A note of the same fundamental pitch sounds quite different when played on a violin, a trumpet, or a flute. This is because all musical instruments produce notes of other frequencies along with the fundamental note. These are called harmonics or overtones. Strings and columns of air (brass and woodwind) produce harmonics whose frequencies are simple numerical multiples of the fundamental, whereas drum skins and solid objects (bell, cymbal and triangle) produce harmonics with no simple relationship, so their sounds are musically discordant. The character of an instrument's tone depends on the presence and relative intensity of these harmonics and is called the timbre of that sound. Fourier's theorem shows that any sound wave can be built up using a sufficient number of pure tones of the correct frequencies and amplitudes. It is these harmonically related frequencies whose amplitudes are controlled by the drawbars on large organs, which allow the player to build up any sound he wishes.

Let us examine first the waveforms which can be produced electronically.

The Sine Wave: this is the basic pure tone, the most simple form of vibration possible. Mathematically the sine wave describes the oscillations of a pendulum or any similar object. Musically, the only instrument which produces a sine wave is a tuning fork or a softly blown flute.

The Triangle Wave: This is the waveform produced by an integrator ramping alternately positive and negative by equal amounts and at the same rate. Analysis of its harmonics shows that it contains only odd multiples of the fundamental frequency, i.e. a triangle wave of frequency 100 Hz contains sine waves of 100, 300, 500, 700, 900...Hz (f, 3f, 5f, 7f, 9f...).

The amplitude of each harmonic is inversely proportional to the square of its harmonic number, i.e. if the fundamental is at relative amplitude A, then 3f is at A/9, 5f is at A/25, 7f is at A/49. This can also be expressed by saying that the harmonics decrease by 12 dB per octave. Thus although it contains all the odd harmonics, most of them are too faint to be heard; however beats between adjacent harmonics may contribute to the overall sound. Something approaching a triangle wave is produced when a string is plucked exactly at its mid-point. Figure 1 shows how the triangle wave can be built up.

Square and Rectangle (pulse) Waves: These are all members of the same family and a simple rule explains their harmonic content. The ratio of the proportions of high and low, or on and off is called the duty cycle or mark:space ratio. Thus a wave which is high for 25% of the time has a duty cycle of 25% and a mark:space ratio of 1:3. The reciprocal of the duty cycle is called the L-number, and for a duty cycle of 25% (=¼), L=4. A rectangular wave contains all the harmonics except those divisible by the L-number, in this case 4f, 8f, 12f etc. The amplitudes of the high harmonics are very strong.

The square wave is a special case; its duty cycle is 50%, giving L=2, so it contains only odd harmonics. Their amplitudes are inversely proportional to the harmonic number (3f=A/3, 5f=A/5 etc.). This means that the harmonics fall off at 6dB per octave (compare with the triangle wave). See Figure 2.

The Sawtooth (ramp) Wave: This very useful waveform contains all odd and even harmonics, and their amplitudes are inversely proportional to the harmonic number, just like the square wave (6dB per octave fall-off). In a good quality sawtooth up to about the thirtieth harmonic will be detectable. See Figure 3.

Interconversion of waveforms

Most organs use bistable dividers and therefore have available large numbers of harmonically related square waves, but no sawtooths. Proper synthesis of many instrumental tones requires even as well as odd harmonics i.e. sawtooths but these can be created electronically by staircasing. A 200 Hz square wave contains odd multiples of 200 Hz (600, 1000, 1400 etc.) which are even harmonics of a 100 Hz tone; similarly a 400 Hz square wave contains also 1200, 2000, 2800... Thus by adding together square waves of 100 Hz at amplitude A, 200 Hz at A/2, and 400 Hz at A/4 a staircase with 8 steps is produced, which contains all the harmonics required in a 100 Hz sawtooth except for the 8th, 16th, 24th, etc., and they are all of the correct amplitude. A simple circuit for doing this can be made from a 741 op-amp. See Figure 4(a). A similar circuit can be used to produce a square wave from two sawtooths an octave apart; all the harmonics of the higher sawtooth are even harmonics of the lower one, so by subtracting 2f at amplitude A/2 from fat A, only the odd harmonics are left, and this constitutes a square wave. See Figure 4(b).

Consider now the sounds of actual instruments: their timbre depends not only on the harmonics present in the waveform, but also on the effect of the formant. This is a band of frequencies, fixed for each instrument but varying even between different specimens of the same type of instrument, and any harmonics falling into this frequency band sound at a loudness level higher than that expected for their harmonic number. This effect is a form of resonance, and it explains why a Stradivarius has a richer tone than a cheap fiddle - the formant of the Strad is unusually wide and extends to higher frequencies. The same effect can be applied to an electronically produced sound by the use of a resonant band pass filter with adjustable centre frequency and 'Q'.