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<?xml version="1.0" encoding="utf-8" standalone="yes"?>
<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom" xmlns:content="http://purl.org/rss/1.0/modules/content/">
<channel>
<title>The Conscious Pasticcio</title>
<link>http://localhost:1313/</link>
<description>Recent content on The Conscious Pasticcio</description>
<generator>Hugo -- gohugo.io</generator>
<language>en-us</language>
<lastBuildDate>Wed, 01 May 2024 09:36:24 +0200</lastBuildDate><atom:link href="http://localhost:1313/index.xml" rel="self" type="application/rss+xml" />
<item>
<title>Topological Audio Analysis</title>
<link>http://localhost:1313/posts/tda/acustopology/</link>
<pubDate>Wed, 01 May 2024 09:36:24 +0200</pubDate>
<guid>http://localhost:1313/posts/tda/acustopology/</guid>
<description>[Epistemic status:] the closure of this post is not compact
This is hard to explain, at least from an epistemological point of view. Lets say I needed an excuse to dig deeper into a subject that bugged me for the last five months.
Topological Data Analysis (TDA), as I&rsquo;ve come to understand it, is the belief that data can hold some internal geometric structure that can be exploited through topologist-single-blade-swiss-army-knife homology.</description>
<content:encoded><![CDATA[<div class="alert-warning alert">
<div class="alert-heading">
<p>[Epistemic status:] the closure of this post is not compact</p>
</div>
</div>
<p>This is hard to explain, at least from an epistemological point of view. Lets say I needed an excuse to dig deeper into a subject that bugged me for the last five months.</p>
<p>Topological Data Analysis (TDA), as I’ve come to understand it, is the belief that data can hold some internal geometric structure that can be exploited through topologist-single-blade-swiss-army-knife homology.</p>
<p>Don’t blame me for the following rant, blame yourself.</p>
<h2 id="the-rough-idea">The rough idea</h2>
<p>So the idea is straightforward:</p>
<ol>
<li>Take any audio file; convert it into its spectrogram.</li>
<li>Take the spectrogram and convert it into a point cloud.</li>
<li>Take the point cloud and compute its Vietoris-Rips complex.</li>
</ol>
<p>There are several flows already.</p>
<p>Here is a list of self-directed objections: <a href="#faq"><strong>FAQ</strong></a></p>
<h2 id="regardlessly-i-wrote-it-anyway-dot"><strong>Regardlessly, I wrote it anyway.</strong></h2>
<p>Sit tight.</p>
<h3 id="first-of-all-how-the-heck-tda-works">First of all, how the heck TDA works?</h3>
<p>Nice and simple. You have your favorite <a href="/ox-hugo/pointcloud.png">point cloud</a>, right?</p>
<p>Then all you need to do is to grow (hyper-)spheres around them. This is, of course, tailored to the dimensionality of your dataset and the distance you impose on the surrounding space.</p>
<h4 id="vietoris-rips">Vietoris-Rips</h4>
<p>Now — hold on.</p>
<blockquote>
<p>\(X \) = Programming a visual representation of the Vietoris-Rips complex</p>
<p>I couldn’t do \(X \) by myself in a reasonable amount of time. So I found out someone somewhere has already done a decent approximation of what I imagined \(X \) would be and the MIT licence allows me to COPYPASTA \(X \) .</p>
</blockquote>
<p><strong>CREDITS:</strong> <a href="https://github.com/cristobal-montecino/VietorisRipsVisualization">Cristobal Montecino</a>. Thanks Bro.</p>
<iframe src="https://cristobal-montecino.github.io/VietorisRipsVisualization/" width="600" height="400"></iframe>
<h3 id="three-stages">Three Stages</h3>
<h4 id="from-music-to-image">From Music to Image</h4>
<p>Ok. Now, how can we massage the information given from an audio file into something we can parse? We have to decide the dimensionality of our data structure, ie we need to choose the what are the type of <strong>indipendent</strong> information about the file we want to keep track of.</p>
<p>Here is an interactive notebook that computes the spectrogram of a given audio file, curtsy of binder:</p>
<p><a href="https://mybinder.org/v2/gist/theconsciouspasticcio/936369fbc913b479413418585f0cc8ee/HEAD">
<img loading="lazy" src="https://mybinder.org/badge_logo.svg" alt="Binder" /></a></p>
<p>or if you fancy google (you need to login to play around):</p>
<p><a href="https://colab.research.google.com/gist/theconsciouspasticcio/f4002b1201f28de634ed45ee4bae01c6/acustopology.ipynb">
<img loading="lazy" src="https://colab.research.google.com/assets/colab-badge.svg" alt="Google Colab" /></a></p>
<p>the code:</p>
<div class="highlight"><pre tabindex="0" style="color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#f92672">import</span> librosa
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> librosa.display
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> matplotlib.pyplot <span style="color:#66d9ef">as</span> plt
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> scipy.signal
</span></span><span style="display:flex;"><span><span style="color:#f92672">from</span> scipy.io.wavfile <span style="color:#f92672">import</span> write
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> numpy <span style="color:#66d9ef">as</span> np
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># import audio file</span>
