TwiddleLab / Math / Trig and Waves

Project: Make a Synth Note

Lesson 11 of 11 · 8 min · Free, no account needed

What you will learn

• A note at f is a sum of sine waves at f, 2f, 3f...: the harmonics.
• How much of each harmonic you add is the timbre: same pitch, different sound.
• A real note fades like e^(-kt): bigger k, faster fade.
• Digital sound is the same sine, sampled: sample n plays at time n ÷ 44100 seconds.

Start the lesson

How the lesson goes

1. Build an instrument nobody has heard before.

   From nothing but sine waves. First, three mystery notes: tap each one.

2. Step 1: set the root note.

   Move the lever to 220 Hz, the note A3. The card shows its period.

3. Step 2: add overtones.

   Raise the bars 2f to 5f. Turn on at least two and press Play.

4. Harmonics and timbre.

   The harmonics of a note at f are the sine waves at f, 2f, 3f, 4f and so on.

   The timbre is the recipe: how much of each harmonic. Same pitch, different recipe, different instrument.

5. Match each recipe to its wave.

   Tap a recipe, then the wave it makes.

   Each harmonic adds a small sine with n wiggles per cycle of f. Only f gives a smooth sine. Odd harmonics keep the bottom half a mirror of the top half; adding the even ones too makes the wave lean like a saw.

6. Step 3: make the note fade.

   Loudness falls like e^(-kt). Set k so only 10% is left after 1 second.

7. The root is 220 Hz. What frequency is the third harmonic?

   Harmonics are whole-number multiples of the root: 3f = 3 × 220 = 660 Hz.

8. Challenge: copy the sound.

   Make your blue lines cover the dashed target: the wave shape and the fade. Compare when ready.

9. Step 4: your own instrument.

   Set the bars and the fade any way you like, pick a name and Keep it.

10. This code plays the note an octave too high.

    A computer plays sound as 44,100 numbers a second. Find the right time for sample n.

    With 44,100 samples a second, sample n is at t = n ÷ 44100 s. Using 22050 doubles every t, which doubles the frequency: 440 Hz, one octave up.

11. Which overtone is in this wave?

    It is f plus one overtone. Tap that overtone's bar.

    A harmonic nf goes through n cycles while f goes through one, so 3f adds three bumps per cycle.

12. Your instrument, made of sine waves.

    Press Play my tune and watch the wave change with every note.

Previous lesson: Rotation with Trig · Part of the course: Trig and Waves

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