Theories of Consonance and Dissonance
Concepts of Consonance and Dissonance
Briefly, these terms are not considered today to be absolute, but are a way of classifying sounds relative to each other. Which sounds are consonant and which dissonant in relation to each other, change with cultural environment, time and taste. Within Western culture, the use of these terms differs depending on musical training.
The Helmholtz Beat Theory of Consonance and Dissonance.
As we have just seen, a complex tone is characterised by its harmonic or overtone structure. When two complex tones are played together (as an interval), the harmonics of each tone are present in the stimulus arriving at the ear of the listener. For some combinations the harmonic frequencies match, for others they do not.
Looking at the diagrams in the handout, we can see that the unison matches exactly, (although in practice this depends upon the instruments used), and this interval is considered to be the most consonant. Next, the perfect 5th shows some matched and some mismatched frequencies. The whole tone shows a mismatch for all frequencies. Also, these frequencies can be close enough together so that discernible beats can result.
The beat theory of Helmholtz maintains that it is the beats that our auditory system responds to, and as a result, the whole-tone interval is more dissonant than the perfect 5th which in turn is more dissonant than the octave that is more dissonant than the unison, which presumably is more dissonant that a single complex tone, which is more dissonant than a single sine tone - this being the most consonant of sounds.
Analysis of these and other intervals reveals that in the terms of this theory an indication of how dissonant an interval is can be judged by how far along the harmonic sequence one has to go before a match up of harmonic frequencies occurs.
As one proceeds down the table the number of mismatched harmonics increases and so does the dissonance. In beat theory then, mismatched harmonics are considered the cause of the dissonance.
In the 19th Century Helmholtz tried to explain consonance and dissonance in harmony entirely in terms of beats. He thought that intervals were consonant if there were no or few beats between the partials. For dissonant intervals, he proposed that the partials of different tones were so close together in frequency that the beating between them was perceived as dissonance.
Audio examples and further discussion.????
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.
Back to main course The Physics And Psychophysics Of Sound & Music
Back to The Psychophysics of Loudness