92 Chapter 6
a quantity like 8 is comparable to integers, still “ the mind is in this case constantly
perplexed ” by its irrationality. Music should rely on arithmetic rather than geometric ratios
because “ the most agreeable to the soul is neither that which is perceived most easily nor
that which is perceived with the greatest difficulty; it is that which does not quite gratify
the natural desire by which the senses are carried to the objects, yet is not so complicated
that it tires the senses. ” Because “ variety is in all things most pleasing, ” music relies on
variation.^11
Where Plato assumed the priority of arithmetic, Descartes allows different mathematical
objects to generate various pleasures. Though he began his innovative world-project by
recasting the ancient theory of music, as Cohen observed, his little treatise is perhaps the
most conservative such work of its century.^12 His Compendium combines the ancient topics
of mathematical music with the nascent mathematical physics: the numerical patterns of
music theory lead directly to the perceptible, observable world. He begins with rhythm,
showing how relative simplicity of ratio governs our experience, how easily we hear two
even notes against one, whereas five notes against one are “ almost impossible to sing. ”
Exactly similar processes govern our awareness of the ordering of a melody as we count
up its phrases, until “ our imagination proceeds to the end, when the whole melody is finally
understood as the sum of many equal parts. ”^13 Each beat physically makes us “ dance and
sway, ” accompanying “ each beat of the music by a corresponding motion of our body. ”
The downbeat of a measure, when “ the sound is emitted more strongly and clearly ... has
a greater impact on our spirits ” by sheer visceral force, so that “ even animals can dance
to rhythm if they are taught and trained, for it takes only a physical stimulus to achieve
this reaction. ”^14
Music arouses various affects precisely through various meters: “ A slower pace arouses
in us quieter feelings such as languor, sadness, fear, pride, etc. A faster pace arouses faster
emotions such as joy, etc. ”^15 Triple meters (like 3/4) “ occupy the senses more ” than duple
ones (like 2/4), “ since there are more things to be noticed in them, ” namely three rhythmic
units, rather than two. Even a solitary military drum can affect us by its beat, demonstrat-
ing the felt reality of pure rhythm. By treating “ number or time in sound ” before pitchFigure 6.2
Descartes ’ s comparison of arithmetic and geometric proportions from his Compendium : “ The example of a
proportion of lines [2:3:4] is more easily distinguished by the eyes than this one [2: 8 :4]. ”