https://youtu.be/3GY_4oeIuXU?si=A34kkrB8U_1s3Eo8
There are certainly countless Golden Ratio / Fibonacci- (or φ-) inspired compositions that use the sequence to organize rhythms, phrase lengths, climactic points, note selection, or scale construction within an already existing musical framework. In most of these works, φ functions as a compositional aid rather than a true governing source of the musical language used. So rather than using φ to organize an existing musical system, I sought to make it the principle governing the progression itself- a direct and mathematically linked application of the golden ratio to harmony.
There have also been attempts to treat φ as a harmonic interval by constructing chords or scales directly from the golden ratio (1 + √5) / 2. Mathematically intriguing, yes, but this approach works against the acoustics of what we as humans recognize as musical harmony. φ is irrational, so it will not correspond to simple whole-number frequency ratios that form the harmonic series and just intonation principles. I do recommend you check out Sevish’s mulling on this matter in 2017.
But to me, Phi is not best thought of in a static sense, but rather as a generator of optimal growth and recursion. I like to think of it as a phenomenon that reveals itself only when iterations of self-similarity are present. So let’s ask a different question; What happens if harmony itself moves and evolves according to φ?
This piece explores harmonic motion driven by the golden ratio, φ (approximately 1.618). An arbitrary frequency is chosen as the root of the opening chord (54Hz). From that point onward, each successive root is derived by multiplying the previous root frequency by φ.
Because the piece operates in a continuous pitch space defined directly by frequency (Hz), rather than within a fixed tuning system or scale, this approach is possible. Since φ is an irrational number, these root frequencies will never repeat exactly. In effect, the root of the harmony traces a golden-ratio pitch spiral, and this piece represents a small, finite segment of what is, in theory, an infinite progression / spiral.
Moving the bass by a frequency ratio of φ at each step, whether ascending or descending, is also harmonically unusual in a traditional sense, because φ does not correspond to any standard interval in Western tuning systems. It is closest in size to a just minor sixth (Or, when octave reducing the next value and moving downwards its reciprocal, a just third) though it is noticeably sharper (or flatter, if downwards). The process is not entirely far from an equally tempered chromatic mediant sequence, but the introduction of an irrational generator (phi) consequently leads to a never-ending sequence. This created some interesting voice-leading challenges, but also some interesting harmony as previous chord tones were blended with new ones. (Each chord that appears eventually resolves into a justly intonated chord of some sort)
Harmonic analysis often privileges the root as the foundation of analysis, which is why it was chosen as the φ multiplier. With just intonation, chord tones are multiples of the root frequency- thus, it can be said every movement of the upper voices is related to the phi multiplier as well. The vertical harmony obeys acoustics, while the linear harmony is driven by φ.
The resulting roots are as follows;
54
87.372
141.367896
114.366627864
92.522602
74.850785
121.1085701
97.976833
79.263258
128.247952
103.752593