tones

We find the regularly repeating vibrations of air particles pleasant and call them tones. Regularly repeating vibrations are also called periodic vibrations. We can produce tones using a voice (when we sing) or by playing musical instruments.

Figure: Noise is produced by random vibrations while a tone is produced by periodic vibrations

Basic properties of tone

The basic properties of a tone are loudness, duration, frequency and timbre. The loudness of a tone is the measure of the amplitude of vibrations. That is, louder tones have bigger amplitudes. The duration of a tone is the measure of the time that passes between its beginning and its ending. The frequency of a tone is the number of periods of vibration that occur in one second.  We say that different instruments produce different timbres or colors of tones. 

Thus, the two tones of the same loudness, duration and frequency, which are played on two different musical instruments, have different timbres. A timbre corresponds to a shape of periodic vibrations. The previous figure shows one possible shape of the vibrations of a tone.

Sinusoidal vibrations 

The simplest periodical vibrations are sinusoidal vibrations which follow the math sinus function.

Figure: Sinusoidal vibration

Timbre and spectrum

We said that different instruments (like the piano and violin) produce tones with different timbres.

The timbre of a tone is determined by different characteristics. Real instruments produce quite complex tones. Their complexity is related to the exact nature of how the sound is produced by the musical instrument. For instance, the piano produces its tones by striking the wires with hammers. The lower and higher piano keys do not produce the exact same sounds, because their wires are differently constructed. Also, the tone of the piano has a very short beginning part which is more like noise than a tone. This is the sound of the hammer’s impact on the wire. Other musical instruments also have their typical beginning of tones. It was experimentally proved that if we remove this beginning then it is hard to recognize which musical instrument was the source of the sound. 

In these chapters we are focused on the middle, stable part of the tone which is really a periodic vibration. When we press a piano key, this stable part starts after the very short beginning part and lasts as long as we keep the piano key pressed.

Mathematically, any periodic vibration of the frequency f can be constructed as the sum of sinusoidal vibrations of the frequencies f, 2f, 3f, 4f… The frequency f is called fundamental frequency. The sinusoidal vibration of frequency f is called fundamental while sinusoidal vibrations of frequencies 2f, 3f, 4f… are called harmonics. Each sinusoidal vibration has its own amplitude. If we plot the amplitudes, we get a spectrum. Different musical instruments have different spectrums. The fundamental frequency is equal to the frequency of the tone.

Figure: A spectrum of periodic vibration

Even the stable part of a tone can change over time. For instance, the amplitude i.e. loudness of the stable part of a piano tone is not constant over time, it decays slowly. Every piano player knows that. If a piano key is held long enough, at the end, the tone will decay completely.