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.
What is harmonics?
Harmonics (German Harmonik) understands itself as a holistic way of thinking and experiencing, based on the correspondence of sound, number and form, in the spirit of Pythagoras, Johannes Kepler and Hans Kayser. It rests on scientific standards but goes beyond them, taking in perception, experience and cultural exchange. This all-embracing view comes from a time when the sciences had not yet separated from religion. Today the scientist speaks of causal chains and chance, the philosopher of meaning and the theologian of creation and providence, three languages for one world. Harmonics seeks to bring them together again and shows, in the number scheme of the lambdoma, how mathematics, music, philosophy and theology interlock. In Germany the Harmonik-Zentrum Deutschland, among others, cultivates this tradition.
The diagram
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 later commentators drew spreading apart like the two arms of a lambda. In the 19th century Albert von Thimus extended the scheme into the full table and named it the lambdoma, and in the 20th 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. Teal: 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. The cells “divisible” (yellow) then become cells “common factor, not divisible” (teal), while the teal ones stay teal. What stays symmetric is the pattern of new (red) and repeated (yellow and teal) cells.
Yellow and teal 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.
Further reading
A book that covers the whole range of harmonic research and goes far beyond it is Willibald Limbrunner, Zahl Seele Kosmos. Wie die universelle Vierheit den Kosmos und unser Bewusstsein prägt (“Number, soul, cosmos: how the universal fourfold shapes the cosmos and our consciousness”, Synergia Verlag, in German). It treats the harmonic tetractys in the form 6 : 8 : 9 : 12, carefully derived, and follows its correspondences through history and science: Platonic solids and other symmetries, the structure of atoms compared with the overtone series, the intellectual line from the Pythagoreans via Platonists, Kabbalists, Rosicrucians and Freemasons to Kayser, with figures such as Robert Fludd, Kepler, Johann Valentin Andreae, Jakob Böhme, C. G. Jung and Carl Friedrich von Weizsäcker. Another focus is the fourfold in human consciousness, for instance in Jung’s fourfold mandala and in correspondences between hearing music and seeing colours. The book leads into molecular biology and particle physics and ends, consistently, in metaphysics.
It complements the arithmetic-geometric tetractys 1 + 2 + 3 + 4 = 10 that this site is about in an ideal way. Both areas of the tetractys belong together.
A vivid introduction to the basics is also given by the German video series Harmonikale Grundlagen und Lambdoma by the organ builder Henny Jahn, author of Weltformel Lambdoma. Using lambdoma diagrams she shows that this number matrix holds far more than a few notes, and expressly invites viewers to draw, check and feel everything for themselves.