The tone ratio of every note in the Pythagorean scale is simultaneously the arithmetic, harmonic & geometric mean of the six notes that surround it in the hexagonal lattice of tone ratios created by the fourth face of the Tetrahedral Lambda. Pairs of blue arrows connect the arithmetic mean of two tone ratios, pairs of green arrows connect the harmonic mean of two tone ratios and pairs of red arrows connect the geometric mean of two tone ratios. This beautiful symmetry is absent from the Table of Nichomachus, the traditional, mathematical scheme known to historians of music. This is because it is based upon the first face of the Tetrahedral Lambda (the musical basis of Plato's cosmology), which is necessarily incomplete because it is but the first stage of generation of this more mathematically complete object. Only with the fourth and last face is the mathematical harmony of the Pythagorean scale finally revealed in a symmetric way. The Tetrahedral Lambda, however, has far more than musical significance. This pattern of 20 integers is the arithmetic counterpart of sacred geometries, as this website will reveal. It means that the mathematical relationships between the musical notes is written into sacred geometries as well. This has to be the case because sound and space are two mediums that the Divine Artist uses to express the same, universal archetypes.

 

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