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Beats Hoodie

TWO CLOSE NOTES PULSE | Pixelated Physics

\[x = \cos(2\pi f_1 t)+\cos(2\pi f_2 t) = 2\cos(\pi\Delta f\, t)\cos(2\pi\bar f\, t)\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Why do two nearly matched notes throb? Add two tones of frequencies f₁ and f₂ and you get one tone at their average pitch, wrapped in a slowly swelling envelope. The loudness peaks once every 1/Δf seconds, where Δf is the difference, so 440 Hz and 442 Hz beat twice a second. Musicians tune until the beats stop. Black pixel-art hoodie; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Two tones with nearly equal frequencies add up to a single tone at their average frequency whose loudness swells and fades. The loudness pulses at the difference of the two frequencies.

Two pure tones close in frequency do not sound like two steady notes. They pulse.

The sum identity is exact: cos(2πf₁t)+cos(2πf₂t) = 2 cos(πΔf t) cos(2πf̄ t), with Δf = |f₁−f₂| and f̄ = (f₁+f₂)/2.

The slow factor cos(πΔf t) is the envelope; its zeros arrive every 1/Δf seconds. That spacing is the beat period. The envelope itself oscillates at Δf/2.

On the hoodie the filled cyan wave is the carrier under the beat envelope; amber beads mark the envelope zeros.

Beats are the ear’s ruler for small frequency differences — tune two strings until the pulses stop.

Equations

\[x = \cos(2\pi f_1 t)+\cos(2\pi f_2 t) = 2\cos(\pi\Delta f\, t)\cos(2\pi\bar f\, t)\]
\[\Delta f = |f_1-f_2|,\quad \bar f=(f_1+f_2)/2\]
SymbolMeaningUnit
\(x\) summed pressure / displacement signal —
\(f_1, f_2\) frequencies of the two tones \(\mathrm{Hz}\)
\(\Delta f =\) f_1-f_2 —
\(\bar f\) average (carrier) frequency \(\mathrm{Hz}\)
\(t\) time \(\mathrm{s}\)

Sources

  1. Halliday, Resnick & Walker, Fundamentals of Physics, Ch. on sound / beats
  2. French, A. P., Vibrations and Waves (MIT Introductory Physics), Ch. on superposition
  3. Wikipedia, Beat (acoustics)

Source Helmholtz, H. von (1863). Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik. Braunschweig: F. Vieweg.

Field notes

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