</span></span><span style="display:flex;"><span>audio_file_path <span style="color:#f92672">=</span> <span style="color:#e6db74">'piano.mp3'</span>
</span></span><span style="display:flex;"><span>y, sr <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>load(audio_file_path, sr<span style="color:#f92672">=</span><span style="color:#66d9ef">None</span>) <span style="color:#75715e"># Load the file as is with its original sampling rate (`sr=None`)</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Assuming y and sr are already defined from loading the audio</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Compute the Mel-scaled spectrogram</span>
</span></span><span style="display:flex;"><span>S <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>feature<span style="color:#f92672">.</span>melspectrogram(y<span style="color:#f92672">=</span>y, sr<span style="color:#f92672">=</span>sr, n_fft<span style="color:#f92672">=</span><span style="color:#ae81ff">8192</span>, hop_length<span style="color:#f92672">=</span><span style="color:#ae81ff">128</span>, n_mels<span style="color:#f92672">=</span><span style="color:#ae81ff">256</span>)
</span></span><span style="display:flex;"><span>S_dB <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>power_to_db(S, ref<span style="color:#f92672">=</span>np<span style="color:#f92672">.</span>max)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Plot the Mel spectrogram</span>
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>figure(figsize<span style="color:#f92672">=</span>(<span style="color:#ae81ff">12</span>, <span style="color:#ae81ff">8</span>))
</span></span><span style="display:flex;"><span>librosa<span style="color:#f92672">.</span>display<span style="color:#f92672">.</span>specshow(S_dB, sr<span style="color:#f92672">=</span>sr, x_axis<span style="color:#f92672">=</span><span style="color:#e6db74">'time'</span>, y_axis<span style="color:#f92672">=</span><span style="color:#e6db74">'mel'</span>)
</span></span><span style="display:flex;"><span><span style="color:#75715e"># plt.colorbar(format='%+2.0f dB')</span>
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>title(<span style="color:#e6db74">'Mel Spectrogram'</span>)
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>xlabel(<span style="color:#e6db74">'Time'</span>)
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>ylabel(<span style="color:#e6db74">'Frequency'</span>)
</span></span><span style="display:flex;"><span><span style="color:#75715e"># plt.tight_layout()</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Save the Mel spectrogram to a file</span>
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>savefig(<span style="color:#e6db74">'mel_spectrogram.png'</span>)
</span></span></code></pre></div><h4 id="from-image-to-data">From Image to DATA</h4>
<!--list-separator-->
<ul>
<li>
<p>the point cloud</p>
<div class="highlight"><pre tabindex="0" style="color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#f92672">import</span> librosa
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> librosa.display
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> matplotlib.pyplot <span style="color:#66d9ef">as</span> plt
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> scipy.signal
</span></span><span style="display:flex;"><span><span style="color:#f92672">from</span> scipy.io.wavfile <span style="color:#f92672">import</span> write
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> numpy <span style="color:#66d9ef">as</span> np
</span></span><span style="display:flex;"><span><span style="color:#f92672">from</span> mpl_toolkits.mplot3d <span style="color:#f92672">import</span> Axes3D
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># import audio file</span>
</span></span><span style="display:flex;"><span>audio_file_path <span style="color:#f92672">=</span> <span style="color:#e6db74">'piano.mp3'</span>
</span></span><span style="display:flex;"><span>y, sr <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>load(audio_file_path, sr<span style="color:#f92672">=</span><span style="color:#66d9ef">None</span>) <span style="color:#75715e"># Load the file as is with its original sampling rate (`sr=None`)</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Assuming you have your audio loaded in `y` and its sampling rate in `sr`</span>
</span></span><span style="display:flex;"><span>S <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>feature<span style="color:#f92672">.</span>melspectrogram(y<span style="color:#f92672">=</span>y, sr<span style="color:#f92672">=</span>sr, n_fft<span style="color:#f92672">=</span><span style="color:#ae81ff">2048</span>, hop_length<span style="color:#f92672">=</span><span style="color:#ae81ff">512</span>, n_mels<span style="color:#f92672">=</span><span style="color:#ae81ff">128</span>)
</span></span><span style="display:flex;"><span>S_dB <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>power_to_db(S, ref<span style="color:#f92672">=</span>np<span style="color:#f92672">.</span>max)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Generate the time and Mel frequency axes</span>
</span></span><span style="display:flex;"><span>times <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>times_like(S, sr<span style="color:#f92672">=</span>sr, hop_length<span style="color:#f92672">=</span><span style="color:#ae81ff">512</span>)
