i>free edge oscillation</i>, <i>reeds</i>, and <i>vibrating
lips</i>. Here, we will only focus on <i>free edge oscillation</i>, since
this is most relevant to the concept of edgetones.
</p>
</div>
<div id="outline-container-orge1e179e" class="outline-4">
<h4 id="orge1e179e"><span class="section-number-4">2.1.1</span> Free edge oscillation</h4>
<div class="outline-text-4" id="text-2-1-1">
<p>
In <i>free edge oscillation</i>, a steady flow of air needs to hit a
sharp object head-on (see Figure <a href="#org610a829">6</a>).
</p>
<div id="org610a829" class="figure">
<p><img src="./images/airwayedge.png" alt="airwayedge.png" />
</p>
<p><span class="figure-number">Figure 6: </span>A narrow stream of air passes through an <b>airway</b> and hits a sharp <b>edge</b> head-on</p>
</div>
<p>
When we talk about a "steady flow of air," we are talking about
<i>laminar flow</i>. Laminar flow is when a fluid moves in smooth
layers (laminae) and each layer is moving in the same direction as
the whole fluid, meaning there are no cross-currents or
eddies.<sup><a id="fnr.20" class="footref" href="#fn.20">20</a></sup> This flow
is laminar inside of the airway, but at some distance from the
airway it becomes <i>turbulent flow</i>. Turbulent flow is when the
motion of a fluid is chaotic and changing. The laminar flow gets a
certain distance into the unconstrained air and loses its
structure, and becomes turbulent.
</p>
<p>
If we place a sharp edge at approximately the distance from the
airway at which the flow naturally becomes turbulent, we force the
flow to pick one side of the edge. The eddies will increase in
intensity on that side and cause the flow to flip to the other
side (see Figure <a href="#org2b662c2">7</a>). This phenomenon will repeat in a
periodic fashion.
</p>
<div id="org2b662c2" class="figure">
<p><img src="./images/edgetone.png" alt="edgetone.png" />
</p>
<p><span class="figure-number">Figure 7: </span>Edgetone illustration from textbook</p>
</div>
<p>
We can describe the frequency of this periodic fluctuation as a
ratio between the the velocity of the air flow, \(v\), and the
distance between the airway and the edge, \(d\).
</p>
<p>
\[f \propto {\frac v d}\]
</p>
<p>
This equation, however, is not 100% correct - the frequency is not
continuous. As velocity increases, at a certain point, frequency
will have a jump discontinuity (See Figure <a href="#org361132d">8</a>). I
do not fully understand why this happens, but it is commonly used
by musicians to acheive higher frequencies.<sup><a id="fnr.21" class="footref" href="#fn.21">21</a></sup> Notice the slope of the lines - this
is not very convenient for musicians, since it is hard to control
your air velocity so precisely. We will see a solution to this in
the <b>edgetones</b> section.
</p>
<div id="org361132d" class="figure">
<p><img src="./images/stepwisepitch.png" alt="stepwisepitch.png" />
</p>
<p><span class="figure-number">Figure 8: </span>Frequency increases stepwise.</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orgd2174a3" class="outline-3">
<h3 id="orgd2174a3"><span class="section-number-3">2.2</span> Sound modifier</h3>
<div class="outline-text-3" id="text-2-2">
<p>
Many wind instruments have a long tube called the <i>bore</i> that
houses the air column. The air column vibrates at resonant modes
that depend on the length<sup><a id="fnr.22" class="footref" href="#fn.22">22</a></sup>, \(L\), of the bore.
\(c\) is the speed of sound in air.
</p>
<p>
\[f_n = {\frac {nc} {2L}}\]
</p>
</div>
<div id="outline-container-org9fc5b9c" class="outline-4">
<h4 id="org9fc5b9c"><span class="section-number-4">2.2.1</span> Edgetones</h4>
<div class="outline-text-4" id="text-2-2-1">
<p>
In a <i>woodwind</i> instrument, whose sound source is an airway
followed by an edge, the gap between the airway and edge is one of
the two open ends of the tube. We learned that tubes with open ends
allow resonant modes with antinodes at the ends. This means that
after the initial "transient" part of the sound, the vibration of
the air column will <i>induce</i> a vibration at the edge. This is
called an <b>edgetone</b>.
