the second hundred · metaphor 207
There are two honest ways to remember a life. One keeps the mood — the whole blend of feelings that ran through a decade, every recurring tone still sounding, but with no dates attached. The other keeps the calendar — the exact night the dread arrived, the afternoon the joy broke open, each surge pinned to its own moment.
His memory worked the first way: ask him about those years and he could tell you which feelings were present — the tenderness, the low fear, the flashes of anger — all of it true, none of it dated. Hers worked the second way: she knew when each one had struck, to the week. Two faithful records of the same shared life. They disagreed about only one thing — whether the timing was worth keeping.
Mathematics keeps both kinds of memory, and has spent two centuries learning the price of each. One reading of a span tells you which rhythms it holds — faithfully, and without a single timestamp. The other tells you when each arrived, at the cost of never being sure of its exact pitch. The instrument below runs both at once, on a signal you compose.
The two memories
Read a stretch of time — a decade, a signal, a song — and ask what rhythms it holds. The Fourier answer is a spectrum: a tall spike at every frequency that is present, its height the strength of that rhythm across the whole span. It is complete and honest — but built from waves that never begin or end, so it cannot say when anything happened. Slide the surge to a different night and the spectrum does not flinch: the pitch was always there, is the only thing it reports, and a pitch has no address.
The wavelet answer keeps the dates. Instead of an endless wave, it uses a little packet — a few oscillations inside a fading envelope, the Morlet — and slides that packet along the timeline, stretching and squeezing it to match slow rhythms and fast ones in turn. Where the packet lines up with something in the signal, it rings; where it doesn't, it stays quiet. The result is a map with two axes: time across, pitch up. A brief surge shows up as a bright blob sitting exactly over the moment it struck — while a slow, stretched packet catches the background tone but blurs across the whole window, a low feeling that is everywhere, undatable.
What to notice
Every panel is computed live from the same signal s[n]: the left is a real discrete Fourier transform, the right a real Morlet wavelet transform of the very same samples — nothing scripted. Grab the surge and drag it, or use the when it struck slider. The gold spectrum barely moves: its peaks report every pitch present, dateless, wherever the surge sits. The periwinkle blob tracks your hand across the map — and the readout wavelet · when is the argmax of that live heatmap, a number read straight off the transform for which moment was loudest.
Now change the surge's pitch: the blob climbs or drops with it. Push the pitch high and the blob grows thin in time but tall in pitch — sharp about when, vague about what. Drop the background low and its band smears across the whole width — you can name the feeling but never date it. The presets: one late surge, two events, and a hidden early spike the spectrum's tallest peak walks right past while the wavelet finds it at once.
Back to the life
This is the couple in the epigraph, made exact. He carried the spectrum — the full inventory of what the relationship contained, every recurring feeling accounted for, none of it dated. She carried the scalogram — the same feelings, but each fixed to the week it arrived. Neither is wrong; neither is more loving. They are two lossless ways to encode one shared signal, and they part company on a single question: is the timing part of the truth, or a detail you may discard?
Most arguments about memory are really this question in disguise. You were unhappy that whole year is a spectrum claim — a feeling, integrated, dateless. No, only from March is a scalogram claim, a surge with an address. Both can be true of the same year; they simply measure different things.
The mapping, exactly
| Mathematics | Life |
|---|---|
| the signal s[n] | A stretch of a life or a relationship, laid out along time. |
| a frequency | A recurring feeling or theme — a rhythm the life keeps returning to. |
| the Fourier spectrum | Knowing which feelings were present across the whole span, with no dates attached. |
| the transient burst | A particular surge — the night it struck: brief, intense, and located. |
| a wavelet scale | A feeling's character — its pitch, how fast or slow it runs. |
| the scalogram's time axis | When each feeling arrived, marked to the week. |
| time–frequency localization | A memory that pins each surge to its moment instead of dissolving it into the whole. |
| the uncertainty trade-off | The sharper you date a feeling, the vaguer its exact pitch — and the reverse. |
| two transforms, one signal | Two faithful memories of one life, differing only in whether they kept the calendar. |
Read as life lessons
A memory that says which and not when has lost nothing but the timestamps. The feeling it reports was really there, all of it. Do not mistake a spectrum's silence about dates for a lie.
What runs quietly under everything smears across the whole window. The background mood has no single moment — asking when it began is asking the wrong panel. Some things were simply always on.
A fast, brief surge lands on a sliver of time. Those are the memories with dates — the ones a scalogram catches and a spectrum drowns. Different feelings ask to be remembered in different ways.
The honest model
The signal is built live: a background tone A₀·sin(2π f₀ n/N) spanning the window, plus a Gabor burst A_b·sin(2π f_b n/N)·exp(−((n−t₀)/w)²) localized at time t₀, optionally a second burst and light noise. The left panel is a direct discrete Fourier transform, S(k)=Σ s[n]e^{−2πikn/N}, drawn as |S(k)|. Shifting the burst in time multiplies S by a pure phase and leaves |S| untouched — which is exactly why the gold panel holds still.
The right panel is a continuous wavelet transform with a Morlet mother ψ(t)=e^{iω₀t}e^{−t²/2}, ω₀=5.5, evaluated over 48 scales: W(a,b)=Σ_n s[n]·ψ*((n−b)/a)/√a. Each scale a is a pitch (small a = high, squeezed; large a = low, stretched); each column b is a moment. The heatmap is |W(a,b)|; the reported event is its brightest cell in the fast band — a live argmax, not a caption. Both panels read the same s[n]; the contrast between them is the whole point.
Where the metaphor tears
They only redistribute the blur. There is a hard floor: the product of your uncertainty in time and in frequency cannot go below a fixed limit. Pin a surge to the instant and its pitch goes vague — the bright blob for a fast event is thin in time but tall in pitch, spanning several rhythms at once. Nail the pitch and the timing smears — the slow background is exact in tone but spread across the whole window. You cannot have both sharp. Naming this sharpens the metaphor: knowing exactly when a feeling struck and exactly what it was are rival luxuries, and any real memory has paid for one with the other.
The Morlet packet is tuned to catch oscillating surges. Swap it for a wavelet shaped like a step and the transform would flag sudden shifts instead; shaped like a spike, it would flag jolts. The "when" you read off is partly an artifact of the lens you brought — change the lens and different moments light up. A memory works the same way: it finds the kinds of events it was shaped, by temperament or grievance, to look for, and calls the rest background.
Both transforms are lossless re-encodings of the signal — but the years were never a signal. Reducing a decade to tones-and-times, dated or not, is already the metaphor's first and largest loss. A marriage is not a sum of oscillations any more than a face is a sum of pixels; the double reading is a beautiful argument about a map, and the territory has quietly left the room.