</span></span><span style="display:flex;"><span>mel_freqs <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>mel_frequencies(n_mels<span style="color:#f92672">=</span><span style="color:#ae81ff">128</span>, fmin<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>, fmax<span style="color:#f92672">=</span>sr<span style="color:#f92672">/</span><span style="color:#ae81ff">2</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Prepare x (time), y (frequency), and z (intensity) for 3D plotting</span>
</span></span><span style="display:flex;"><span>x <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>tile(times, len(mel_freqs))
</span></span><span style="display:flex;"><span>y <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>repeat(mel_freqs, len(times))
</span></span><span style="display:flex;"><span>z <span style="color:#f92672">=</span> S_dB<span style="color:#f92672">.</span>flatten()
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Plotting</span>
</span></span><span style="display:flex;"><span>fig <span style="color:#f92672">=</span> plt<span style="color:#f92672">.</span>figure(figsize<span style="color:#f92672">=</span>(<span style="color:#ae81ff">10</span>, <span style="color:#ae81ff">7</span>))
</span></span><span style="display:flex;"><span>ax <span style="color:#f92672">=</span> fig<span style="color:#f92672">.</span>add_subplot(<span style="color:#ae81ff">111</span>, projection<span style="color:#f92672">=</span><span style="color:#e6db74">'3d'</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Scatter plot for the 3D point cloud</span>
</span></span><span style="display:flex;"><span>scat <span style="color:#f92672">=</span> ax<span style="color:#f92672">.</span>scatter(x, y, z, c<span style="color:#f92672">=</span>z, cmap<span style="color:#f92672">=</span><span style="color:#e6db74">'viridis'</span>, s<span style="color:#f92672">=</span><span style="color:#ae81ff">1</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Adding labels and title</span>
</span></span><span style="display:flex;"><span>ax<span style="color:#f92672">.</span>set_xlabel(<span style="color:#e6db74">'Time (s)'</span>)
</span></span><span style="display:flex;"><span>ax<span style="color:#f92672">.</span>set_ylabel(<span style="color:#e6db74">'Frequency (Hz)'</span>)
</span></span><span style="display:flex;"><span>ax<span style="color:#f92672">.</span>set_zlabel(<span style="color:#e6db74">'Intensity (dB)'</span>)
</span></span><span style="display:flex;"><span>ax<span style="color:#f92672">.</span>set_title(<span style="color:#e6db74">'3D Point Cloud of Mel Spectrogram'</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Add a color bar to indicate the intensity values</span>
</span></span><span style="display:flex;"><span>cbar <span style="color:#f92672">=</span> fig<span style="color:#f92672">.</span>colorbar(scat, ax<span style="color:#f92672">=</span>ax)
</span></span><span style="display:flex;"><span>cbar<span style="color:#f92672">.</span>set_label(<span style="color:#e6db74">'Intensity (dB)'</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>savefig(<span style="color:#e6db74">'mel_pointcloud.png'</span>)
</span></span></code></pre></div></li>
</ul>
<!--list-separator-->
<ul>
<li>
<p>interactive plot</p>
<div class="highlight"><pre tabindex="0" style="color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#f92672">import</span> librosa
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> numpy <span style="color:#66d9ef">as</span> np
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> plotly.graph_objects <span style="color:#66d9ef">as</span> go
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Load your audio and compute the Mel spectrogram</span>
</span></span><span style="display:flex;"><span>y, sr <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>load(<span style="color:#e6db74">'piano.mp3'</span>)
</span></span><span style="display:flex;"><span>S <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>feature<span style="color:#f92672">.</span>melspectrogram(y<span style="color:#f92672">=</span>y, sr<span style="color:#f92672">=</span>sr, n_fft<span style="color:#f92672">=</span><span style="color:#ae81ff">2048</span>, hop_length<span style="color:#f92672">=</span><span style="color:#ae81ff">512</span>, n_mels<span style="color:#f92672">=</span><span style="color:#ae81ff">128</span>)
</span></span><span style="display:flex;"><span>S_dB <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>power_to_db(S, ref<span style="color:#f92672">=</span>np<span style="color:#f92672">.</span>max)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Generate the time and Mel frequency axes</span>
</span></span><span style="display:flex;"><span>times <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>times_like(S, sr<span style="color:#f92672">=</span>sr, hop_length<span style="color:#f92672">=</span><span style="color:#ae81ff">512</span>)
</span></span><span style="display:flex;"><span>mel_freqs <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>mel_frequencies(n_mels<span style="color:#f92672">=</span><span style="color:#ae81ff">128</span>, fmin<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>, fmax<span style="color:#f92672">=</span>sr<span style="color:#f92672">/</span><span style="color:#ae81ff">2</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Prepare the data</span>
