On the sensations of tone as a physiological basis for the theory of music, by Hermann Helmholtz

Part 1 of Helmholtz’s book opens with an explanation that the sensation of hearing is unique to any other human sense in that only the ear can be stimulated by such precise exterior conditions of vibration. On page 128, Helmholtz explains that the ear detects simpler elements of detected waves and vibrates sympathetically with those waves. He continues to describe the workings of the ear and in Parts 2 and 3 he moves on to describe sounds in terms of waveform in relation to music.

This work became a major influence on composers during the twentieth century especially for its annunciation that western, or twelve-tone tuning systems seem to have become accepted largely in spite of tonal quality. To be more specific, the central message of the book is that frequencies generated by two or more tones can cause by-products that are disharmonious to the ear.

When two or more compound tones are sounded at the same time beats may arise from the combinational tones as well as from the harmonic upper partials. (Helmholtz 1885 p.197)

The beats that Helmholtz is referring to here and all through his book are the beats or pulsations of a vibration or oscillation. Visualised in waveform the vibrations can be understood in terms of the frequency of the wave. When two or more tones are sounded at intervals a combination frequency is sounded. Helmholtz explains this principle through diagrams, illustrations and explanations of the mechanics of numerous musical instruments. He focuses his point by talking about intonation and tuning methods. He explains that some of the beats produced by composite frequencies within twelve-tone equal temperament happen to fall outside of this framework. Helmholtz shows that these by-products are dissonant when perceived in relation to the tuning system from which they derive.

It could be said that Helmholtz has addressed aesthetics from a mathematical perspective when you consider that dissonance is an unfavourable vibration and Helmholtz is stating that vibrations can be deemed to be unfavourable when they produce results that noticeably fall outside the framework of tuning used in Western Europe from around the sixteenth century.

HELMHOLTZ, H. V., & ELLIS, A. J. (1885). On the sensations of tone as a physiological basis for the theory of music. London, Longmans, Green.