Advanced Musical Geometry

Advanced Musical Geometry: Fractals and Phi

Musical ratios like the 2:1 octave and 3:2 perfect fifth can extend into advanced geometric forms, revealing connections to fractals and mathematical constants like phi (the golden ratio, ~1.618). These visualizations explore recursive patterns, spiral growth, and harmonic spacing to uncover these relationships.

1. Fractal Circle: Recursive Perfect Fifths

The 3:2 perfect fifth ratio (~216° on a circle) is recursively subdivided, creating a fractal-like pattern of smaller intervals, showing how musical ratios can form self-similar structures.

2. Golden Spiral: Phi and Octaves

A spiral grows by the golden ratio (phi, 1.618) per turn, compared to an octave spiral (2:1). Musical intervals like the perfect fifth (3:2) are placed on both, highlighting phi’s subtle influence in harmonic spacing.

3. Harmonic Series: Geometric Progression

The harmonic series (1:1, 2:1, 3:2, 4:3, etc.) is plotted as a geometric progression. We look for phi-like ratios (e.g., consecutive harmonics) to see if anomalies emerge in the spacing.