Musical Offerings, Fall 2018

70 True ⦁ Intonation and Modulation modulation sometimes changes these intervals. C to F is 503 - 76 = 427 cents. This is extremely sharp and shows why meantone temperament allows for only limited modulation. René Descartes, considered to be the father of modern philosophy, analyzed the way humans perceive sound, and how those perceptions determine dissonance or consonance. In his work, Compendium musicae, Descartes begins with eight preliminaries which summarize these ideas. Several of these preliminaries relate directly to temperament. The fourth preliminary states that “an object is perceived more easily by the senses when the difference of the parts is smaller. ” 30 In terms of temperament, this means that the simplest ratios sound best; the pure major third (5/4) will always sound more consonant than the Pythagorean major third (81/64). Descartes also pointed out that, “Among the sense objects, the most agreeable to the soul is neither that which is perceived most easily nor that which is perceived with the greatest difficulty. ” 31 This means that in addition to the beauty of simple proportions, there must be some variety. A pure open fifth will sound beautiful, but it can sound bland when it is compared to a complete triad. Descartes observed that there must be a trade-off between simple ratios and interesting complexities. This description of pleasing sound would form a basis for the discussions that followed. Consider the work of Marin Mersenne, known for his contributions to music and mathematics. In 1636, he published his studies on acoustics in the book Harmonie universelle . His idea was that consonance, or “sweetness,” is determined theoretically by the simplest ratio . 32 Consequently, the unison is the sweetest and most agreeable sound. However, Mersenne knew that the most simplistic ratio idea is not universally true in practice. As Roger Grant pointed out, “In this scheme, the natural seventh should be more consonant than the fourth compound octave (16:1), which again contradicts conventional knowledge and musical experience.” 33 To compensate for this difference between theory and reality, Mersenne made a second stipulation that “the most agreeable 30 Larry M. Jorgensen, “Descartes on Music: Between the Ancients and the Aestheticians,” British Journal of Aesthetics 52, no. 4 (October 2012): 409, doi : 10.1093/aesthj/ays041 . 31 Ibid. 32 Roger Mathew Grant, “Ad infinitum: Numbers and Series in Early Modern Music Theory,” Music Theory Spectrum no. 1 (2013): 67, doi : 10.1525/mts.2013.35.1.62 . 33 Ibid.