# symmetry-fourier-three-tone-packet.mp4

Chapter section: The Fourier Structure of Waves

Caption (verbatim):

No separate caption in the chapter.

Image description (verbatim):

Three-tone chord packet and its Fourier decomposition

## Before the animation

Chapter lines 800–800:

If you strike a chord on a piano, some complicated function of time describes how the sound pressure reaches your ear. It starts soft, gets louder, softens again. It has discernible main tones, but also a clutter of overtones that comprise the timbre of the piano. While you hear a clear tonal structure, a plot of the sound pressure level over time reaching your ear would completely obscure that structure. 

## After the animation

Chapter lines 806–806:

But we know something about this random-seeming function reaching your ear. We know it is comprised of mostly 3 pitches, and some overtones. If we plot the sound pressure level as a function of these pitches rather than as a function of time, the structure of our plot clearly reveals what we hear naturally with our ear. What we call a "pitch" is the frequency of a pure **mode**. The pattern of sound pressure over time is the sum, or "composition," of these modes. We can transform back and forth between the wave number and position representations. Typically we label the wave number representation with a tilde:

Chapter lines 808–810:

```math
\psi(x)\;\longleftrightarrow\;\tilde{\psi}(k)
```