Many ideas in the world are superficial constructions created to simplify and avoid the complexities of life. What's more, some of these ideas go largely unexplored and unquestioned. A concept in music known as the circle of fifths (COF) could qualify as one of these ideas. Indeed, a number of musical theorists elegantly simplify music into this single circle, and it has become a foundational symbol of most modern music and theory. While the COF has been useful to a degree, like most complexity-simplifying concepts, might it stifle deeper understanding?
First off, what is the circle of fifths? It is a repeated cycle of one of the most, or if not the most, important musical intervals known as the "perfect fifth." Musicians throughout history have analyzed where, and how, fifths shed light on music. In modern music theory, when the fifth of every tone is taken consecutively, it circles back around to the same tone. At least, it seems that way. Curiously, it takes twelve successive fifths to complete a circle. Hence, the result is the twelve-tone scale found in most modern music. This, in theory, makes music more understandable and practical, because any musical pattern or song can then be taken through only twelve permutations (keys) before returning to square one.
While the COF makes sense conceptually, the subject in question is this: does a consecutive series of fifths actually make a perfect circle? Surprisingly, mathematics shows otherwise. A true fifth originates from the third harmonic of a tone, which is measured at 702 cents. If a cycle of twelve true fifths is taken when measured at 702 cents, it very nearly shuts into a circle. The key word in the last immediate sentence is: nearly. When fully calculated, it is off from a perfect circle by a mere 23.46 cents, an interval that baffled Greek musicians over two thousand years ago. This seemingly minute difference is so important that the infamous circle symbol associated with the fifth completely depends on it.
Consequently, if the math doesn't hold up to support the circle of fifths theory, then what symbol might best represent the concept of a series of fifths? If a closer look into nature is taken, one eventually finds a perfect fit: the spiral. A spiral always remains open, as do successive fifths. A spiral can expand or contract, as can successive fifths. A spiral can theoretically extend indefinitely, as can successive fifths. The parallels are clear.
Symbols are powerful due to humanity living largely visual-based lives. With a simple change of musical imagery, the very nature of music becomes dramatically more mysterious and complex. When the spiral starts to symbolize the musical fifth, society may begin to visualize music in its true, boundless form. Indeed, the spiral can lead musicians away from circles, and straight into the unknown, spiraling as deep as the imagination allows.
Spiral of Fifths