The Geometry of sound in time
Musical Frequency Ratios as Geometric Forms
Music and geometry are deeply connected through frequency ratios. An octave (e.g., C to the next C) has a 2:1 frequency ratio, meaning the higher note’s frequency is twice the lower. A perfect fifth (e.g., C to G) has a 3:2 ratio. These ratios can be visualized as geometric shapes, where the octave forms a complete cycle (like a circle or spiral), and the perfect fifth is a proportional segment within that cycle. Below are three views of this geometry.
1. Circle: Proportional Angles
The octave (2:1) is a full circle (360°), and the perfect fifth (3:2) is an angle of ~216°, derived from log₂(3/2) × 360°.
2. Spiral: Layered Growth
The octave forms a full spiral loop (2:1 ratio doubles the radius), and the perfect fifth marks a point at the 3:2 ratio along the spiral’s arc.
3. Line: Ratio Segments
The octave (2:1) and perfect fifth (3:2) are shown as line lengths, directly reflecting their frequency ratios as geometric proportions.