The first page said "there are only twelve notes". But why twelve, and why these twelve? The answer comes from how strings vibrate. This page has a little more maths than the others; you can skip it and come back later, but it explains a lot.
On this page, notes get a number after the letter, like A4 or A2. A4 is the A in the middle of the piano. A2 is two octaves lower. The number tells you which octave a note is in; more on that on the next page.
An interval is the distance between two notes. Some intervals sound smooth when the two notes are played together, others clash. To see why, we start with a single string.
A plucked string doesn't vibrate at just one . Along with its main note it vibrates in halves, thirds, quarters and so on. That adds quieter, higher sounds called overtones, at 2, 3, 4, 5 … times the frequency. Here is A2 (110 Hz) with its first overtones (1× is the note itself):
Neighbouring overtones give us the intervals that sound smoothest together. Their frequencies are in simple ratios, so their own overtones line up. Twice the frequency (2:1) is the you already know. Musicians call 3:2 a 5th, because from C up to G it spans five letters, C D E F G (7 ). In the same way 4:3 is a 4th (four letters, C to F: 5 half steps) and 5:4 a major 3rd (three letters, C to E: 4 half steps). We'll meet these intervals properly in the Scales chapter.
After the octave, the 5th (3:2) is the smoothest interval. So let's build notes from it: start on C and keep going up a 5th, moving each new note down by octaves so it stays in one octave: C, G, D, A, E, B, F♯ … (Musicians draw this as the circle of fifths, which gets its own chapter later.)
| 5ths up | Note | Half steps above C | Off from the equal note |
|---|---|---|---|
| 0 | C | 0 | ±0.0 cents |
| 1 | G | 7 | +2.0 cents |
| 2 | D | 2 | +3.9 cents |
| 3 | A | 9 | +5.9 cents |
| 4 | E | 4 | +7.8 cents |
| 5 | B | 11 | +9.8 cents |
| 6 | F♯ | 6 | +11.7 cents |
| 7 | C♯ | 1 | +13.7 cents |
| 8 | G♯ | 8 | +15.6 cents |
| 9 | D♯ | 3 | +17.6 cents |
| 10 | A♯ | 10 | +19.6 cents |
| 11 | E♯ | 5 | +21.5 cents |
| 12 | B♯ | 0 | +23.5 cents |
(A cent is a hundredth of a half step; most people can't hear a difference below about 5 cents. The last column shows how far each note is from the same note on a modern piano; more on that below.)
Look at what the first few 5ths give us:
So the natural notes are seven neighbours on the circle of fifths, and the five sharp/flat notes are the next five. That's why twelve: it's where stacking 5ths comes back around.
"Almost" is the catch. Twelve pure 5ths overshoot seven octaves by a little, about 23.5 cents, a quarter of a half step. B♯ comes out slightly higher than C. Listen to the two together; the slow wobble is the mismatch:
The fix used on almost every instrument today is equal temperament: make all twelve half steps exactly the same size, so that twelve of them make exactly an octave. Each half step multiplies the frequency by the 12th root of 2, about 1.0595: 5.9% higher each time. (That's the "about 6%" from Intro to Notes.)
The 5th is then a tiny bit narrower than pure, by only 2 cents, so little that almost nobody notices. The 3rds are off more, but we're used to that sound. In the table, "perfect" is just the traditional name for the 4th and the 5th. The minor 3rd (6:5, 3 half steps, like C to E♭) is the major 3rd's smaller cousin. Compare:
| Interval | Pure ratio | Equal | Difference | Listen |
|---|---|---|---|---|
| Octave | 2:1 (2.000) | 2.000 | ±0.0 cents | |
| Perfect 5th | 3:2 (1.500) | 1.498 | −2.0 cents | |
| Perfect 4th | 4:3 (1.333) | 1.335 | +2.0 cents | |
| Major 3rd | 5:4 (1.250) | 1.260 | +13.7 cents | |
| Minor 3rd | 6:5 (1.200) | 1.189 | −15.6 cents |
What we gain: a song can start on any of the twelve notes and sound equally in tune, so music can move freely between them. We'll see how useful that is in the Keys chapter.
One note has to be fixed, and the rest follow. Today that's A4 = 440 Hz. For every other note, take 440 Hz and multiply it by 1.0595 once for each half step up (or divide, going down). This app calculates every frequency it plays this way. Middle C (C4) is the C nearest the middle of the piano: