Algebra and Trigonometry 2e · §7.3 Unit Circle
OpenStax · Open textbook · §7.3
Read §7.3 for radians and unit-circle coordinates; use §8.1 from the same book for graphing sine/cosine.
Open the sourceHarmony as a path, not a frozen shape
Rotations create waves; progressions require a state space.
Prerequisites: 08 · A matrix is an action, 11 · Eigenvectors and the sign of a geometry, 25 · Hidden states and observable shadows
Understand
Objectives
- Predict what changing a weight does versus a frequency.
- Prove the common-period claim from frequency ratios.
- Separate waveform phases from pitch-class chord encodings.
Start with something you can see
In the harmonics lab, keep the second frequency at 2 and set the second contribution a to zero. Predict which part of the signal disappears and whether the rotating state stops having its second circle. Now choose √2. Predict whether the whole state can return to its starting point at a positive time. The Fourier companion shows how different frequencies contribute to one observed signal. [1]
Give the idea a precise name
Each rotating pair has a phase that advances at its own angular frequency. The state contains all three pairs, but the observed wave is one weighted scalar. An observation weight can hide a pair without stopping its motion.
z(t)=(cos t,sin t,cos ft,sin ft,cos 3t,sin 3t); y(t)=cos t+a cos(ft)+b sin(3t)
For f=2 the frequency ratios are rational and the full state repeats at 2π. It traces one closed orbit, not a filled three-dimensional torus. With f=√2, a common period T would have to make both T and √2T integer multiples of 2π, impossible for positive T. The first and third phases remain locked: one irrational mode does not make all three phases independent.
Work one small world
Set a=0.5 and b=0.25 at t=0: the signal is 1.5 and the state begins (1,0,1,0,1,0). Change a to zero: the observed signal at t=0 becomes 1, but the state is unchanged. The full locked state first returns at 2π; with f=√2 it has no common positive period.
| Lab step | What changes | What stays |
|---|---|---|
| Set a=0 | Scalar observation | Second rotating pair |
| Set f=√2 | Second rotation and full-state period | First and third frequencies, locked 1:3 |
Counterexample to inferring the hidden period from the signal: with only the cos 2t component observed, its period is π, while a state that also contains (cos t,sin t) first returns at 2π. A closed four-chord path likewise does not determine a Möbius or Klein-bottle ambient space; those require specified identifications.
THE BRIDGEThe next lesson asks which distinctions survive compression rather than a scalar observation.
WHERE THIS IDEA STOPSA waveform, pitch-class set, voiced chord and perceived harmony are different representations.
Experiment
Set f=2, a=0.5, b=0.25 and time t=0. Predict y=1.5. Set a=0 and compare. Then select √2 and step Time t; Play / pause is optional. Read Full state repetition rather than inferring a period from the finite plotted wave.
Are you changing the underlying motion, or only the weights through which you observe it?
Check & explain
For (cos t,sin t,cos 2t,sin 2t), what is the smallest positive full-state period, in multiples of π?
When f=√2, which statement follows?
- The full state has no common positive period
- The orbit fills an entire three-torus
- The third pair stops rotating
Teach back
Define a four-chord sequence using pitch-class sets C={0,4,7}, G={2,7,11}, Am={0,4,9}, F={0,5,9}, with semitones mod 12. Name the distance you choose and what this encoding forgets.
- A missing dynamics/observation distinction: vary a without changing z.
- A missing period proof: compare the first and irrational modes.
- An unsupported topology claim: define an ambient space and identifications first.
Read deeper
Mathematics for Machine Learning
Deisenroth, Faisal & Ong · Freely readable textbook
Read §§4.5–4.6, pp.119–134 for decomposition as a later extension.
Open the sourceBut what is the Fourier Transform? A visual introduction
3Blue1Brown · Grant Sanderson · Video · 20 min
Start at 2:10; watch the short 2:10–4:40 core or the complete 20-minute video. The one scalar is not a unique inverse of its components.
Open the sourceLecture 31: Eigenvectors of Circulant Matrices: Fourier Matrix
Gilbert Strang · MIT OpenCourseWare · Lecture · 53 min
Watch the full 53-minute lecture after working the rotating-pair example; its OCW page refers to the same lecture.
Open the source