</span></span><span style="display:flex;"><span>x <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>tile(times, len(mel_freqs))
</span></span><span style="display:flex;"><span>y <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>repeat(mel_freqs, len(times))
</span></span><span style="display:flex;"><span>z <span style="color:#f92672">=</span> S_dB<span style="color:#f92672">.</span>flatten()
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Define the dB threshold for filtering out negligible values</span>
</span></span><span style="display:flex;"><span>db_threshold <span style="color:#f92672">=</span> <span style="color:#f92672">-</span><span style="color:#ae81ff">60</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Create a mask to filter out values below the threshold</span>
</span></span><span style="display:flex;"><span>mask <span style="color:#f92672">=</span> z <span style="color:#f92672">></span> db_threshold
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Apply the mask to x, y, and z</span>
</span></span><span style="display:flex;"><span>x_filtered <span style="color:#f92672">=</span> x[mask]
</span></span><span style="display:flex;"><span>y_filtered <span style="color:#f92672">=</span> y[mask]
</span></span><span style="display:flex;"><span>z_filtered <span style="color:#f92672">=</span> z[mask]
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Create a Plotly scatter3d object with the filtered data</span>
</span></span><span style="display:flex;"><span>fig <span style="color:#f92672">=</span> go<span style="color:#f92672">.</span>Figure(data<span style="color:#f92672">=</span>[go<span style="color:#f92672">.</span>Scatter3d(
</span></span><span style="display:flex;"><span> x<span style="color:#f92672">=</span>x_filtered,
</span></span><span style="display:flex;"><span> y<span style="color:#f92672">=</span>y_filtered,
</span></span><span style="display:flex;"><span> z<span style="color:#f92672">=</span>z_filtered,
</span></span><span style="display:flex;"><span> mode<span style="color:#f92672">=</span><span style="color:#e6db74">'markers'</span>,
</span></span><span style="display:flex;"><span> marker<span style="color:#f92672">=</span>dict(
</span></span><span style="display:flex;"><span> size<span style="color:#f92672">=</span><span style="color:#ae81ff">2</span>,
</span></span><span style="display:flex;"><span> color<span style="color:#f92672">=</span>z_filtered, <span style="color:#75715e"># Use filtered intensity for color mapping</span>
</span></span><span style="display:flex;"><span> colorscale<span style="color:#f92672">=</span><span style="color:#e6db74">'Viridis'</span>,
</span></span><span style="display:flex;"><span> opacity<span style="color:#f92672">=</span><span style="color:#ae81ff">0.8</span>,
</span></span><span style="display:flex;"><span> colorbar<span style="color:#f92672">=</span>dict(title<span style="color:#f92672">=</span><span style="color:#e6db74">'Intensity (dB)'</span>, bgcolor<span style="color:#f92672">=</span><span style="color:#e6db74">'black'</span>)
</span></span><span style="display:flex;"><span> )
</span></span><span style="display:flex;"><span>)])
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Update the layout for a better viewing experience in dark mode</span>
</span></span><span style="display:flex;"><span>fig<span style="color:#f92672">.</span>update_layout(
</span></span><span style="display:flex;"><span> margin<span style="color:#f92672">=</span>dict(l<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>, r<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>, b<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>, t<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>),
</span></span><span style="display:flex;"><span> paper_bgcolor<span style="color:#f92672">=</span><span style="color:#e6db74">'black'</span>,
</span></span><span style="display:flex;"><span> plot_bgcolor<span style="color:#f92672">=</span><span style="color:#e6db74">'black'</span>,
</span></span><span style="display:flex;"><span> scene<span style="color:#f92672">=</span>dict(
</span></span><span style="display:flex;"><span> xaxis<span style="color:#f92672">=</span>dict(
</span></span><span style="display:flex;"><span> title<span style="color:#f92672">=</span><span style="color:#e6db74">'Time (s)'</span>,
</span></span><span style="display:flex;"><span> backgroundcolor<span style="color:#f92672">=</span><span style="color:#e6db74">"black"</span>,
</span></span><span style="display:flex;"><span> color<span style="color:#f92672">=</span><span style="color:#e6db74">'white'</span>,
</span></span><span style="display:flex;"><span> gridcolor<span style="color:#f92672">=</span><span style="color:#e6db74">'gray'</span>
</span></span><span style="display:flex;"><span> ),
</span></span><span style="display:flex;"><span> yaxis<span style="color:#f92672">=</span>dict(
</span></span><span style="display:flex;"><span> title<span style="color:#f92672">=</span><span style="color:#e6db74">'Frequency (Hz)'</span>,
</span></span><span style="display:flex;"><span> backgroundcolor<span style="color:#f92672">=</span><span style="color:#e6db74">"black"</span>,