</p>
<p>
An edgetone is a form of coupling, like sympathetic vibrations in
strings. It forms a feedback loop, and most interestingly, it
causes the source to vibrate at the resonant mode of the
bore. This is significant, because if you remember from Figure
<a href="#org361132d">8</a> and the corresponding equation, the frequency of
the source depended on air velocity and distance, which are <i>not</i>
properties of the bore! The vibration of the air column has a high
enough amplitude to overcome the eddies of the turbulent flow at
the edge and force the flow to oscillate at a frequency dictated
by the properties of the bore.
</p>
<p>
Does this mean that the air velocity and edge distance have no
effect on the output frequency of the instrument? Not quite. While
it is true that for small changes in velocity, the frequency
remains constant (dictated by the resonant mode of the bore), the
jump discontinuities will have an effect on the output
frequency. The steps in frequency at the edge will help <i>select</i>
which resonant mode of the bore will have the highest
amplitude. Examine Figure <a href="#orgfa60e4c">9</a>. There are now
<i>ranges</i> of values of \({\frac v d}\) that result in the same
frequency.<sup><a id="fnr.23" class="footref" href="#fn.23">23</a></sup>
</p>
<div id="orgfa60e4c" class="figure">
<p><img src="./images/stepwisepitchedgetone.png" alt="stepwisepitchedgetone.png" />
</p>
<p><span class="figure-number">Figure 9: </span>The edgetones flatten pitch change locally, but still react to the jump discontinuities</p>
</div>
</div>
</div>
</div>
<div id="outline-container-org18aa0d3" class="outline-3">
<h3 id="org18aa0d3"><span class="section-number-3">2.3</span> Modifying the modifier</h3>
<div class="outline-text-3" id="text-2-3">
<p>
Unlike an acoustic guitar, which typically leaves the soundboard
the same and changes the sound source, woodwinds usually come with
the ability to dynamically modify the acoustic properties of the
bore. The way they do this is with <i>holes</i>. Holes in the bore force
nodes at those locations because they fix the pressure at the hole
location to be approximately equal to the atmospheric pressure
outside of the bore.
</p>
</div>
</div>
</div>
<div id="outline-container-orgae6cce0" class="outline-2">
<h2 id="orgae6cce0"><span class="section-number-2">3</span> Bibliography</h2>
<div class="outline-text-2" id="text-3">
<ul class="org-ul">
<li><a href="http://mattdailis.github.io/simulating-strings/resources/physics_of_vibrating_strings.pdf">The physics of vibrating strings - Giordano, Gould, Tobochnik</a></li>
<li><a href="http://mattdailis.github.io/simulating-strings/resources/karplus-strong.pdf">Digital Synthesis of Plucked-String and Drum Timbres, Karplus and Strong</a></li>
<li><a href="http://mattdailis.github.io/simulating-strings/resources/response-variation.pdf">Response Variation in a Group of Acoustic Guitars - Mark French</a></li>
<li><a href="http://mattdailis.github.io/simulating-strings/resources/Vitsten.pdf">Simple model for low-frequency guitar function</a></li>
<li><a href="http://mattdailis.github.io/simulating-strings/resources/significant_peak.pdf">Frequency Response Function Of A Guitar: A Significant Peak - Samo Sali</a></li>
<li><a href="http://mattdailis.github.io/simulating-strings/resources/soundboard-review.pdf">Mathematical Modelling and Acoustical Analysis of Classical Guitars and Their Soundboards</a></li>
<li><a href="http://mattdailis.github.io/simulating-strings/resources/loadedstring/StringWave.html">Loaded String simulation source code</a></li>
</ul>
</div>
</div>
<div id="outline-container-orga882c5f" class="outline-2">
<h2 id="orga882c5f"><span class="section-number-2">4</span> Appendix</h2>
<div class="outline-text-2" id="text-4">
</div>
<div id="outline-container-org947f5a1" class="outline-3">
<h3 id="org947f5a1"><span class="section-number-3">4.1</span> Program listing</h3>
<div class="outline-text-3" id="text-4-1">
<p>
The following is the source code for the octave program I wrote for
the simulating strings section.
