The phenomenon of consonance and dissonance has occupied the minds of scientists throughout the ages including names such as Pythagoras, Descartes and Euler. However, it appears that it was Stumpf who was responsible for the shift away from correlating number ratios to consonance/dissonance perception towards an experimental psychological approach which ultimately resulted in Hofmann‑Engl's cognitive sonance model.
This paper aims at uncovering the historical pathway in regards to the phenomenon of consonance and dissonance. Here, particularly the works of Descartes, Euler and Stumpf appear to have found little recognition within historical musicology although their contributions are undoubtedly of great importance. Further, the design of Hofmann‑Engl's cognitive sonance model would have been unthinkable without these predecessors. At the same time, the paper will aim at disseminating the main mathematical aspects of this sonance model as well as providing the experimental evidence of it’s validity.
Since Pythagoras simple ratios were considered to produce consonant sounds and complex ratios dissonant sounds. With Descartes we find that the dichotomy between consonance and dissonance had been abolished. Further, Euler pointed out in his later works that the ear might adjust deviations from exact ratios. Stumpf then abandoned the idea of ratios altogether in favor of the Verschmelzungsgrad. Here, Stumpf understood the Verschmelzungsgrad (fusion degree) as the inability of laypeople to hear two simultaneously played sounds as two instead of as one sound. The idea of the Verschemlzungrade then prompted Hofmann‑Engl to produce a sonance model where higher consonance correlates to chords which produce clearer roots and higher dissonance to chords of ambivalent roots. He further tested this sonance model within experiments where he asked participants to state which of two played chords would be more dissonant. Working under the assumption that the higher the difference in sonance of those two chords the more participants would select the “correct” chord, he found that the experiments largely confirmed this hypothesis.
The main implication for a functional sonance model is it's implementation within musical analysis. However, it might prove equally useful in comparing music in a ethnomusicologically fashion. Needless to say that it can help the composer control the consonance/dissonance aspect of her/his music actively.
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