Error and conjecture in music theory
Hans Kayser saw the tetractys mainly in the numbers 6, 8, 9 and 12, and considered the ten a simplification. The case against that view.
If the tetractys is granted any scientific meaning at all, it is usually in music theory. The leading voice of this view in the twentieth century was Hans Kayser (1891–1964), founder of modern Harmonik, a doctrine that sees a basic pattern of the world in the numerical ratios of tones.
Kayser’s discovery on Raphael’s tablet
On the small tablet in Raphael’s School of Athens Kayser found the numbers 6, 8, 9 and 12. They combine the arithmetic, geometric and harmonic proportions and contain every basic interval: octave 6 : 12, fifth 6 : 9 and 8 : 12, fourth 6 : 8 and 9 : 12, whole tone 8 : 9. In them he saw the true, “inner” meaning of the Pythagorean fourness.
The familiar form 1 + 2 + 3 + 4 = 10, on the other hand, he regarded as the version for the uninitiated. This is reported by Theo Reiser in his 1967 booklet Das Geheimnis der pythagoräischen Tetraktys (The secret of the Pythagorean tetractys). Reiser refers to a letter from Kayser written in 1963 and builds his own studies on this view. The number ten no longer appears in them.
The objection
There is a contradiction here. On the same tablet where Kayser found his proportion 6 : 8 : 9 : 12, Raphael painted, directly below it, the triangle with the number X: the tetractys with its sum of ten. Anyone who reads meaning into the tablet cannot take one part seriously and pass over the other just because it has no harmonic explanation.
Then there is the evidence of the ancient sources: the Pythagorean Philolaus explicitly praises the power of the decad (see The ancient sources). A tetractys conceived that comprehensively cannot be exhausted by the study of intervals. It ought, so the thesis goes, to have something to say even where number theory is most puzzling: in the distribution of the primes.
The interface
The musical relationships themselves are not in question. They can be followed and heard (try them under Number and sound). But there is one place where Kayser’s harmonics and the geometry of star polygons meet: the lambdoma. The same table that orders all tonal ratios in harmonics orders all divisibilities in number theory, and the figures of the simplexes can be fanned out into it. More under Division table and The lambdoma.