</p>
<div class="org-src-container">
<pre class="src src-octave"><span style="color: #8D8D84;">#</span><span style="color: #8D8D84; font-style: italic;">pkg load signal</span>
<span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">damping</span>(x<span style="color: #006FE0;">,</span> dampingTime<span style="color: #006FE0;">,</span> bitRate)
y <span style="color: #006FE0;">=</span> 0.5 <span style="color: #006FE0;">^</span> (x <span style="color: #006FE0;">/</span> (dampingTime <span style="color: #006FE0;">*</span> bitRate))<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">createDamping</span>(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> dampingTime)
y <span style="color: #006FE0;">=</span> arrayfun(@(x) damping(x<span style="color: #006FE0;">,</span> bitRate<span style="color: #006FE0;">,</span> dampingTime)<span style="color: #006FE0;">,</span> [1 <span style="color: #006FE0;">:</span> bitRate <span style="color: #006FE0;">*</span> duration])<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> <span style="color: #006699;">playSound</span>(vector<span style="color: #006FE0;">,</span> bitRate)
player <span style="color: #006FE0;">=</span> audioplayer (vector<span style="color: #006FE0;">,</span> bitRate<span style="color: #006FE0;">,</span> 16)<span style="color: #006FE0;">;</span>
play (player)<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">while</span>(isplaying(player))
<span style="color: #0000FF;">endwhile</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">puretone</span>(bitRate<span style="color: #006FE0;">,</span> seconds<span style="color: #006FE0;">,</span> frequency<span style="color: #006FE0;">,</span> phaseShift<span style="color: #006FE0;">=</span>0)
y <span style="color: #006FE0;">=</span> sinewave(bitRate <span style="color: #006FE0;">*</span> seconds<span style="color: #006FE0;">,</span> bitRate<span style="color: #006FE0;">/</span>frequency<span style="color: #006FE0;">,</span> phaseShift)<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">createtone</span>(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> frequency<span style="color: #006FE0;">,</span> dampingFactors)
y <span style="color: #006FE0;">=</span> puretone(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> frequency) <span style="color: #006FE0;">.*</span> dampingFactors<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">createharmonics</span>(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> fundamental<span style="color: #006FE0;">,</span> weights)
dampingFactors <span style="color: #006FE0;">=</span> createDamping(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> 0.3)<span style="color: #006FE0;">;</span>
M <span style="color: #006FE0;">=</span> []<span style="color: #006FE0;">;</span>
<span style="color: #8D8D84; font-style: italic;">## Build up matrix where each row is another harmonic</span>
<span style="color: #0000FF;">for</span> index <span style="color: #006FE0;">=</span> 1 <span style="color: #006FE0;">:</span> length(weights)
M <span style="color: #006FE0;">=</span> [M<span style="color: #006FE0;">;</span> weights(index) <span style="color: #006FE0;">*</span> createtone(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> fundamental <span style="color: #006FE0;">*</span> index<span style="color: #006FE0;">,</span> dampingFactors)]<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfor</span>
<span style="color: #8D8D84; font-style: italic;">## Collapse them at the end</span>
S <span style="color: #006FE0;">=</span> sum(M)<span style="color: #006FE0;">;</span>
y <span style="color: #006FE0;">=</span> S <span style="color: #006FE0;">/</span> max(S)<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> v <span style="color: #006FE0;">=</span> <span style="color: #006699;">ffttest</span>(bitRate<span style="color: #006FE0;">,</span> X<span style="color: #006FE0;">,</span> checkFreq)
