Golden Music, Golden Ratio, Divine Ratio

Golden Ratio is often used in Music

An idea often under emphasized and under appreciated in music theory and composition is the golden ratio. Sometimes also called the divine ratio, it is mathematically defined as a relationship between two numbers a and b such that (a+b)/a = a/b. It is also sometimes algebraically expressed as x2-x-1=0. Although it has no rational solutions, the approximations are 1.618:1 or 1:0.618. However it is written, it roughly translates to numbers like 5 and 3, 8 and 5 and so forth. If you are mathematically inclined, you may already know or deduce that the entire Fibonacci Series can be derived by multiplying each previous member of the series by increasingly more accurate approximations of this ratio (i.e. 5×1.618~=8 and 8×1.618~=13…). In fact, the sequence of the ratios between consecutive Fibonacci numbers converges to the actual ratio (symbolized by the Greek letter phi).

So how does this natural phenomenon relate to music? Besides the basics, like 13 keys in a piano octave, 8 white and 5 black, or 8 tones in a scale where the 5th and the 3rd form the foundation of major chords, there are numerous examples in the music we love. Early classical music (perhaps until Bach or even later) was basically all about cadences which simply traversed to the dominant and then resolved back to the tonic. Although that quickly became incredibly boring, the good pieces usually put the dominant right at phi. In other words, if the cadence was 8 measures long, the dominant was reached in the 5th measure and the resolution to the tonic took 3 measures.

Soon thereafter, others took this idea to new heights. Latter classical music boasts a variety of examples of clever usages of phi and the Fibonacci Series. Debussy, Bartok, Schubert and many others have used this phenomenon very successfully on much larger scales. Modern musicians and composers also utilize the golden ratio and the Fibonacci series. Tool’s Lateralus is by far the best example of such usage, as the song is literally filled with Fibonacci (lyrics, rhythm, melody, meaning, etc.). Others include Pink Floyd, Bob Dylan, Billy Joel, possibly Frank Sinatra and more. Sometimes the usage is intentional, sometimes incidental (especially when it pertains to complex rhythms like 5/4, which get naturally subdivided along the golden ratio (2+3 or 3+2) or even more interestingly, composite rhythms where the beat changes among measures, but taken in larger scopes conforms to some more abstract system). No matter how it presents itself, the golden mean inevitably makes the music more appealing to the listeners. So, the next time you think about writing a piece, some lyrics, or even a drum section, consider the divine ratio and how it could make your creation better.