Musikteoretiske og akustiske aspekter ved konsonans og dissonans

Abstract

The more than 2000-year-old fascination with the mathematical aspects of music is the starting point. The preoccupation with the numeric relations of the intervals seems to have been nourished by the idea that it will be possibie to get beyond the surface onto »the thing in itself«. In section 1.1 the theoreticai and historicaI concepts of consonance and dissonance are described. The acceptance of thirds and sixths in the music of the 14th century leads to the standard grouping of the musical intervals into three categories: perfect consonances, imperfect consonances, and dissonances. Around 1400 the strict dissonance treatment is established, i.e.: the demand that a dissonance on rhytmically strong notes is tied over from the previous chord and resolved by step, usually downwards. This stylistic feature remains amazingly constant through the history of Western music; until c. 1900, in fact, when the dissonances are emancipated. In our century the decisive aspect of a chord is not that it is joined with the neighbouring chords by a specific voice leading technique, but rather its purely sonic appearance. It is amazing that this radical change of style has not led to a corresponding revolution of consonance and dissonance on the theoretical level, a new concept of harmony. A survey of the Renaissance music theorist Zarlino's account of the numeric relations and the musical character of the intervals concludes this section.

The acoustic and psychological concept of consonance and dissonance is surveyed in section 1.2: The discovery of oscillations in the Renaissance; later the discovery of the partial tone series; through the numerous theories of harmony in the 19th century; to the psycho-acoustical research of the last 20 years, in which the concept »roughness« of perceived dissonance is the keyword.

The acoustical prerequisites for consonance and dissonance are discussed in the 2nd section: First a number of elementary acoustical phenomena (2.1), secondly the auditory functions including the transformation of sound energy into nerve impulses as uncovered by the latest audio-physiological research (2.2). These relations fumish us with a new way of understanding the resolving power of the ear, i.e. its capability of discriminating several simultaneous stimuli (2.3). The ear's frequency analysis of two simultaneous sine waves can, according to the circumstances result in complete resolution, no resolution, or partial resolution. The critical bandwidths (2.4) are audio-psychologically defined frequency bands describing the resolving power of the ear in quantitative terms. They are of great significance in understanding the neural coding and thus the perception of sound. Combination tones including difference tones (2.5) are perceived tones not physically present in the original signal. Their phenomenology is well-known but their origin is not fully explained as yet. Beats of mistuned consonances for sinuiodal tones (2.6) have earlier been explained by the presence of aural harmonics. Today there is an inclination to explain them by the ear's sensibility to variations in the cyclic waveform. They are an example of how the resolving power of the ear works on compund waveforms.

The residual pitch (2.7) is, in short, the phenomenon that the fundamental of a harmonic spectrum is perceived even if it is not physically present. It has been explained both with reference to the resolved tones of the spectrum and by the unresolved part of it. The residual mechanism may have something to do with the gestalt-formation of the perception of chords.

Roughness (2.8) is a concept from the newer psycho-acoustical research established in order to be able to describe the perception of simultaneous tones without being burdened by the theoreticai and historicai implications of musical consonance and dissonance. The roughness of two simultaneously sounding simple tones is closely related to the critical bandwidth. There is maximum roughness at c. 1/4 of the critical bandwidth. These matters are investigated in great detail by Plomp and others, while Kameoka and Kuriyagawa have established a well-defined scale for roughness with general perception-theoretical considerations as the point of departure. This has enabled us to mathematically calculate the roughness of a given sound. These calculations are expanded to include complex tones with given partials in different registres.

It turns out that the roughness of several simultaneous complex tones cannot be explained by the interaction of their fundamentals alone. One must calculate the joint effect of all the partials grouped in pairs as they clash within the individual critical bandwidths. It is thus possibie to understand the resolving power of the ear as the audio-psychological prerequisite of roughness. Combination tones may add to the roughness but they cannot be regarded as a determining factor.

The third section is concerned with the relation between the consonance/dissonance concept on the one hand and purely acoustical phenomena on the other. First the concept of roughness is related to music with strict dissonance treatment (3.1). The empirical investigations of roughness based on comparative listening: Different intervals in different registres and with different timbres are compared. In music with strict dissonance treatment the way of listening is also comparative: The dissonance is compared with the subsequent consonance. But as the resolution of the dissonance is stepwise without change in »instrumentation«, one does not compare intervals and chords in different registres and with different timbres. As regards register and timbre roughness then is another thing than consonance / dissonance in music with strict dissonance treatment. The two phenomena do not aequivalate such as most authors of modern acoustic literature have it.

Register and timbre will often endow a musical gestalt with a high degree of roughness without making it a musical dissonance, i. e. without necessitating dissonance treatment. It would, however, be wrong to believe that register and instrumentation in music with strict dissonance treatment are arranged without regard to their influence on the sonic appearance of the music. A statisticaI investigation of two musical works has uncovered that the partial tone clashes are kept within limits which agree nicely with the critical bandwidths. Within a given register and with a given timbre there will normally be a good correlation between musical consonance / dissonance classification and the psycho-acoustical roughness with, however, certain exceptions, e. g. the fourth. In music after 1900 with free dissonance treatment (3.2) the musical point of a chord is not the expected resolution of its formal dissonances but primarily its »sonic appearance«, in which not only its tone combination but also register and instrumentation play an important part. Roughness, which in fact describes the individual harmony without regard to the surrounding voice leading, is here a very adequate concept for a theoreticaI description of the sonic organism of music and it might be a basis for a new musical concept of harmony on the premises of this music. Descriptions and analyses of new music based on the tradition al concept of consonance and dissonance are normally so poor that it seems clear that this concept is outdated. One may safely conclude that the consonance / dissonance concept is inseparable from the rules of dissonance treatment. In 3.3 it is concluded that the consonance / dissonance concept is a cultural phenomenon, that the sharp distinction between consonance and dissonance is not reflected in the sensory perception, but if uniform register and timbre are presupposed, consonance / dissonance normally aequivalate with roughness. Under this aequivalence the physical and physiological conditions for consonance and dissonance are naturally the same as for roughness. The deficient resolving power of the ear may thus be regarded as the fundamental generator of the phenomena.

It is (by the way) difficult to establish a simple correlation between the musical intervals as described by their simple frequency relations and the degree of roughness because, a.o. things, it necessitates the definition of a »normal« spectrum. If the resolving power of the ear was perfect, neither consonance nor dissonance would exist. The two concepts are interdependent.