The Transform of a Finite List
Last timeTwo Numbers for Every Rhythm
With both probes bundled into one turning arrow, the transform of a list of readings is a single finite sum. It inverts exactly, and the count of numbers balances on both sides.
Two things are now in hand. There is an operation that extracts the amount of one
chosen rhythm, and each rate needs two of them. This lesson bundles the pair into
one object, writes the whole transform as a single sum, and settles the accounting.
One arrow instead of two probes
The two probes are never used apart and their answers are always read as one arrow,
so carrying them separately is clerical work. A single turning arrow does both jobs:
multiply the reading by it, and the across part of the product is what the cosine
probe would have contributed while the up part is what the sine probe would have
contributed.
This is where complex numbers enter, and they are not a deeper idea. They are
notation for two numbers that always travel together, adopted because multiplying
two arrows adds their angles, which makes the whole of the next lesson one line long.
- one arrow, for the rate that completes k whole cycles across the record. Its length is how much of that rhythm there is and its direction is where in the cycle the rhythm stood at the first reading
- step through every reading once, multiply, and add. Nothing is continuous and no limit is taken: this is a finite sum of N products, which is also why the whole transform costs N of these
- the readings, used as they came. They are real numbers here, which is what forces the mirror symmetry the counting depends on
- the probe arrow at the nth reading. It starts pointing along the across direction and turns by k full turns over the course of the record, so faster rates are faster turns and nothing else changes
Only half the rates are new
Two numbers per rate looks like twice as many as the record can possibly support,
and the resolution is that the rates are not all different questions.
A probe turning faster than half a cycle per reading is only ever looked at at the
reading instants, and at those instants it passes through exactly the values a
slower probe turning the other way passes through. The two are indistinguishable by
any amount of data from this record.
The lesson stops here
2 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents