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The number 24 (=1×2×3×4)*
is at the centre of the fourth face of the Tetrahedral Lambda. The diagram displays the lattice of tone
ratios, starting with 1, the fundamental, that are created by dividing every number in the tetractys and outside it
by 24. Using any other number in or outside the tetractys as divisor would have created the same lattice of
tone ratios. Overtones are shown in yellow circles, red lines connect octaves (×2), green lines connect perfect
fourths (×4/3) and blue lines
connect perfect fifths (×3/2). The tone interval of 9/8 is also indicated by the orange line joining
the centre of the tetractys (coloured grey) to one corner. The tone ratios 27/16 of note A and 243/128 of note B
are similarly defined by, respectively, indigo and violet diagonals extending from the number 1 to corners of
larger triangles. Successive notes of the scale for each octave are joined by dashed lines. They zigzag between an octave, the seventh note of the octave and its perfect
fourth, i.e., between the extremities of the Pythagorean scale and its
midpoint.
* The number 6, the centre of the Lambda tetractys, is the fourth overtone and 24 is the tenth overtone. Integers 6, 8, 12 & 24 at the centres of the four faces have the ratios 1, 3/2, 4/3, 2, 3 & 4 expressed by the Pythagorean integers 1, 2, 3 & 4.
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