The lambdoma – harmonics and the tetractys
The table of all tonal ratios. In harmonics it orders the intervals, in number theory the divisibilities, and in it the two readings of the tetractys meet.
The lambdoma is a coordinate system tilted by 45°. Both axes carry the numbers from 1 to infinity, and each cell holds the fraction of numerator and denominator. The triangular shape to which it owes its name (after the Greek letter Λ) is meant to make its inner symmetry visible.
The basic idea comes from Plato’s Timaeus: there the world soul is ordered by two series of numbers, 1, 2, 4, 8 and 1, 3, 9, 27, which spread apart like the two arms of a lambda. In the twentieth century Hans Kayser made it the central table of his harmonics.
The lambdoma up to 8 × 8. Each cell plays the tone 220 Hz × numerator/denominator. Blue: octave, fifth and fourth. Gold: the diagonal on which every ratio equals 1.
Two tones at right angles
Musically, the lambdoma crosses two series of tones: the row gives the overtones of a fundamental, the column its undertones. Cells with the same tone lie on straight lines through the corner (1/2, 2/4, 3/6 …). This produces the symmetry that harmonicists read as the blueprint of the world.
The interface with number theory
The same table is the division table of number theory. Colour its cells by divisibility and you see the “matrix code” of the four start figures:
The lambdoma coloured by divisibility. Yellow: divisible. Green: common factor, not divisible. Red: coprime, a new ratio.
Swap numerator and denominator, and the two halves of the table swap their reciprocals as well. Only one thing changes: the cells “divisible” (yellow) become cells “common factor, not divisible” (green), and vice versa. The symmetry is preserved.
Yellow and green cells show fractions that repeat. Red cells show new ratios. In a lambdoma thought of as infinite, their density is 6/π².
The lambdoma is therefore the bridge between the two readings of the tetractys: the musical one after Kayser, with the ratios 1 : 2, 2 : 3, 3 : 4, and the geometric–number-theoretical one with the sum of ten (see Error and conjecture in music theory). In the lambdoma it should be possible to show that both mean the same thing, as Philolaus describes it in his writing on the decad.