Lie Brackets and Harmony of Spheres
that $[X;Y] = 0$ if $Y$ is invariant under the action given by the flow of $X$. From the opposition condition, $[X;Y] = -[Y;X]$, it immediately follows that $[X;Y] = 0$ if and only if $[Y;X] = 0$ and therefore we have that $Y$ is invariant under the action given by the flow of $X$ if and only if $X$ is invariant under the action given by the flow of $Y$. In the case where $Y$ is invariant under the action given by the flow of $X$ it is therefore possible to know the action of the virtuality $Y$ in the final state $q$ thanks to the sole knowledge of the action of the virtuality $Y$ in the initial state $p$ and the law of the flow of $X$. In symbols $d_p L_t(Y(p)) = Y(q)$, where $q = X(t;p) = L_t(p)$. We believe it is not difficult to discern in these relations the ultimate reasons for certain predictive capacities, which lose their aura of mystery once traced back to the geometric nature of a domain of Existence.
The Harmony of the Spheres: The Circle, Harmonics and Fourier Series (Part 1) (Solis)
One of the strengths of contemporary mathematics is its capacity for abstraction, which often allows us to identify the essence hidden behind multiple apparently distinct phenomena. In this article we will analyze through the tools of modern mathematics an aspect of a classical theme such as that of the harmony of the spheres. The theme, in itself very vast, would deserve a monograph to be properly developed from a historical, mathematical and esoteric point of view. Reserving a more organic analysis for future publications, we want here to at least highlight the relationships that exist between the Circle, musical harmonics and their application to the harmony of the spheres. We do not consider ourselves here to present anything particularly original, but rather something that every scholar of the liberal arts should know, deeply understand and internalize. At a certain point in each person's initiatic development, it becomes evident how the arts of the Quadrivium (Mathematics, Geometry, Music and Astronomy) are nothing other than different ways of studying the same identical metaphysical reality: the Number. Indeed, Arithmetic is the study of mutual relations between Numbers; Geometry is the study of Number in space; Music is the study of Number in movement, that is in time, and Astronomy, or better initiatic Astrology, is the Geometry of causes, that is the study of the causality of Number.
Musical Harmonics and Regular Polygons
Natural harmonics in music, regular polygons in geometry, the harmonic succession in mathematical analysis, the cyclotomic equation in algebra are all symbolically equivalent concepts that express the natural progression and propagation that proceeds through the resonance of refined elements. When a note is played, for example by setting a string on a piano into oscillation, this oscillation, amplified by the resonance chamber of the instrument, is transmitted through the air and reaches the listener's ear which translates the mechanical oscillation into an electrical signal. This, subsequently, will be translated into an auditory perception dependent on the oscillation frequency of the string, will produce psychic activity and eventually also spiritual activity. For practical purposes a sound is never pure, but is constituted by an amalgam of accessory sounds that are more acute and less intense. These accessory sounds are called harmonics and are very important in determining the body, clarity or harshness of the resulting sound. Natural harmonics are those accessory sounds that are naturally produced by a string instrument or a brass instrument, as well as by the human voice. The characteristic of these harmonics is that they have a frequency exactly multiple of the fundamental frequency. Their formation can be easily understood by analyzing the dynamics of an oscillating string constrained at both ends. When a violin string fixed at both ends is set into vibration by the pinch of a finger, in addition to the oscillation...