Acoustics

Every sound begins with something vibrating.

On a guitar, what vibrates is the string. When the musician plucks it, it vibrates from side to side, and while it vibrates there is sound. On a saxophone, what vibrates is the reed, a thin blade of cane fixed to the mouthpiece: the air blown into the instrument makes it vibrate against the mouthpiece. In the voice, what vibrates are the vocal folds in the larynx, which close and open as the air passes through them.

The string, the reed and the vocal folds make the same movement: they go back and forth quickly, and this movement produces sound. Acoustics is the science that studies sound (Lacerda 1966, ch. XLIII).

Vibration

A vibration is a back-and-forth movement around a resting position. One complete vibration is the whole path: to one side, to the other and back to the starting point (Lacerda 1966, ch. XLIII). A plucked string repeats this path, shorter each time, until it stops.

From the instrument to the ear

The vibration passes to the surrounding air and propagates as a sound wave. The air itself barely moves; what propagates is the vibration. When the wave reaches the ear, it makes the eardrum vibrate in the same pattern, and the ear perceives this vibration as sound (Lacerda 1966, ch. XLIII).

In air, sound propagates at about 340 metres per second, roughly a kilometre every three seconds. That is why thunder arrives after the lightning.

Regular vibrations produce a musical sound, which has a definite pitch. Irregular vibrations, such as those of a knock on a table, produce noise (Lacerda 1966, ch. XLIII).

Frequency and pitch

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

Frequency determines pitch: the more vibrations per second, the higher the sound (Lacerda 1966, ch. XLIII). For Blatter as well, pitch as perceived by the ear corresponds to frequency as measured by physics (Blatter 2007, art. 1).

The frequency of a note does not depend on the instrument. A given A has the same frequency whether played on a violin or a clarinet or sung by a tenor (Lacerda 1966, ch. XLIII).

When the frequency doubles, the note is the same, one octave higher:

Hear the passage♩ 84

These three As have frequencies of 220, 440 and 880 Hz. Each has twice the frequency of the one before.

Amplitude and intensity

Plucking the string harder or more gently does not change the frequency, and so does not change the note. It changes how far the string moves from its resting position. That distance is the amplitude, and it determines intensity: the greater the amplitude, the louder the sound (Lacerda 1966, ch. XLIII).

Frequency and amplitude are independent. Frequency determines pitch; amplitude, intensity (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 decreases and its frequency stays the same. That is why a piano note grows softer without changing pitch.

Limits of hearing

The ear does not perceive every frequency. Below a certain limit, the vibration can be felt but is not heard as sound; above another, it is no longer heard. The books give different limits:

Source	Lowest sound	Highest sound
Lacerda	32 Hz	8,200 Hz
Med	16 Hz	about 4,100 Hz

Part of the difference comes from the criteria. Lacerda gives the limits of what the ear normally perceives (Lacerda 1966, ch. XLIII). Med gives as the upper limit the highest sound still recognised as a note and, as the lower limit, the sound of the largest organ pipe (Med 1996, ch. XXXVIII).

Current references give a range of 20 Hz to 20,000 Hz for young ears, and the upper limit falls with age. Above a few thousand hertz sound is still heard, but its pitch becomes hard to identify.

Reference note

For different instruments to sound in tune with one another, musicians tune to a reference note with a fixed frequency. The instrument that produces this note is the tuning fork (Lacerda 1966, ch. XLIII).

The reference note is an A. Lacerda gives 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 to 442 or 444 Hz for a brighter sound (Med 1996, ch. XXXVIII). Electronic tuners use 440 Hz as their factory default.

Harmonic series and timbre

In practice, a note is almost never a single vibration. The string vibrates as a whole and, at the same time, in halves, thirds and quarters, and each of these parts adds a higher, softer sound. Blatter describes the result as the sum of many simple waves (Blatter 2007, art. 1). These sounds form the harmonic series, and their relative intensity determines the timbre of each instrument.