</span></span><span style="display:flex;"><span> color<span style="color:#f92672">=</span><span style="color:#e6db74">'white'</span>,
</span></span><span style="display:flex;"><span> gridcolor<span style="color:#f92672">=</span><span style="color:#e6db74">'gray'</span>
</span></span><span style="display:flex;"><span> ),
</span></span><span style="display:flex;"><span> zaxis<span style="color:#f92672">=</span>dict(
</span></span><span style="display:flex;"><span> title<span style="color:#f92672">=</span><span style="color:#e6db74">'Intensity (dB)'</span>,
</span></span><span style="display:flex;"><span> backgroundcolor<span style="color:#f92672">=</span><span style="color:#e6db74">"black"</span>,
</span></span><span style="display:flex;"><span> color<span style="color:#f92672">=</span><span style="color:#e6db74">'white'</span>,
</span></span><span style="display:flex;"><span> gridcolor<span style="color:#f92672">=</span><span style="color:#e6db74">'gray'</span>
</span></span><span style="display:flex;"><span> )
</span></span><span style="display:flex;"><span> )
</span></span><span style="display:flex;"><span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Save the plot as an HTML file</span>
</span></span><span style="display:flex;"><span>html_file_path <span style="color:#f92672">=</span> <span style="color:#e6db74">'filtered_mel_spectrogram_3d_dark.html'</span>
</span></span><span style="display:flex;"><span>fig<span style="color:#f92672">.</span>write_html(html_file_path)
</span></span></code></pre></div><p>This is the output:</p>
<iframe src="https://theconsciouspasticcio.github.io/acustic-top1/filtered_mel_spectrogram_3d_dark.html" width="700" height="450"></iframe>
</li>
</ul>
<h4 id="from-data-to-topology">From DATA to Topology</h4>
<p>Ok since my 20yo laptop could melt any instant, let’s try to be reasonable and downsample the previous result. This code just divide the space into little cubes and takes the mean value of all the points within.</p>
<div class="highlight"><pre tabindex="0" style="color:#f8f8f2;background-color:#272822;-moz-tab-size:4;-o-tab-size:4;tab-size:4;"><code class="language-python" data-lang="python"><span style="display:flex;"><span><span style="color:#f92672">from</span> ripser <span style="color:#f92672">import</span> ripser
</span></span><span style="display:flex;"><span><span style="color:#f92672">from</span> persim <span style="color:#f92672">import</span> plot_diagrams, PersistenceImager
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> numpy <span style="color:#66d9ef">as</span> np
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> matplotlib.pyplot <span style="color:#66d9ef">as</span> plt
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> librosa
</span></span><span style="display:flex;"><span><span style="color:#f92672">import</span> plotly.graph_objects <span style="color:#66d9ef">as</span> go
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Load your audio and compute the Mel spectrogram</span>
</span></span><span style="display:flex;"><span>y, sr <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>load(<span style="color:#e6db74">'piano.mp3'</span>)
</span></span><span style="display:flex;"><span>S <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>feature<span style="color:#f92672">.</span>melspectrogram(y<span style="color:#f92672">=</span>y, sr<span style="color:#f92672">=</span>sr, n_fft<span style="color:#f92672">=</span><span style="color:#ae81ff">2048</span>, hop_length<span style="color:#f92672">=</span><span style="color:#ae81ff">512</span>, n_mels<span style="color:#f92672">=</span><span style="color:#ae81ff">128</span>)
</span></span><span style="display:flex;"><span>S_dB <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>power_to_db(S, ref<span style="color:#f92672">=</span>np<span style="color:#f92672">.</span>max)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Generate the time and Mel frequency axes</span>
</span></span><span style="display:flex;"><span>times <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>times_like(S, sr<span style="color:#f92672">=</span>sr, hop_length<span style="color:#f92672">=</span><span style="color:#ae81ff">512</span>)
</span></span><span style="display:flex;"><span>mel_freqs <span style="color:#f92672">=</span> librosa<span style="color:#f92672">.</span>mel_frequencies(n_mels<span style="color:#f92672">=</span><span style="color:#ae81ff">128</span>, fmin<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>, fmax<span style="color:#f92672">=</span>sr<span style="color:#f92672">/</span><span style="color:#ae81ff">2</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Prepare the data</span>
</span></span><span style="display:flex;"><span>x <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>tile(times, len(mel_freqs))
</span></span><span style="display:flex;"><span>y <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>repeat(mel_freqs, len(times))
</span></span><span style="display:flex;"><span>z <span style="color:#f92672">=</span> S_dB<span style="color:#f92672">.</span>flatten()
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Define the dB threshold for filtering out negligible values</span>
