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

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Abstract: Purpose: The Pythagorean Commarefers to an ancient Greek musical, mathematical tuning method that defines aninteger ratio of exponential coupling constant harmonic law of two frequenciesand a virtual frequency. A Comma represents a physical harmonic system that isreadily observable and can be mathematically simulated. The virtual harmonic isessential and indirectly measurable. The Pythagorean Comma relates to twodiscrete frequencies but can be generalized to any including infinite harmonicsof a fundamental frequency, vF.These power laws encode the physical and mathematical properties of theircoupling constant ratio, natural resonance, the maximal resonance of the powersof the frequencies, wave interference, and the beat. The hypothesis is that thePythagorean power fractions of a fundamental frequency, vF are structured by the same harmonic fraction systemseen with standing waves. Methods: The Pythagorean Comma refers to theratio of (3 2)12 and 27 that is nearly equal to 1. AComma is related to the physical setting of the maximum resonance of the powersof two frequencies. The powers and the virtual frequency are derived simulatingthe physical environment utilizing the Buckingham Π theorem, array analysis,and dimensional analysis. The powers and the virtual frequency can begeneralized to any two frequencies. The maximum resonance occurs when theirdimensionless ratio closest to 1 and the virtual harmonic closest to 1 Hz. ThePythagorean possible power arrays for a vF system or any two different frequencies are evaluated. Results: Thegeneralized Pythagorean harmonic power law for any two different frequenciescoupling constant are derived with a form of an infinite number of powersdefining a constant power ratio and a single virtual harmonic frequency. Thispower system has periodic and fractal properties. The Pythagorean power lawalso encodes the ratio of logs of the frequencies. These must equal or nearlyequal the power ratio. When all of the harmonics are powers of a vF the Pythagorean powers aredefined by a consecutive integer series structured in the identical form asstandard harmonic fractions. The ratio of the powers is rational, and all ofthe virtual harmonics are 1 Hz. Conclusion: The Pythagorean Comma powerlaw method can be generalized. This is a new isomorphic wave perspective that encompassesall harmonic systems, but with an infinite number of possible powers. It isimportant since there is new information: powers, power ratio, and a virtualfrequency. The Pythagorean relationships are different, yet an isomorphicperspective where the powers demonstrate harmonic patterns. The couplingconstants of a vF Pythagorean power law system are related to the vFs raised to the harmonic fraction series whichaccounts for the parallel organization to the standing wave system. This newperspective accurately defines an alternate valid physical harmonic system.

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