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.
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Audio examples and further discussion.