L <span style="color: #006FE0;">=</span> length(X)<span style="color: #006FE0;">;</span>
Y <span style="color: #006FE0;">=</span> fft(X)<span style="color: #006FE0;">;</span>
P2 <span style="color: #006FE0;">=</span> abs(Y <span style="color: #006FE0;">/</span> L)<span style="color: #006FE0;">;</span>
P1 <span style="color: #006FE0;">=</span> P2(1<span style="color: #006FE0;">:</span>(L <span style="color: #006FE0;">/</span> 2) <span style="color: #006FE0;">+</span> 1)<span style="color: #006FE0;">;</span>
checkIndex <span style="color: #006FE0;">=</span> checkFreq <span style="color: #006FE0;">*</span> L <span style="color: #006FE0;">/</span> bitRate<span style="color: #006FE0;">;</span>
[checkVal<span style="color: #006FE0;">,</span> checkIndex2] <span style="color: #006FE0;">=</span> max(P1(checkIndex <span style="color: #006FE0;">-</span> 5 <span style="color: #006FE0;">:</span> checkIndex <span style="color: #006FE0;">+</span> 5))<span style="color: #006FE0;">;</span>
v <span style="color: #006FE0;">=</span> checkVal<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #8D8D84;">## </span><span style="color: #8D8D84; font-style: italic;">location must be between 0.0 and 1.0 </span>
<span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">kink</span>(L<span style="color: #006FE0;">,</span> location)
k <span style="color: #006FE0;">=</span> location<span style="color: #006FE0;">;</span>
x <span style="color: #006FE0;">=</span> (0 <span style="color: #006FE0;">:</span> L)<span style="color: #006FE0;">;</span>
y1 <span style="color: #006FE0;">=</span> x <span style="color: #006FE0;">/</span> (k <span style="color: #006FE0;">*</span> L)<span style="color: #006FE0;">;</span>
y2 <span style="color: #006FE0;">=</span> (1 <span style="color: #006FE0;">-</span> (x <span style="color: #006FE0;">/</span> L)) <span style="color: #006FE0;">/</span> (1 <span style="color: #006FE0;">-</span> k)<span style="color: #006FE0;">;</span>
y <span style="color: #006FE0;">=</span> [y1(1<span style="color: #006FE0;">:</span>k<span style="color: #006FE0;">*</span>L)<span style="color: #006FE0;">,</span> y2(k<span style="color: #006FE0;">*</span>L<span style="color: #006FE0;">+</span>1<span style="color: #006FE0;">:</span>L)]<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #8D8D84;">## </span><span style="color: #8D8D84; font-style: italic;">Returns the first 10 harmonic weights for the given pluck location</span>
<span style="color: #0000FF;">function</span> y <span style="color: #006FE0;">=</span> <span style="color: #006699;">getHarmonicWeights</span>(pluckLocation)
v <span style="color: #006FE0;">=</span> kink(1000<span style="color: #006FE0;">,</span> pluckLocation)<span style="color: #006FE0;">;</span>
X <span style="color: #006FE0;">=</span> abs(fft(v <span style="color: #006FE0;">-</span> mean(v)))<span style="color: #006FE0;">;</span>
y <span style="color: #006FE0;">=</span> (X <span style="color: #006FE0;">/</span> max(X))(2<span style="color: #006FE0;">:</span>10)<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> <span style="color: #006699;">filtertest</span>()
sf <span style="color: #006FE0;">=</span> 800<span style="color: #006FE0;">;</span> sf2 <span style="color: #006FE0;">=</span> sf<span style="color: #006FE0;">/</span>2<span style="color: #006FE0;">;</span>
data<span style="color: #006FE0;">=</span>[[1<span style="color: #006FE0;">;</span>zeros(sf<span style="color: #006FE0;">-</span>1<span style="color: #006FE0;">,</span>1)]<span style="color: #006FE0;">,</span>
sinetone(25<span style="color: #006FE0;">,</span>sf<span style="color: #006FE0;">,</span>1<span style="color: #006FE0;">,</span>1)<span style="color: #006FE0;">,</span>
sinetone(50<span style="color: #006FE0;">,</span>sf<span style="color: #006FE0;">,</span>1<span style="color: #006FE0;">,</span>1)<span style="color: #006FE0;">,</span>
