Consider the superposition of two harmonic waves at different frequencies:

The resultant y = y₁ + y₂ (purple curve) is periodic, but it is certainly not harmonic. Of course, to be harmonic a disturbance must only have one frequency, and y does not.

• (a) Six sinusoidal wave forms.
• (b) The result of combining these wave forms.
• (c) The associated spectrum. Each vertical line represents the amplitude of the corresponding harmonic in (a). Note that f₂= 2 f₁, f₃=3 f₁, f₄ = 4 f ₁, and so forth.

Example:

• Pure tone or pure note: sinus curve.
• Sound (music): superposition of several pure tones and harmonics, complicated but regular pattern.
• Noise: irregular pattern
• Bang: spike

Superposition (sum) of 2 tones with slightly different frequencies results in beats.
(The difference is less than about 10 Hz.)

Note that at t₁ and t₃ the signals are 180^o out-of-phase, while at t₂ they are in-phase.

When two signals of different but nearly equal frequencies (f₁ and f₂) superimpose, they create a beat pattern, which is true of sound as well as light. The intensity rises and falls with a frequency equal to the difference (f₁ - f₂).

In equation form:

Here the absolute value, i.e. | f₁ - f₂|, means that either f₁ can be larger than f₂ or vice versa. The beat frequency does not distinguish which of the two frequencies is the larger one.

The number of beats per second, or beat frequency, equals the difference in frequency between the two sources.

The auditory sensation that allows to distinguish between sounds is timbre. Listening to a flute, a trumpet, a saxophone, or a tuning fork, each producing the same note (same pitch) at the same loudness, there is little difficulty telling one from the other. The attribute that allows for this distinction is the timbre, which depends primarily on the waveform - that is, the frequencies present, their relative phases, and amplitudes. When the same tones made by these various instuments are picked up by a microphone and displayed on an oscilloscope, they are seen to be substantially different. Similarly, the same note produced by a piano and a singer have distinctively different frequency spectra and the ear immediately distinguishes the vocalist from the accompaniment, even when the sounds are made at the same moment.

As example the frequency spectrum for the A₃ (220 Hz) note sounded by a piano and an alto voice are shown here.

An actual tone will contain frequencies higher than the fundamental (or first harmonic), and these are called overtones . An overtone need not be a whole-numbered multiple of the fundamental; that is, it need not be a harmonic.
Remember: Harmonics are exact multiples of the fundamental.

Overtones add richness to the sound

It is the number of overtones (whether they are harmonics or not) and their relative amplitudes that more than anything else determines timbre.

Sound Quality

Amplitudes of harmonics

being added to the fundamentals

Attack and decay of a note

determine the characteristic of a tone

Formant region of an instrument

is the frequency region which is emphasized relative to other harmonics. (This gives the unique sound to an instrument.)

Vibrato

Periodic changes in the pitch of a musical tone.

Tremelo

Periodic changes in the amplitude of a musical tone.

Chorus effect

Many instruments play the same part in unison. -->
The sound quality is quite different from the individual instruments due to superposition of many similar tones with slightly different frequencies and tone qualities. ( sample)

Free - Open-source audio recording and modification progrom
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Import a wav file, e.g.
Effects: (Highlight region) e.g.
-- amplify
-- change pitch
-- equalizer
-- .....

Here are some samples to download:

-- Flute Solo
-- Cymbals
-- Keith Jarrett: The Köln Concert

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