Press a key. Four things happen at once: you hear it, you see its name, you see its frequency in hertz, and you see the actual wave that your ear is receiving. They are four views of one single thing — and being able to move between them is most of what your essay is about.
Frequency = how many times the air pushes your eardrum each second. That is the whole of what "pitch" is.
Press a key below.
A4 = 440.00 Hz
A5 = 880.00 Hz
one octave up = × 2
Same note letter. Twice the frequency. Twice as many cycles in the same window — look at the two curves above.
Keyboard shortcuts, if you like them: A W S E D F T G Y H U J runs left to right across the twelve keys, the same way a real keyboard is laid out. Nothing plays until you ask it to.
This is the same twelve keys written as a list. Look down the right-hand column before you read any further — every step multiplies by the same number.
| Note | Frequency (Hz) | Ratio to the note below | Ratio to the first note |
|---|
Because of what the two waves are, not because anyone decided it. When a note is at double the frequency, exactly two of its cycles fit inside one cycle of the lower note. Every single time. The two waves line up again and again, forever, and never drift apart.
Your ear is extremely good at noticing that kind of lining-up. When two sounds keep agreeing, you hear them as belonging together — so strongly that we give them the same letter name. A4 and A5 are different sounds that your brain files under one word. That is a remarkable thing for a brain to do, and it is why the psychology half of your pathway is not a decoration on the maths half.
The octave is 2:1 — the simplest possible ratio other than 1:1. The next simplest is 3:2, and it sounds almost as agreeable. Musicians call it a perfect fifth: C to G.
| Interval | Pure whole-number ratio | As a decimal | What 12 equal steps actually gives | Off by |
|---|---|---|---|---|
| Octave | 2 : 1 | 2.0000 | 2.0000 | 0 cents — exact |
| Perfect fifth | 3 : 2 | 1.5000 | 1.4983 | −1.96 cents |
| Perfect fourth | 4 : 3 | 1.3333 | 1.3348 | +1.96 cents |
| Major third | 5 : 4 | 1.2500 | 1.2599 | +13.7 cents |
| Minor third | 6 : 5 | 1.2000 | 1.1892 | −15.6 cents |
A cent is 1/100 of a semitone — 1/1200 of an octave. Most people start noticing a mistuning somewhere around 5–10 cents on a sustained note.
Read that table again and notice what it is confessing. On a piano, the octave is perfect and every other interval is slightly wrong. The major third is nearly 14 cents sharp — that is audible. We accepted that deal so that all twelve keys would work equally well, so that you could change key in the middle of a piece and not need a different instrument.
That trade is your essay. "Why twelve notes?" has a mathematical answer, and it is not "because twelve is pretty". It is: twelve is the smallest number of equal steps that lands close enough to the simple ratios your ear wants. Try the arithmetic yourself — 27/12 = 1.4983 against 3/2 = 1.5000. Two cents. Nobody hears two cents. That near-miss is the entire foundation of Western music, and it is a coincidence of arithmetic.
And the cost of the deal has a name — the Pythagorean comma. Stack twelve perfect fifths: (3/2)12 = 129.746. Stack seven octaves: 27 = 128. They do not meet. They miss by about 1.36%, roughly 23.5 cents — a quarter of a semitone, comfortably audible. Twelve-tone equal temperament is what you get when you take that error and smear it evenly across all twelve steps so it never piles up anywhere. You met this in the M6 challenge problems. That is Unit 3.
The question to bring back: the friendly pair has a simple whole-number ratio and the unfriendly one has an irrational number. Your ear worked that out without being told. How? Nobody handed it a calculator.
Do not look this up yet. Come with your own guess first — a guess you made and then tested is worth more than an answer you were given, and it is also exactly what the evidence section of your essay needs.
Every tone here is a pure sine wave — one frequency, nothing else. Real instruments are never that: a real piano note is a fundamental plus a stack of harmonics, which is why a piano and a flute at the same pitch sound different, and why your oscilloscope at home would show something far more jagged than these smooth curves. That is the M6 territory. This page deliberately shows you the clean skeleton first, so that when the real thing turns out to be messier you know exactly which part is the mess.