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      <text>5.4.3 (Application - music theory)</text>
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      <text><![CDATA[<h1>Trigonometry in music theory</h1>
<p>One of the more prevalent applications of trigonometry is in the production of music. Every sound we hear including the sound of a car, talking and music reach our ears as sound waves. Sound is the result of a vibration of an object, this produces sound waves which travel through the air from the source of the sound bouncing off object such as walls, furniture and people until the sound reaches our ears. This is also true for music. Many concert halls are designed in such a way that they reflect the sound waves into our ears in an optimal way. This is what is meant when people refer to the acoustics of a room. The study of waves and music is called harmonic theory or <em>harmonic analysis</em>. We have seen the the sine and cosine functions have the form of a wave. Many signals such as music are sent in the form of waves which are a series of sin waves that have been composed with each other. Therefore to understand the waves, we need to understand the sine functions.</p>
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<h3>Waves from a piano</h3>
<p>Music is defined to be the arrangement of notes in a pleasing sequence. We will see that in terms of mathematics, it is actually ratios of sine and cosine waves. Interestingly, we are being constantly bombarded with sound waves, however humans can only hear waves of frequency \(20\) to \(20 000\) Hz. Consider the diagram below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Wave%20diagram.png" alt="wave diagram" width="671" height="341" /></p>
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<p>The <strong>frequency</strong> of a wave is defined as \(1/period\). It is measured in Hertz which is equal to one occurrence of a period per second. For example if a wave oscillates 120 times in a minute then it has a frequency of \(2\) Hz. Thus if we were to increase the frequency of a sine function, we decrease the period of each oscillation. As the frequency gets higher, we hear a higher-pitched note. As an example, let us consider the waves produced by notes of a piano. A normal piano produces notes in range of around \(25 - 4200Hz\). The middle note - 'middle c' is around \(260 hz\), we can see this in the following diagram.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/frequency%20piano.jpg" alt="frequency of a piano" width="588" height="231" /></p>
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<p>Therefore, the curve produced by the note middle c has the equation \(y = \sin (520 \cdot x)\). Indeed, every note is determined by the size of the curve of the corresponding sine wave. When we play a chord on the piano, we hit multiple notes at once, and the note produced is a sum of the sine waves of the individual notes. It is due to famous work by Fourier that we know that any wave can be expressed as a (possibly infinite) sum of sine and cosine waves. Indeed the different types of chords appear in different ratios of sine functions. We do not yet understand fully why some of these combinations sound 'good' to us and produce music humans enjoy, whereas others produce noise that can be jarring for the human ear. This is a problem at the crossroads, of mathematics, neuroscience, and music.</p>
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*/<p>In music an <b>octave</b> is the interval between one musical pitch and another with double its frequency. For example if a note has a frequency \(120\)Hz then the note one octave above has frequency \(240\). The number of octaves between two frequencies can be calculated using the formula \(\mbox{number of octaves} = \log_2(\frac{f_2}{f_1})\), for frequencies \(f_2, f_1\).</p>]]></text>
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