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Puzzle Foundry · PF-03 · Signal / Spectrum / Percept Instrument

The Note That Is Not There

Four tones enter. A fifth note is never played. Yet the whole waveform can repeat at that absent note’s period—and your ear may still organize the sound around it. Build the signal, slide its ghost by one period, and watch the impossible-looking overlap.

SIGNAL · SPECTRUM · PERIODICITY
AUDIO OPTIONAL · SILENT ROUTE COMPLETE
First · Watch the recipe become a waveform

See the ingredients. Then break intuition.

No prior theory is required. Tap through the three recipe states, then operate the full signal below. The explanation follows the spectacle.

Fourier analysis asks a simple question: can a complicated-looking repeating signal be described as a sum of simpler sine/cosine waves? The spectrum is the ingredient list. The waveform is what all those ingredients look like when added together through time.
Guided mini-lab · watch the recipe become a waveform

1. Build the signal in three steps

220 Hz only. The spectrum has one line at 220 Hz. The waveform repeats every 1/220 second, about 4.55 ms. In this easy case, the line frequency and repeat rate feel like the same idea.

2. Read the ingredient list

110
220
330
440
550 Hz
A line means that frequency is physically in the constructed signal. An empty slot means that frequency is not one of the synthesized sinusoidal ingredients.
220 = 1 × 220. One ingredient, one obvious repeat rate.
Your turn · two tiny checks before the real puzzle

If the spectrum has lines at 220 Hz and 440 Hz, what does that directly tell you?

Choose one. We will explain why.

If a waveform lines up perfectly with itself after a 9.091 ms shift, what have you directly tested?

Choose one. This distinction is the center of the puzzle.

When both ideas are clear: the spectrum tells you the ingredients; a time shift tells you whether the whole waveform repeats.

SIGNAL / SPECTRUM

What frequencies are physically present?

Fourier analysis decomposes the waveform into sinusoidal components. Think ingredient list, not “what pitch do I hear?”

signal = component 1 + component 2 + ...
TIME / PERIOD

When does the whole shape repeat?

A period is a shift in time that makes the complete waveform land back on itself.

s(t + T) = s(t)
PERCEPT / INFERENCE

What does a listener report?

Your auditory system can organize a complex signal into a perceived pitch. That report is real evidence about perception—but it is not a spectral inventory.

percept ≠ spectral inventory
Now the strange case: build the signal only from 220, 330, 440, and 550 Hz. Notice the spacing: each one is an integer multiple of 110 Hz—2×, 3×, 4×, 5×—but 110 itself is not an ingredient. The puzzle asks what survives anyway.
Third · Operate

The constructed signal

The canonical construction has all four partials active. You may mute them for exploration, but the canonical inference uses 220/330/440/550 all nonzero.

Time view
f₀ reference110 Hz
T₀ = 1/f₀9.091 ms
Shift overlapnot tested
empty
110 Hz
220
330
440
550
Math lens · the notation is part of the instrument

Construction

s(t)=Σ⁵ₙ₌₂ aₙ cos(2πn f₀t + φₙ),   f₀=110 Hz

There is no n=1 term. So there is no synthesized 110 Hz sinusoid.

Why T₀ still works

s(t+T₀)=s(t),   T₀=1/f₀

Each component picks up an integer number of full cycles: 2πn. The whole waveform therefore repeats after 9.091 ms.

Why this is the common fundamental period

gcd(2,3,4,5)=1

With all four constructed harmonics nonzero, there is no smaller common integer divisor that shortens the canonical period.

The firewall

perceived pitch ≠ physical Fourier line

A report near 110 Hz is an empirical perceptual observation. It does not manufacture a missing spectral component or prove a 110 Hz source.

REVELATION · THE ABSENT FUNDAMENTAL

THE NOTE IS NOT THERE.

The amber ghost has shifted by exactly 9.091 ms—and it lands on the blue waveform. The signal repeats at the period associated with 110 Hz even though no 110 Hz sinusoid was put into the spectrum.

SPECTRUM110 Hz is absent.

The constructed ingredients are 220, 330, 440 and 550 Hz.

TIME9.091 ms survives.

All four ingredients return to the same phase after 1/110 second, so the full shape repeats.

PERCEPTIONYour ear may supply the center.

A listener can report a pitch near 110 Hz. That is a perceptual fact—not proof of a hidden 110 Hz source.

Fourth · Classify the evidence

What kind of claim does each observation support?

This is the heart of the puzzle. You are sorting evidence by what produced it—not hunting for green buttons. Each answer explains the measurement behind it.

Observation A · The 110 Hz spectrum slot is empty.
What kind of evidence is this?
Observation B · The waveform overlays itself after +9.091 ms.
What does this establish most directly?
Observation C · A listener reports a pitch near 110 Hz.
What is that report evidence about?
Claim D · “Therefore a physical 110 Hz oscillator must be present.”
How should the instrument classify that inference?
Evidence matrix
QuestionWhat we can sayStatus
Physical 110 Hz component?Canonical spectrum contains 220/330/440/550 only.awaiting classification
Repeats after 1/110 s?Test with +T₀ waveform overlay.awaiting time test
Pitch near 110 reported?Possible empirical listener report; not guaranteed.awaiting classification
110 Hz source proven?Would require source evidence not supplied here.awaiting classification
No correctness criterion depends on hearing the missing fundamental. The visual/numeric route is complete.
Fifth · Transfer check

Keep the predicates separate

Four bounded checks. Immediate feedback. No free-text storage and no answer payload sent anywhere.

Does “no 110 Hz spectral line” imply “no 1/110 s periodicity”?

Does “I perceive a pitch near 110 Hz” by itself prove a physical 110 Hz sinusoid?

Which observation directly supports the repeat-time claim?

Which category does the sentence “I hear a pitch near 110 Hz” belong to?

Understanding matrix0 / 4

THE NOTE IS NOT THERE.

Absent as a 110 Hz Fourier line. Present as a 1/110-second temporal structure. Potentially reported as a pitch near 110 Hz. None of those facts, alone, proves a physical 110 Hz source.

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