sinetone(100<span style="color: #006FE0;">,</span>sf<span style="color: #006FE0;">,</span>1<span style="color: #006FE0;">,</span>1)]<span style="color: #006FE0;">;</span>
[b<span style="color: #006FE0;">,</span>a]<span style="color: #006FE0;">=</span>butter ( 1<span style="color: #006FE0;">,</span> 50 <span style="color: #006FE0;">/</span> sf2 )<span style="color: #006FE0;">;</span>
filtered <span style="color: #006FE0;">=</span> filter(b<span style="color: #006FE0;">,</span>a<span style="color: #006FE0;">,</span>data)<span style="color: #006FE0;">;</span>
<span style="color: #0000FF;">endfunction</span>
<span style="color: #0000FF;">function</span> <span style="color: #006699;">main</span>()
bitRate <span style="color: #006FE0;">=</span> 44100<span style="color: #006FE0;">;</span>
weights <span style="color: #006FE0;">=</span> getHarmonicWeights(0.15)<span style="color: #006FE0;">;</span>
duration <span style="color: #006FE0;">=</span> 2.0<span style="color: #006FE0;">;</span>
A3 <span style="color: #006FE0;">=</span> createharmonics(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> 220<span style="color: #006FE0;">,</span> weights)<span style="color: #006FE0;">;</span>
C3 <span style="color: #006FE0;">=</span> createharmonics(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> 523.26 <span style="color: #006FE0;">/</span> 2<span style="color: #006FE0;">,</span> weights)<span style="color: #006FE0;">;</span>
E3 <span style="color: #006FE0;">=</span> createharmonics(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> 330<span style="color: #006FE0;">,</span> weights)<span style="color: #006FE0;">;</span>
E2 <span style="color: #006FE0;">=</span> createharmonics(bitRate<span style="color: #006FE0;">,</span> duration<span style="color: #006FE0;">,</span> 330 <span style="color: #006FE0;">/</span> 2<span style="color: #006FE0;">,</span> weights)<span style="color: #006FE0;">;</span>
aMinor <span style="color: #006FE0;">=</span> [A3<span style="color: #006FE0;">,</span> (C3 <span style="color: #006FE0;">+</span> E3) <span style="color: #006FE0;">/</span> 2<span style="color: #006FE0;">,</span> E2<span style="color: #006FE0;">,</span> (C3 <span style="color: #006FE0;">+</span> E3) <span style="color: #006FE0;">/</span> 2]<span style="color: #006FE0;">;</span>
song <span style="color: #006FE0;">=</span> [aMinor]<span style="color: #006FE0;">;</span>
playSound(song<span style="color: #006FE0;">,</span> bitRate)
<span style="color: #8D8D84; font-style: italic;">## Uncomment the next line to save the audio to a file</span>
<span style="color: #8D8D84; font-style: italic;">## audiowrite("with_damping.wav", song, bitRate);</span>
<span style="color: #0000FF;">endfunction</span>
main()
</pre>
</div>
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</div>
<div id="footnotes">
<h2 class="footnotes">Footnotes: </h2>
<div id="text-footnotes">
<div class="footdef"><sup><a id="fn.1" class="footnum" href="#fnr.1">1</a></sup> <div class="footpara"><a href="https://mattdailis.github.io">https://mattdailis.github.io</a></div></div>
<div class="footdef"><sup><a id="fn.2" class="footnum" href="#fnr.2">2</a></sup> <div class="footpara"><a href="https://toomuchidle.com/">https://toomuchidle.com/</a></div></div>
<div class="footdef"><sup><a id="fn.3" class="footnum" href="#fnr.3">3</a></sup> <div class="footpara">44100 is a common sampling frequency because of the Sony CD
standard: <a href="https://en.wikipedia.org/wiki/44,100_Hz">https://en.wikipedia.org/wiki/44,100_Hz</a></div></div>
<div class="footdef"><sup><a id="fn.4" class="footnum" href="#fnr.4">4</a></sup> <div class="footpara"><a href="https://en.wikipedia.org/wiki/Additive_synthesis">https://en.wikipedia.org/wiki/Additive_synthesis</a></div></div>
