Acoustics

Pluck a guitar string and watch it. For a moment it is a blur: the string is moving back and forth so quickly that the eye cannot follow. Then the blur narrows, and when it stops, the sound stops with it.

Acoustics is the science of sound (Lacerda 1966, ch. XLIII), and that blur is where all of it begins.

A vibration

A string at rest is straight. Pull it to one side and let go, and it tries to return to where it was. It does not stop there: its own tension carries it past the resting point to the other side, from where it swings back again. Each trip is a little shorter than the last, until the string is still.

One complete trip, out to one side, across to the other and back, is a vibration (Lacerda 1966, ch. XLIII). A string, a drumhead, the column of air inside a flute and the vocal folds in your throat all make sound the same way: they vibrate.

From the string to the ear

A vibrating string pushes the air next to it, and that air pushes the air next to it. The air itself barely moves; what travels is the push, passed along as a sound wave. When the wave reaches your ear it sets the eardrum vibrating in the same pattern, and the sensation that produces is what we call sound (Lacerda 1966, ch. XLIII).

The push takes time to travel. In air it covers about 340 metres per second, roughly a kilometre every three seconds. That is why thunder arrives after the lightning: the light reaches you almost at once, the sound seconds later.

Not every vibration gives a note. When the movement repeats regularly, the ear hears a musical sound, one with a pitch you can sing back. When it is irregular, as when you knock on a table, the ear hears noise (Lacerda 1966, ch. XLIII).

Frequency: how high

The number of vibrations in one second is the frequency, measured in hertz (Hz) (Lacerda 1966, ch. XLIII). A string that completes 440 vibrations every second sounds at 440 Hz.

Frequency decides pitch. The more vibrations per second, the higher the sound (Lacerda 1966, ch. XLIII). Blatter puts it the same way: pitch, as we hear it, corresponds to frequency as a physicist measures it (Blatter 2007, art. 1).

A note always has the same frequency, whatever produces it. The A that a violin, a clarinet and a tenor sing together is one frequency sung by three different sources (Lacerda 1966, ch. XLIII).

The most useful fact about frequency for music is what an octave is. Double the frequency and you get the same note an octave higher:

Hear the passage

These three As vibrate at 220, 440 and 880 Hz. Each is twice the one below.

Amplitude: how loud

A string plucked gently and a string plucked hard vibrate the same number of times per second, so they sound the same note. What changes is how far the string travels from its resting point on each swing. That distance is the amplitude, and it decides the intensity of the sound: the wider the swing, the louder the sound (Lacerda 1966, ch. XLIII).

Lacerda insists on keeping the two apart, and it is worth repeating: frequency decides how high a sound is; amplitude decides how strong it is (Lacerda 1966, ch. XLIII). A loud note is not a high note, and a soft note is not a low one.

As a note dies away, its amplitude shrinks and its frequency stays put. That is why a piano note gets quieter without going flat.

Where hearing stops

The ear does not catch every vibration. Very slow ones are felt as a shaking rather than heard as a note, and very fast ones are not heard at all. The books give different limits:

Source	Lowest sound heard	Highest sound
Lacerda	32 vibrations per second	8,200 vibrations per second
Med	16 vibrations per second	about 4,100, the highest the ear can still identify as a note

The figures differ partly because they answer different questions. Lacerda gives the range the ear normally perceives (Lacerda 1966, ch. XLIII); Med's upper figure is the highest sound still recognisable as a musical pitch, with the lowest produced by the largest organ pipe (Med 1996, ch. XXXVIII). Today the range usually quoted for young ears is from about 20 Hz to about 20,000 Hz, and it narrows at the top with age. Above a few thousand hertz we still hear sound, but naming its note becomes hard, which is Med's point.

One agreed A

If every orchestra tuned to a different reference, a piece would sound higher in one city than in another. So musicians agree on one note and one frequency to tune to, and a small instrument that produces it, the tuning fork (Lacerda 1966, ch. XLIII).

The agreed note is an A. Lacerda gives it as 435 Hz, the older standard (Lacerda 1966, ch. XLIII). Med gives 440 Hz, the standard adopted internationally in the twentieth century, and notes that some orchestras tune higher, at 442 or 444 Hz, for a brighter sound (Med 1996, ch. XXXVIII). 440 Hz is the A that electronic tuners use by default today.

What comes next

A real note is almost never a single vibration. A string vibrates as a whole and, at the same time, in halves, thirds and quarters, and each of those smaller vibrations adds a faint higher sound. Blatter describes the result as many simple waves added together (Blatter 2007, art. 1), and the particular mixture is what gives each instrument its colour. That mixture is the harmonic series.