Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

3 Birkhoff’s Aesthetic Measure.

George David Birkhoff (1884-1944) who worked brilliantly on the three body problem, differential equations, general relativity among other areas and was an honorary member of the Mathematical Society of México, and who contributed also to the cultural development of México. In 1924 he retakes some ideas (see also Birkhoff, 1929, 1932, 1933, 1945) he had several years before, but which he did not develop because he had to focus on pure mathematics studies. (Sounds familiar?) He thought that the melody depended on the order of the notes as heard by the ear. He thought that certain order relations could be established, kept by the notes, so to choose the best melodies. For him, the Fundamental Problem of Aesthetics was to determine, for a class of objects, the specific characteristics on which the aesthetic value depends.

Birkhoff considers that there are three consecutive phases for the aesthetic experience: first, a preliminary effort of attention, which is necessary to perceive an object and that is proportional to the complexity $C$ of the object; second, a pleasurable sensation or aesthetic measure $M$ which rewards the preliminary effort; and third, a certification that the object possesses a harmony, symmetry or order $O$ which seems to be a necessary condition, if not sufficient, for the aesthetic experience. So, Birkhoff proposes the formula $M = O/C$ with which he expresses the aesthetic measure as the effect of the density of the order relations compared to complexity. He inquires the daring of the formula and provides certain historical justifications. Aesthetics deal with aesthetic pleasure and with the objects that produce it. So, we have classes of objects that can be compared with respect to their aesthetic value (objects in different classes can not be compared). Then, the fundamental problem of analytic aesthetics is to determine the aesthetic factors and their relative importance. To perceive an aesthetic object requires certain adjustments, and the effort sensation or tension that goes with it appears as the sum of all tensions to the diverse automatic adjustments. So, if $A, B,C,...$ represent the adjustments, each with tensions $a,b,c,...,$ and if these are realized $r,s,t,...$ times, he considers the sum $C = ra+ sb+ tc+ ...$ as the complexity.

From another side, to the order $O$ correspond certain associations that intervene in the act of perception. For example, symmetry would be an association. If $L,M, N, ...$ are associations of various kinds, each one with index of sensation $l,m,n,...$ which happens $u,v,w, ...$ times, then we can consider the total of sensations (positive or negative) $O = ul+ vm + wn + ...$ as the order of the object. Hence, the intuitive estimation of the quantity of order $O$ inherent to the aesthetic object, compared to its complexity, yields us its aesthetic measure. Obviously, this mathematical theory can only apply to objects whose aesthetic factors are essentially mathematical or formal. There are other factors that are beyond this theory, as for example, the associations about the signification of a beautiful poem. Birkhoff enumerates how some thinkers have perceived the presence of mathematical elements in art. It is interesting to see what they thought.

In contrast to the hedonistic, mystic or moralist theories, the analytic theory concentrates in providing a quantitative solution to the fundamental problem mentioned above. It would seem that aesthetics, if it has to be considered scientific, has