How does a function sound?

Periodic functions are a great model for sounds. Sounds are physical vibrations in the air, which can be described very well with the right functions. Physically, the independent variable is always time t. However, Math-Nodes can also output sounds if you choose x as the independent variable, for example. In the following examples, we will take a look at how you can generate sounds with functions and change their pitch and sound step by step.

1 The first sound

To generate a sound in Math-Nodes, you need at least 3 cards: One card for the independent variable, one function machine, and the audio card. The sine function in this example oscillates back and forth \(440\) times per second, so it goes through \(2 \pi\) for \(t = 1\) 440 times. The resulting sound has a frequency of \(440\) Hz. Click on Sound On/Off in the audio machine, start the animation of the independent variable, and see if you can hear anything.

2 The influence of parameters

If you have made it this far, you have created your first sound. Great! Here is your first research assignment: Investigate the influence of the parameters \(a,b,c,d\) on what you hear, i.e., your auditory impression. To do this, change the values of the parameters on the corresponding cards.

3 Linking machines and analyzing with the graph module

In this task, you will first compare the function \(f(t) = a \cdot \sin(b \cdot 2 \pi \cdot t)\) graphically with the function \(g(f(t)) = (a \cdot \sin(b \cdot 2 \pi \cdot t))^2\) and then audibly. Do you have a suspicion about the influence of the function \(g\) on the sound? If you want to listen to the functions, you need to increase the frequency (parameter \(b\)). Humans can only perceive sounds as tones starting from about \(50\) Hz. It is best to choose a frequency of around \(300\) Hz.

4 Why does it sound different?

The sine functions \(f\) and \(g\) are each modulated with the functions \(a\) and \(b\). Formulate hypotheses in the workbook about how the sounds of \(f\) and \(g\) differ. Then check your hypotheses.

5 How does the function sound?

Design a patch that corresponds to the function equation in the equation card.

Before listening to the sound, first make assumptions about the auditory impression and the graph in the workbook.

    6 Word to sound

    Create a patch for each description so that the resulting sound (from \(t=0\)) corresponds to the following descriptions. Write down the function equations in the workbook.

    a) The sound starts quietly and gets louder and louder.

    b) The pitch of the sound fluctuates slightly around the tuning fork A \(440\) Hz.

    c) The sound starts loud, then gets quieter, and after a while gets louder again. In addition, the sound should get steadily higher.

    d) The pitch starts very high, drops very quickly, then rises more and more slowly and remains almost constant.

    e) The sound starts very loud at \(t=0\), is not audible at \(t=1\), and then gets louder more and more slowly.

    7 Sound modeling

    a) Listen to the sound and describe in the workbook what you hear. Pay attention to pitch and volume. What remains the same? What changes?

    b) What could the corresponding graph look like? Sketch your idea in the workbook. Use appropriate axis labeling.

    c) Design a patch in Math-Nodes that sounds exactly like the examined audio file. In the workbook, provide the resulting function equation including the parameters.