Essay

What is the relationship between the Golden Ratio and the idea of beauty we have been attempting to explore in music and mathematics? In The Geometry of Art and Life, Mattel Ghyka, before providing a subtle analysis of the Divine Proportion and other ratios, described the nature of ratio itself. Ratio, he noted, is different from proportion. It means that a comparison is being made. It is "the quantitative comparison between two things or aggregates belonging to the same kind or species." In other words, a ratio is an internal comparison within a given universe. It involves a mapping between two objects in that universe and a number. We can have a ratio between lines (measuring length) or spaces (measuring area); we can have a ratio between dangers (measuring probabilities of disaster); we can even have a ratio between sounds (measuring frequencies). These worlds of lines and dangers and sounds are mapped into the world of numbers. A ratio, then, requires a "space" of objects, and it requires a measure that we use to comprehend these objects.

Thus, establishing a ratio is already a sophisticated act of the mind. It requires comparison between one object and another, a measurement of their relationship, and an assessment of their differences and similarities. We can go further: it is no accident that creating a ratio is nearly the archetypal act of rationality. It is an act of reason that is not divisive but connective, showing relations between distinct objects.

Because creating a ratio means giving a number to these relations, it also allows us to compare different worlds, not just objects within a single world. We can identify ratios from different worlds through their numbers. The ratio of 2, for example, could apply to lengths of a line, populations in various countries, or the frequency of pitches on a piano. We can say that the ratio of the lengths of two lines is equivalent to the ratio of two frequencies of vibration, thus making a connection between the lengths of vibrating strings and pitch. When compared in this way ratio becomes something more profound; it is being treated as a proportion.

A proportion can show the similarities between different worlds; it can demonstrate similarities and relationships that are shared; in a sense, what we have been attempting to find in our explorations of music and mathematics are shared "proportions." The proportion, rhetorically, is an analogy, which has its roots in the word analogia, originally used to refer to proportion. As Ghyka pointed out, first the Greeks and Romans and then the Gothic and Renaissance builders used repeated proportions - analogies - in their architecture. A ratio between dimensions was never accidental and always echoed other ratios throughout the plan. Structures contained internal resonances, echoes, and repetitions. The term symmetry meant much more than simple mirror identity between two parts; it referred to repeated proportions.

Vitruvius, the great Roman theoretician of architecture, wrote, "Symmetry resides in the correlation by measurement between the various elements of the plan, and between each of these elements and the whole." Vitruvius compared the symmetry of building with the symmetry of the human body: "It proceeds from proportion - the proportion which the Greeks called analogia - [it achieves] consonance between every part and the whole." Proportion, he explained, "is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard." A building built with proportion and symmetry in mind is generated by that standard. Vitruvius defined the aesthetic importance of such proportion: "When every important part of the building is thus conveniently set in proportion by the right correlation between height and width, between width and depth, and when all these parts have also their place in the total symmetry of the building, we obtain eurythmy."

This notion of symmetry and eurythmy is close to what we call, in other contexts, harmony. The Divine Proportion is a powerful example of eurythmy because of the clear relationship it displays between the whole and the parts. It may even have been a Pythagorean belief that the proportion extended from the human body (which, as the Renaissance artists and the ancients knew, tends to be divided into "divine" proportions by the navel) to the universe itself (which was supposed to have a structure like one of the five regular solids, the dodecahedron, itself suffused with divine proportions, like its two- dimensional relation, the pentagon). In later centuries, the analogies between parts and other parts, and between parts and the whole, became the foundation of mystical beliefs upon which the notion of magic depends. In magic, one object becomes an analogy for another (like a voodoo doll for the human body); act on part of one object, and the magic acts similarly on the other. "Metaphors for us, but not for Him" is how, we recall, one of the Jewish mystics explained the manner in which similarities and comparisons in the earthly realm are mirrors of divine equivalencies. Proportion, in these beliefs, can become a magical identification of parts and influence.

There have been many attempts to describe or identify the beauty of different types of eurythmy. Ghyka and other writers have made a distinction between ratios based on irrational numbers like or and ratios of rational numbers like 2 or 3. The rational ratios, while ostensibly simpler and more elegant, offer fewer possibilities for internal echoes and analogies. The irrational ratios offer greater complexity on the surface but more profound similarities below. Even the Greeks understood that those irrational relations created more dynamic shapes, with greater interaction between parts and whole. Because the irrational proportions are not easily divisible into simple compartments, because their ratios imply an infinite sequence of relations, the eye and the mind are pulled to their geometric representation (which suggests that beauty may sometimes require a more profound simplicity than one presented on the surface).

It is possible to analyze a form to determine what "types" of proportions and ratios are displayed within it. One scholar established a typology of architectural forms in ancient and Renaissance architecture, an attempt to articulate basic "themes" of relationship and proportion that underlie much of Western art. It resembles the attempt by Heinrich Schenker to find the fundamental melodic line underlying Western tonal music, those archetypal descents 3-2-1. These themes are based on properties of geometric ratios - ways in which rectangles or other shapes can be divided, ways in which parts relate to wholes.