</span></span><span style="display:flex;"><span>db_threshold <span style="color:#f92672">=</span> <span style="color:#f92672">-</span><span style="color:#ae81ff">60</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Create a mask to filter out values below the threshold</span>
</span></span><span style="display:flex;"><span>mask <span style="color:#f92672">=</span> z <span style="color:#f92672">></span> db_threshold
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Apply the mask to x, y, and z</span>
</span></span><span style="display:flex;"><span>x_filtered <span style="color:#f92672">=</span> x[mask]
</span></span><span style="display:flex;"><span>y_filtered <span style="color:#f92672">=</span> y[mask]
</span></span><span style="display:flex;"><span>z_filtered <span style="color:#f92672">=</span> z[mask]
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># # Create a Plotly scatter3d object with the filtered data</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># fig = go.Figure(data=[go.Scatter3d(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># x=x_filtered,</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># y=y_filtered,</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># z=z_filtered,</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># mode='markers',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># marker=dict(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># size=2,</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># color=z_filtered, # Use filtered intensity for color mapping</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># colorscale='Viridis',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># opacity=0.8,</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># colorbar=dict(title='Intensity (dB)', bgcolor='black')</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )])</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Define the downsample by averaging function</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#66d9ef">def</span> <span style="color:#a6e22e">downsample_by_averaging</span>(points, cube_size):
</span></span><span style="display:flex;"><span> min_coords <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>min(points, axis<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>)
</span></span><span style="display:flex;"><span> max_coords <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>max(points, axis<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>)
</span></span><span style="display:flex;"><span> indices <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>floor((points <span style="color:#f92672">-</span> min_coords) <span style="color:#f92672">/</span> cube_size)<span style="color:#f92672">.</span>astype(int)
</span></span><span style="display:flex;"><span> dtype <span style="color:#f92672">=</span> [(<span style="color:#e6db74">'sum'</span>, float, <span style="color:#ae81ff">3</span>), (<span style="color:#e6db74">'count'</span>, int)]
</span></span><span style="display:flex;"><span> grid <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>zeros(indices<span style="color:#f92672">.</span>max(axis<span style="color:#f92672">=</span><span style="color:#ae81ff">0</span>) <span style="color:#f92672">+</span> <span style="color:#ae81ff">1</span>, dtype<span style="color:#f92672">=</span>dtype)
</span></span><span style="display:flex;"><span> <span style="color:#66d9ef">for</span> point, index <span style="color:#f92672">in</span> zip(points, indices):
</span></span><span style="display:flex;"><span> grid[tuple(index)][<span style="color:#e6db74">'sum'</span>] <span style="color:#f92672">+=</span> point
</span></span><span style="display:flex;"><span> grid[tuple(index)][<span style="color:#e6db74">'count'</span>] <span style="color:#f92672">+=</span> <span style="color:#ae81ff">1</span>
</span></span><span style="display:flex;"><span> non_empty_cubes <span style="color:#f92672">=</span> grid[<span style="color:#e6db74">'count'</span>] <span style="color:#f92672">></span> <span style="color:#ae81ff">0</span>
</span></span><span style="display:flex;"><span> averaged_points <span style="color:#f92672">=</span> grid[<span style="color:#e6db74">'sum'</span>][non_empty_cubes] <span style="color:#f92672">/</span> grid[<span style="color:#e6db74">'count'</span>][non_empty_cubes][:, np<span style="color:#f92672">.</span>newaxis]
</span></span><span style="display:flex;"><span> <span style="color:#66d9ef">return</span> averaged_points
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Assuming x_filtered, y_filtered, and z_filtered are defined and contain your point cloud data</span>
</span></span><span style="display:flex;"><span>points <span style="color:#f92672">=</span> np<span style="color:#f92672">.</span>column_stack((x_filtered, y_filtered, z_filtered))
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Combine filtered points and apply downsampling</span>
