The shape of harmony.

Question 1

What is a note?

A vibration that repeats hundreds of times a second.

The simplest one is a sine wave. How fast it repeats is the pitch you hear.

Question 2

What happens when two notes are almost the same?

Two sine waves a few hertz apart drift in and out of step. Together they swell and fade. This is called beating.

Move them further apart and the beating gets faster, until it becomes a harsh buzz.

Move them further still and you hear two separate notes.

Why does the ear buzz?

The beating is just what happens when two waves add.

The roughness comes from your ear. Inside the cochlea is a long membrane. One end shakes for high pitches, the other for low. Each spot along it responds to a narrow band of pitches.

Two tones far apart shake two different spots. Two tones close together shake the same spot, and that spot is shaken by the beating.

A slow shake sounds like a wobble in loudness. Above about twenty shakes a second, it sounds rough.

Each spot’s band is about 100 Hz wide for low notes, and wider for high ones. That is why the curve has the shape it does.

Pure sine waves have nothing special at a fifth or an octave. So where do musical intervals come from?

Question 3

Then why do C and G sound good together?

Real notes are not pure sine waves.

A plucked string vibrates as a whole, in halves, in thirds and in quarters, all at once. Each part adds an overtone at 2×, 3× or 4× the base frequency.

You hear the mix as the sound of the instrument.

So two notes means two sets of overtones. If an overtone of one lands close to an overtone of the other, those two buzz, like the pair in Question 2.

Question 4

Can we measure “sounds good”?

Yes. Below are the overtones of two notes in hertz. C is on the left and the second note is on the right.

If a rung on one side is close to a rung on the other, that pair buzzes. The curve from Question 2 says how much.

If two rungs land on the same frequency, they do not buzz.

Add up the buzz of every pair and you get one number for the two notes.

Try C + C♯. Then try C + G.

The deepest valleys are at simple ratios. The octave is 2:1. The fifth is 3:2. Then the major sixth (5:3), the fourth (4:3) and the thirds (5:4 and 6:5).

These are the intervals music theory teaches. They come straight out of the overtones.

Question 5

Where do scales come from?

Pick notes that sit in valleys and you have a scale.

But the valleys above C are not quite the valleys above D or G. A keyboard tuned perfectly for one key sounds off in another.

Pianos use equal temperament: twelve notes spaced evenly. The posts below are those twelve. Most land in or near a valley. The major third lands 14 cents sharp, a little way up the slope.

Question 6

Would another instrument give another scale?

Yes. Here is the landscape for the metal bar from Question 3.

The valleys move. The octave is no longer a valley. The nearest valley is about 38 cents below it.

So a scale depends on the overtones of the instruments that play it.

Question 7

What about chords?

With a third note, the landscape becomes a map.

Each point is a three-note chord starting on C. Across is the second note. Up is the third.

The contour lines show roughness, like the height of the landscape. The blue lakes are the least rough chords.

The chords in the lakes are the major and minor chords, the suspended chords, and their inversions. These are the everyday chords of Western music.

Question 8

Why does music feel like it’s going somewhere?

Music moves from chord to chord. Each chord in a key has a job: home, leaving, or pulling back.

Play them and look at how rough each one is.

G major is a little smoother than C major, because it sits higher.

Adding F makes G7, and the roughness jumps by half. B and F form a tritone.

B is one step from C, and F is one step from E. So G7 wants to move to C.

Question 9

Is there a logic to the journeys?

Yes, mostly.

Each note in a chord is a voice. When the chord changes, each voice moves to a note of the next chord.

Common progressions let every voice move a small step or stay put. The theorist Dmitri Tymoczko showed this in A Geometry of Music.4

Below are the same chords played two ways. First with each voice moving to the nearest note. Then with every chord in the same shape, so all the voices jump together.

Two other things shape progressions.

Roughness. Progressions usually get rougher away from home and smoother coming back. G7 is the roughest chord here, and it comes just before the last C.

Habit. In Bach’s chorales, the most common move is from G to C. After hearing it enough, you expect it.5 The pop loop I–V–vi–IV never makes that move, so it keeps going round.

Question 10

What turns chords into a song?

Chords are the base. Three things go on top.

Rhythm

When notes happen. It gives the pulse.

Melody

A single line you can sing. On the strong beats it uses notes from the chord. Between them it steps through other notes.

Motif

A short idea that repeats. Here it moves down one step each bar to fit the chord.

Add them one at a time.

The Music box composes whole pieces this way: a motif, a plan of chords, and notes chosen to fit them.