Hugo Riemann's moonshine experiment.

During a silent night in 1875, the young musicologist Hugo Riemann struck a key on his grand piano. He was listening for undertones, which he believed to exist in the sound wave. His nocturnal experiment seemed successful – his aural experience confirmed his experimental hypothesis. These undertones, he would explain later, relate to one sounded tone exactly in the manner of the harmonic or overtone series but extending in the opposite direction. As Example 1.1 shows, where the overtone series extends above a given note (in this case, C two octaves below middle C), the undertone series extends below it (in this case, C two octaves above middle C), in the same integer ratios, to form its exact complement. In hearing these undertones, Riemann believed he had found the natural basis for the minor triad. Since he discovered the works of the physicist-cum-music theorist Arthur von Oettingen in 1869, the young researcher had felt an affinity to the music-theoretical approach that became known as harmonic dualism, which explained the minor triad as the polar opposite of the major triad. Starting from the observation that both major and minor triads contain a perfect fifth and a major third, the dualists explained the major triad upwards from the bottom, and the minor down from the top. In this way, the minor triad is conceptualised as the exact inversion of the major. [ABSTRACT FROM AUTHOR]