</span></span><span style="display:flex;"><span>downsampled_points <span style="color:#f92672">=</span> downsample_by_averaging(points, cube_size<span style="color:#f92672">=</span><span style="color:#ae81ff">8</span>) <span style="color:#75715e"># Adjust cube_size as needed</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># # Plotting the downsampled points with Plotly</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># fig = go.Figure(data=[go.Scatter3d(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># x=downsampled_points[:, 0], # Time</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># y=downsampled_points[:, 1], # Frequency</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># z=downsampled_points[:, 2], # Intensity</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># mode='markers',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># marker=dict(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># size=2,</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># color=downsampled_points[:, 2], # Color by intensity</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># colorscale='Viridis',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># opacity=0.8,</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )])</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># fig.update_layout(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># title="Downsampled 3D Mel Spectrogram",</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># scene=dict(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># xaxis_title='Time',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># yaxis_title='Frequency',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># zaxis_title='Intensity',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># #Update the layout for a better viewing experience in dark mode</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># fig.update_layout(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># margin=dict(l=0, r=0, b=0, t=0),</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># paper_bgcolor='black',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># plot_bgcolor='black',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># scene=dict(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># xaxis=dict(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># title='Time (s)',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># backgroundcolor="black",</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># color='white',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># gridcolor='gray'</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># ),</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># yaxis=dict(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># title='Frequency (Hz)',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># backgroundcolor="black",</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># color='white',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># gridcolor='gray'</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># ),</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># zaxis=dict(</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># title='Intensity (dB)',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># backgroundcolor="black",</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># color='white',</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># gridcolor='gray'</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># )</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># # Save the plot as an HTML file</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># html_file_path = 'downsampled_3D_mel_spectrogram_plot.html'</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># fig.write_html(html_file_path)</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># print(f"Plot saved to {html_file_path}")</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Assuming downsampled_points is available and contains the downsampled point cloud</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Apply Ripser to compute Vietoris-Rips persistence barcodes</span>
</span></span><span style="display:flex;"><span>diagrams <span style="color:#f92672">=</span> ripser(downsampled_points)[<span style="color:#e6db74">'dgms'</span>]
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Plot and save the persistence diagrams</span>
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>figure(figsize<span style="color:#f92672">=</span>(<span style="color:#ae81ff">12</span>, <span style="color:#ae81ff">8</span>))
</span></span><span style="display:flex;"><span>plot_diagrams(diagrams, show<span style="color:#f92672">=</span><span style="color:#66d9ef">False</span>)
</span></span><span style="display:flex;"><span>plt<span style="color:#f92672">.</span>savefig(<span style="color:#e6db74">'persistence_diagrams_downsampled.png'</span>)
</span></span><span style="display:flex;"><span><span style="color:#75715e"># plt.close()</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Initialize PersistenceImager</span>
</span></span><span style="display:flex;"><span>pimgr <span style="color:#f92672">=</span> PersistenceImager(pixel_size<span style="color:#f92672">=</span><span style="color:#ae81ff">0.05</span>)
