The Sound of Sine, Part 2

While playing one note is nice, we want much more variety when creating music! Variety can be introduced by changing the pitch and the timbre of a sound. Manipulating these qualities by changing and combining sine waves is what the field of additive synthesis is about (and gives us the name for the term "synthesizer!")

____________________________________________ Pitch We already saw how changing the period of a sine wave manipulates the pitch - waves with a longer period have a lower pitch, and waves with a shorter period have a higher pitch. Playing several notes with different pitches at the same time builds a chord.

1. In the applet below, try playing each note separately and then all together in the chord below (in western music, this is called the C-major chord) In the final chord (equation j(x), playing all three notes at once), how were the first equations for the three notes combined? What did the new graph look like?

2. In the applet below, try building your own chord by editing the periods of each equations. What were your new three equations you used? How would you describe this chord's sound to somebody?

_______________________________________________ You may have noticed when you were building chords that, when multiple sine equations are added together, we get a more complicated waveform. There are very few musical instruments that manipulate the air into a perfect sine wave - most of the time, there are variations in how the instrument vibrates that create the timbre of an instrument. This is why the same pitch played on a guitar vs on a piano sounds so different - those extra, smaller waves and variations change how the sound is perceived. Combining multiple waves gets a little more computation-heavy than a regular Geogebra applet, so we're going to switch to WolframAlpha to play the sound of our equations (the format of the equation is still the same) In the search bar below, enter play sin(2*pi*250x)

Wolfram Alpha

Now let's try adding some harmonics to the sound - smaller sinewaves that change the overall pattern of the wave Let's take our original period of 250 hertz and multiply it x3 to play a wave at 750 hertz To make sure this wave doesn't overpower our original equation, we'll change the amplitude so it is only 1/3 as loud To add a bit more complexity, let's do that process one more time with a different integer: We'll take our original period of 250 hertz and multiply it x5 to play a wave at 1250 hertz To make sure this wave doesn't overpower our original equation, we'll change the amplitude so it is only 1/5 as loud Now type play sin(2*pi*250x)+1/3*sin(2*pi*750x)+1/5*sin(2*pi*1250x) and listen to the new sound

3. If we wanted to keep going with the pattern we started above, what would the next equation look like?

4. Take a look at the shape of the waveform - if we kept continuing the pattern and adding more and more sine equations using the 1, 1/3, 1/5...rule we started above, what do you think the final waveform will look like?

5. Test your hypothesis by adding additional sine equations to the "sin(2*pi*250x) + 1/3sin(2*pi*750x) + 1/5sin(2*pi*1250x)" equation we started before. Write the final (long!) equation below and describe what the final waveform looked like.

Credits: Synthesis minifigure image: https://www.needpix.com/photo/download/1154059/synthesizer-mixer-music-free-pictures-free-photos-free-images-royalty-free-free-illustrations

The Sound of Sine, Part 2

Wolfram Alpha

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