The Lambdoma
The fundamental harmonic diagram, which Thimus called the “Lambdoma”
because it is drawn in the shape of a Greek lambda (Λ), and to which I
gave the designation “partial-tone coordinates” because it is best for
us to notate it in the familiar form of a coordinate grid, is (like all
harmonic diagrams) not simply a mathematical-logical scheme that could
be drawn with any other symbols. The form of these tone-number groups
is also justified optically through the geometric-arithmetical
arrangement of their contents, as is shown by the construction of the
monochord, on which every tone-value in the diagram can be realized by
means of the “equal-tone lines.” In pure numeric terms, this is a
division canon that rationally subdivides any linear unit.
Regarding the tonal content, these partial-tone coordinates provide,
for the first time in the history of acoustics and music theory, a system of tones,
which science has lacked before now. Since this system is based on the
natural law of the overtone series, and its group-theoretical form also
appears to be based in nature in other ways-and since, on the other hand, all the tone-values in it correspond to forms present in our psyche, as the “monochord test”
demonstrates-this fundamental harmonic diagram contains one of those
rare coincidences of the natural and the psychic, of matter and spirit,
which promises to impart completely different findings from those of a
merely logical (mathematical) formulation, or from a merely
psychological analysis of inner forms and experiences.
“Theology in the form of mathematical figures is taught by Plato, the
Pythagorean scriptures, and Philolaus,” says Proclus in his Commentary
on Euclid.
With this we return once again to Pythagoreanism itself. It is my firm
belief that the only way to orient oneself in the great maze of
Pythagorean traditions and their “authentic” and “inauthentic” theorems
is to starts with the central idea of Pythagoreanism-namely with the
tone-number, and the geometric-tonal configurations developing from it.
I made a first attempt at this orientation in my essay on Pythagoras,
and even in those initial harmonic analyses showed that many of the
previously “obscurest” Pythagorean theorems could be illuminated with
comparative ease; indeed, a few previously considered “foreign” and
“un-Greek,” such as the doctrine of the transmigration of souls, can be
derived directly from the Lambdoma. The reader will find various such
analyses mentioned in this book.
Pythagoreanism had an enormous influence in all of antiquity with its double concept of number and
harmony. This effect spread both into the most problematic individual
phenomena and into the great universal akróatic concepts, as we have
already learned (the harmony of the spheres, for example). Pythagoras
taught in two ways. With one section of his students, the mathematikoi, he approached problems discursively, working through proofs; with the others, the akusmatikoi (= hearers), he taught symbolically by means of concise, easily remembered epigrams (akusmata). Of the many akusmata that remained in circulation throughout antiquity, an especially tenacious one has been preserved in different variations. Reduced to a common denominator, its translation reads: “Number is the wisest of things, and the name of things the next wisest.”