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># # Fit and transform the H1 diagram to a persistence image</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># # Note: diagrams[1] corresponds to H1. Use diagrams[0] for H0.</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># imgs = pimgr.fit_transform(diagrams[1])</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># # Plot and save the persistence image</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># plt.figure(figsize=(6, 6))</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># pimgr.plot_image(imgs[0], ax=plt.gca())</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># plt.savefig('persistence_image_downsampled.png')</span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># # plt.close()</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#75715e"># Check if the desired homology dimension diagram is non-empty</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span><span style="color:#66d9ef">if</span> diagrams[<span style="color:#ae81ff">1</span>]<span style="color:#f92672">.</span>size <span style="color:#f92672">></span> <span style="color:#ae81ff">0</span>:
</span></span><span style="display:flex;"><span> <span style="color:#75715e"># Fit and transform to get the persistence image</span>
</span></span><span style="display:flex;"><span> imgs <span style="color:#f92672">=</span> pimgr<span style="color:#f92672">.</span>fit_transform([diagrams[<span style="color:#ae81ff">1</span>]]) <span style="color:#75715e"># diagrams[1] for H1</span>
</span></span><span style="display:flex;"><span>
</span></span><span style="display:flex;"><span> <span style="color:#75715e"># Plot and save the first (and in this case, only) persistence image</span>
</span></span><span style="display:flex;"><span> plt<span style="color:#f92672">.</span>imshow(imgs[<span style="color:#ae81ff">0</span>], origin<span style="color:#f92672">=</span><span style="color:#e6db74">'lower'</span>, cmap<span style="color:#f92672">=</span><span style="color:#e6db74">'hot'</span>)
</span></span><span style="display:flex;"><span> <span style="color:#75715e"># plt.colorbar()</span>
</span></span><span style="display:flex;"><span> plt<span style="color:#f92672">.</span>savefig(<span style="color:#e6db74">'persistence_image_downsampled.png'</span>)
</span></span><span style="display:flex;"><span><span style="color:#66d9ef">else</span>:
</span></span><span style="display:flex;"><span> print(<span style="color:#e6db74">"No features found in the desired homology dimension."</span>)
</span></span></code></pre></div><h2 id="faq"><strong>FAQ</strong></h2>
<h3 id="why-does-the-spectrogram-of-an-audio-file-capture-its-essence">Why does the spectrogram of an audio file capture its essence?</h3>
<p>It doesn’t. In my opinion is just an interesting perspective you can take. Besides, it actually simplify a lot the next step.</p>
<h4 id="wait-but-what-are-spectrograms">Wait, but what are spectrograms?</h4>
<div style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;">
<iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="allowfullscreen" loading="eager" referrerpolicy="strict-origin-when-cross-origin" src="https://www.youtube.com/embed/7WYw3qoTdU4?autoplay=0&controls=1&end=0&loop=0&mute=0&start=0" style="position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;" title="YouTube video"
></iframe>
</div>
<p>TLDW: Spectrograms are the representation in (aplitude, frequency, time)—space of sound.</p>
<p>They encompass so much information about the sound. The key is that by taking only the notes you are basically throwing away all the information about the timbre, i.e. you are effectively restricting to the first Fourier coefficient and silencing to zero every other one.</p>
<p><span class="underline">But Fourier is nice, we want everything</span>. Hence, the Topological Audio-Data Analysis ( <a href="https://www.youtube.com/watch?v=mLRcBCJlfTI">TADA!</a> )</p>
<h3 id="euclidean-distance-doesn-t-capture-any-musical-information">Euclidean distance doesn’t capture any “musical” information</h3>
<p>\(C_{0}\) is musically, perceivably, “closer” to its harmonics than \(D_{0}\), i.e.</p>
<p>\[[C_0, C_1, G_1, C_2, E_2, G_2]\]
are the pitches of the first six harmonics in the equal temperament tuning system.</p>
<p>or pedantically the first five overtones:
\[[C_1, G_1, C_2, E_2, G_2]\]</p>
<p>and overtones do not have overtones, i.e. are f sine waves. And <a href="https://physics.stackexchange.com/questions/363307/why-is-a-pure-tone-sinusoidal">why is that</a> is f beautiful and Fourier-pilled</p>
<h3 id="what-does-the-information-about-this-topology-tells-about-the-audio">What does the information about this “topology” tells about the audio</h3>
<p>Honestly, I don’t know. But this can be a way to classify the data coming from an audio file. Thinking not that hard about it, you could imagine that no matter the significance of the topological information retrieved in this way, still on a meta-analysis you could find patterns inside classes of similar audio file.</p>
<p>Similarly
>> Chris Dumas video</p>
<h3 id="given-that-the-output-is-mathematically-meaningful-can-you-recover-the-topological-space-from-the-persistent-barcodes">Given that the output is mathematically meaningful, can you recover the topological space from the persistent barcodes?</h3>
<h3 id="have-somebody-else-already-done-it">Have somebody else already done it?</h3>
<p>Kind of:</p>
<div style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;">
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></iframe>
</div>
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