TITLE:
The Generalized Pythagorean Comma Harmonic Powers of a Fundamental Frequency Are Equivalent the Standing Wave Harmonic Fraction System
AUTHORS:
Donald Chakeres
KEYWORDS:
Power Laws, Harmonic Systems, Standing Wave, Harmonic Fractions, Dimensional Analysis, Buckingham Pi Theorem, Pythagorean Comma
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.7,
July
19,
2018
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. Methods: The Pythagorean Comma refers to the
ratio of (3/2)12 and 27 that is nearly equal to 1. A
Comma is related to the physical setting of the maximum resonance of the powers
of two frequencies. The powers and the virtual frequency are derived simulating
the physical environment utilizing the Buckingham Π theorem, array analysis,
and dimensional analysis. The powers and the virtual frequency can be
generalized to any two frequencies. The maximum resonance occurs when their
dimensionless ratio closest to 1 and the virtual harmonic closest to 1 Hz. The
Pythagorean possible power arrays for a vF system or any two different frequencies are evaluated. Results: The
generalized Pythagorean harmonic power law for any two different frequencies
coupling constant are derived with a form of an infinite number of powers
defining a constant power ratio and a single virtual harmonic frequency. This
power system has periodic and fractal properties. The Pythagorean power law
also encodes the ratio of logs of the frequencies. These must equal or nearly
equal the power ratio. When all of the harmonics are powers of a vF the Pythagorean powers are
defined by a consecutive integer series structured in the identical form as
standard harmonic fractions. The ratio of the powers is rational, and all of
the virtual harmonics are 1 Hz. Conclusion: The Pythagorean Comma power
law method can be generalized. This is a new isomorphic wave perspective that encompasses
all harmonic systems, but with an infinite number of possible powers. It is
important since there is new information: powers, power ratio, and a virtual
frequency. The Pythagorean relationships are different, yet an isomorphic
perspective where the powers demonstrate harmonic patterns. The coupling
constants of a vF Pythagorean power law system are related to the vFs raised to the harmonic fraction series which
accounts for the parallel organization to the standing wave system. This new
perspective accurately defines an alternate valid physical harmonic system.