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 C...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.展开更多
We find that π represents dual attributes. One is within the purely mathematical domain and can be derived for example, from infinite series, among several other methods. The other is within a 2D geometric-physical d...We find that π represents dual attributes. One is within the purely mathematical domain and can be derived for example, from infinite series, among several other methods. The other is within a 2D geometric-physical domain. This paper analyzes several physical constants from an analogous perspective where they are defined solely by mathematical and 2D geometric properties independent of any actual physical scaling data. The constants are evaluated as natural unit frequency equivalents of the neutron, electron, Bohr radius, Rydberg constant, Planck’s constant, Planck time, a Black hole with a Schwarzschild radius, the distance light travels in one time unit;and the fine structure constant. These constants are defined within two inter-related harmonic domains. In the linear domain, the ratios of the frequency equivalents of the Rydberg constant, Bohr radius, electron;and the fine structure constant are related to products of 2 and π. In the power law domain, their partial harmonic fraction powers, and the integer fraction powers of the fundamental frequency for Planck time are known. All of the constants are then derived at the point where a single fundamental frequency simultaneously fulfills both domains independent of any direct physical scale data. The derived values relative errors from the known values range from 10-3 to 10-1 supporting the concept and method.展开更多
Quantum gravity and the transformation of a neutron star or the merger of two neutron stars into a black hole are important topics in cosmology. According to the Schwarzschild radius relationship, a black hole arises ...Quantum gravity and the transformation of a neutron star or the merger of two neutron stars into a black hole are important topics in cosmology. According to the Schwarzschild radius relationship, a black hole arises when two times of the gravitational binding energy of the gravitational system, GBE, equal the annihilation energy of its total mass. From a quantum perspective, the integer number of neutrons defines the GBE and mass in the merger of binary pure neutron stars transforming to a black hole. Therefore, one can scale all gravitational binding energy relationships by using neutron mass, energy, distance, time, or frequency equivalents. We define ?of the neutron as the binding energy, 1.4188 × 10󔼹 J, of a virtual system of two neutrons separated by the neutron Compton wavelength. The??divided by a neutron’s rest mass energy represents a fundamental, dimensionless proportionality constant, 9.4252 × 10󔼰, . The square root of , αG, which we introduce here as a coupling constant, is identical in concept to the fine structure constant found in electromagnetic physics, but for gravity. Both αG and ?inter-relate the neutron, proton, electron, Bohr radius, speed of light, Planck’s constant, GBE of the electron in hydrogen, and Planck time. This paper demonstrates a direct conceptual and computational rationale of why the neutron and its negative beta decay quantum products accurately can represent a quantum gravitational natural unit system.展开更多
文摘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.
文摘We find that π represents dual attributes. One is within the purely mathematical domain and can be derived for example, from infinite series, among several other methods. The other is within a 2D geometric-physical domain. This paper analyzes several physical constants from an analogous perspective where they are defined solely by mathematical and 2D geometric properties independent of any actual physical scaling data. The constants are evaluated as natural unit frequency equivalents of the neutron, electron, Bohr radius, Rydberg constant, Planck’s constant, Planck time, a Black hole with a Schwarzschild radius, the distance light travels in one time unit;and the fine structure constant. These constants are defined within two inter-related harmonic domains. In the linear domain, the ratios of the frequency equivalents of the Rydberg constant, Bohr radius, electron;and the fine structure constant are related to products of 2 and π. In the power law domain, their partial harmonic fraction powers, and the integer fraction powers of the fundamental frequency for Planck time are known. All of the constants are then derived at the point where a single fundamental frequency simultaneously fulfills both domains independent of any direct physical scale data. The derived values relative errors from the known values range from 10-3 to 10-1 supporting the concept and method.
文摘Quantum gravity and the transformation of a neutron star or the merger of two neutron stars into a black hole are important topics in cosmology. According to the Schwarzschild radius relationship, a black hole arises when two times of the gravitational binding energy of the gravitational system, GBE, equal the annihilation energy of its total mass. From a quantum perspective, the integer number of neutrons defines the GBE and mass in the merger of binary pure neutron stars transforming to a black hole. Therefore, one can scale all gravitational binding energy relationships by using neutron mass, energy, distance, time, or frequency equivalents. We define ?of the neutron as the binding energy, 1.4188 × 10󔼹 J, of a virtual system of two neutrons separated by the neutron Compton wavelength. The??divided by a neutron’s rest mass energy represents a fundamental, dimensionless proportionality constant, 9.4252 × 10󔼰, . The square root of , αG, which we introduce here as a coupling constant, is identical in concept to the fine structure constant found in electromagnetic physics, but for gravity. Both αG and ?inter-relate the neutron, proton, electron, Bohr radius, speed of light, Planck’s constant, GBE of the electron in hydrogen, and Planck time. This paper demonstrates a direct conceptual and computational rationale of why the neutron and its negative beta decay quantum products accurately can represent a quantum gravitational natural unit system.
