Instrument 7 · the Fourier series

A square wave is a pile of sines

Add one sine and you get a smooth wobble. Add the odd ones, shrinking as 1/k, and a corner appears. That is the whole trick behind every equaliser, every mp3 and every image compressor: any shape is a sum of waves, and the boring ones are the small ones.

The sum so far

The recipe

the shape we want the wave just added the sum of the first N waves

The same statement, with the numbers in it

f (t) = \sum_{k = 1}^{N} a_{k} \sin (2 π k t)

a_{k} = \frac{4}{π k}, k_{odd}

Only the odd harmonics appear, and each is a fixed fraction of its index: 4/π ≈ 1.27, 4/3π ≈ 0.42, 4/5π ≈ 0.25… The recipe is the whole waveform. Nothing else is stored.

Fine tune

Harmonics 5

The shape we are building

The sum so far▶ hear it

The recipe

The sum so far