The Critical Band Theory of Plomp
With a series of experiments on consonance and dissonance judgment, Plomp has concluded that two pure tone frequencies falling outside the critical band (fCB) are judged as being consonant.
These experiments revealed that the most dissonant interval is the one for which the two pure tone frequencies are separated by 25% of the critical band. In fact frequencies differing in the range 5 to 50% of their critical band are typically judged dissonant.
Therefore we are led to conclude that two pure tones whose frequency difference is less than 50% of the appropriate critical bandwidth for those frequencies is a dissonance.
In the following examples we will examine a perfect fifth, two major thirds and a major second.
Example 1: The fifth C = 262 Hz to G = 392 Hz (assuming the tempered scale) has a centre frequency (i.e. average frequency) of 327 Hz. The critical bandwidth fCB at 327 Hz is, from the table, 100 Hz. This means that any two tones within the range 277 to 377 Hz will have the described harsh, rough quality. The frequencies 262 and 392 do not fall within this range, and so the interval is judged consonant.
Example 2: For the major third C = 262 Hz to E = 330 Hz, the centre frequency is 296 Hz for which fCB = 95 Hz. This interval is considered consonant since the frequency difference of the two pure tones is greater than 0.5 fCB.
Example 3: For the major third C = 131 Hz to E = 165 Hz, one octave lower than that in example 2 above, the centre frequency is 148 Hz for which fCB = 90 Hz. This interval is certainly less consonant in terms of the Plomp criterion than that in example 2, and may even be considered a dissonant interval.
Example 4: For the major second C = 262 Hz to D = 294 Hz, the centre frequency is 278 Hz for which fCB = 95 Hz. It is according to the Plomp criterion a dissonance.
The experimental results obtained for pure tones are extended to complex tones. Each complex tone in a musical interval has a 'family' of harmonics. If harmonics up to 6ffund are considered for each complex tone, consonances will have a predominance of neighbouring harmonics whose frequency difference is greater than 0.5 fCB. A dissonance, on the other hand, will have neighbouring harmonics whose frequency difference is less than 0.5 fCB.