The Generalized Pythagorean Comma Harmonic Powers of a Fundamental Frequency Are Equivalent the Standing Wave Harmonic Fraction System

Abstract

Purpose: The Pythagorean Comma refers to an ancient Greek musical, mathematical tuning method that defines an integer ratio of exponential coupling constant harmonic law of two frequencies and a virtual frequency. A Comma represents a physical harmonic system that is readily observable and can be mathematically simulated. The virtual harmonic is essential and indirectly measurable. The Pythagorean Comma relates to two discrete frequencies but can be generalized to any including infinite harmonics of a fundamental frequency, vF. These power laws encode the physical and mathematical properties of their coupling constant ratio, natural resonance, the maximal resonance of the powers of the frequencies, wave interference, and the beat. The hypothesis is that the Pythagorean power fractions of a fundamental frequency, vF are structured by the same harmonic fraction system seen with standing waves.