TECH TALK TA
Even the sound of a musical instrument contains a superposition of several harmon ic tones typical of the instrument. However, an arbitrary sound can be represented as a sum of harmonics, their frequencies and their amplitudes.
The frequency components – the acoustic spectrum – of a sound can be extracted through spectral analysis similar to the spectral analysis of light.
In terms of sound pitch, the human ear perceives the tonal differen ce of two pairs of
tones fa1, fa2 and fb1, fb2 equally, if the ratio – not the difference – of the two pairs is equal,
that is, if we have fa1/fa2 = fb1/fb2.
So, for example, we perceive the transition from 100Hz to 125Hz and the transition from 1000Hz to 1250Hz as an equal change in pitch. This relative tonal impression is reflected in the subdivision of the scale into octaves – a doubling of frequency – and other intervals such as second, third, fourth and fifth, which is commonly used in music.
Weber-Fechner-Law
It is not only this tonal impression that a stimulus R has to be increased by a certain
percentage to be perceived as an equal change in perception . It is true for other human senses as well.
In mathematical terms, the increment of a perception ∆E is proportional to the ratio of the absolute increase of the stimulus ∆R and the stimulus R, so ∆E = k∆R/R where k is a proportionality constant.
Moving towards infinitesimally small variations dE and dR, integration yields E=2.3 k lg (R/R0) where lg is the logarithm to the base 10 and R0 the threshold stimulus, at which the perception starts. This relation is known as the Weber-Fechner-Law.
The perception of loudness also follows the logarithmic Weber-Fechner-Law, since the human ear is faced with the task of perceiving very quiet sounds, such as falling leaves in a quiet countryside, as well as very loud sounds, such as the roaring of a nearby waterfall. Indeed, humans are able to perceive sound −6N/m2
pressures from 20×10 to approximately
200N/m2 where the upper limit is the human pain threshold.
Human hearing covers about seven orders of magnitude of loudness, which is an exceedingly large physical interval. It is therefore handy to
use a logarithmic measure when quantifying sound pressure technically, in stead of the physical sound pressure itself.
The sound pressure level L is defined as: L=20 lg (p/p0)=10 lg (p/p0)2
where p0=20×10 −6N/m 2.
The reference value p0 corresponds to the hearing threshold at a frequency of 1kHz as the hearing threshold depends on the frequency. The specification decibel dB is not a unit but in dicates the use of the logarithm to the base 10. The factor 20 is chosen in a way that corresponds to our perception – if two sounds differ by 1dB, we just perceive a difference in loudness.
Assigning sound pressure levels maps the range of sound pressure covering seven powers of ten to a scale from 0 to 140dB. It is remarkable that even the sound pressure of 200N/m2 related to the sound pressure level at the pain threshold is still only a small fraction (1/500) of the static atmospheric pressure of about 105N/m2. Often, several sound pressure levels have to be summarised to one. Signals originating from sources independent from each other,
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