<div class="footdef"><sup><a id="fn.5" class="footnum" href="#fnr.5">5</a></sup> <div class="footpara"><a href="https://en.wikipedia.org/wiki/Harmonic">https://en.wikipedia.org/wiki/Harmonic</a></div></div>
<div class="footdef"><sup><a id="fn.6" class="footnum" href="#fnr.6">6</a></sup> <div class="footpara">I found this out by plucking a string on my
guitar and looking at the spectrum in the <a href="https://play.google.com/store/apps/details?id=com.ntrack.tuner">n-Track Tuner</a> mobile app</div></div>
<div class="footdef"><sup><a id="fn.7" class="footnum" href="#fnr.7">7</a></sup> <div class="footpara">Mathematical
Modelling and Acoustical Analysis of Classical Guitars and Their
Soundboards</div></div>
<div class="footdef"><sup><a id="fn.8" class="footnum" href="#fnr.8">8</a></sup> <div class="footpara">Intuitively, this is
because a "kink" in a string has nodes at the ends no nodes in
between, which is similar to the shape of the fundamental</div></div>
<div class="footdef"><sup><a id="fn.9" class="footnum" href="#fnr.9">9</a></sup> <div class="footpara">Slides day 23</div></div>
<div class="footdef"><sup><a id="fn.10" class="footnum" href="#fnr.10">10</a></sup> <div class="footpara">Every time I
tried to use octave's <code>plot</code> function on my computer, I got a
segmentation fault…</div></div>
<div class="footdef"><sup><a id="fn.11" class="footnum" href="#fnr.11">11</a></sup> <div class="footpara">The physics of vibrating strings - Giordano, Gould, Tobochnik</div></div>
<div class="footdef"><sup><a id="fn.12" class="footnum" href="#fnr.12">12</a></sup> <div class="footpara">The physics of vibrating strings - Giordano, Gould, Tobochnik</div></div>
<div class="footdef"><sup><a id="fn.13" class="footnum" href="#fnr.13">13</a></sup> <div class="footpara">obtained using the <a href="https://play.google.com/store/apps/details?id=com.ntrack.tuner">n-Track Tuner</a> mobile app</div></div>
<div class="footdef"><sup><a id="fn.14" class="footnum" href="#fnr.14">14</a></sup> <div class="footpara">at first I
thought it was a membranophone, but I suppose there is no tension
involved</div></div>
<div class="footdef"><sup><a id="fn.15" class="footnum" href="#fnr.15">15</a></sup> <div class="footpara"><a href="https://newt.phys.unsw.edu.au/music/guitar/guitarchladni.html">https://newt.phys.unsw.edu.au/music/guitar/guitarchladni.html</a></div></div>
<div class="footdef"><sup><a id="fn.16" class="footnum" href="#fnr.16">16</a></sup> <div class="footpara">"Frequency
Response Function Of A Guitar - A Significant Peak" By Samo Šali</div></div>
<div class="footdef"><sup><a id="fn.17" class="footnum" href="#fnr.17">17</a></sup> <div class="footpara">I
should have started with this! Although the process of discovery
was also very instructive</div></div>
<div class="footdef"><sup><a id="fn.18" class="footnum" href="#fnr.18">18</a></sup> <div class="footpara">karplus-strong.pdf</div></div>
<div class="footdef"><sup><a id="fn.19" class="footnum" href="#fnr.19">19</a></sup> <div class="footpara">A <i>longitudinal wave</i>
oscillates in the same axis as it propagates.</div></div>
<div class="footdef"><sup><a id="fn.20" class="footnum" href="#fnr.20">20</a></sup> <div class="footpara"><a href="https://en.wikipedia.org/wiki/Laminar_flow">https://en.wikipedia.org/wiki/Laminar_flow</a></div></div>
<div class="footdef"><sup><a id="fn.21" class="footnum" href="#fnr.21">21</a></sup> <div class="footpara">musicians call
this technique <i>overblowing</i></div></div>
<div class="footdef"><sup><a id="fn.22" class="footnum" href="#fnr.22">22</a></sup> <div class="footpara">assuming no holes</div></div>
<div class="footdef"><sup><a id="fn.23" class="footnum" href="#fnr.23">23</a></sup> <div class="footpara">This is very convenient for musicians, because it
reduces a continuum of frequencies to a discrete set. Frets on a
guitar perform a somewhat analogous function.</div></div>
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