**Tetraktys**

Four forms of complete arc diagrams with four nodes. In this experiment, the numbers assigned to the nodes are interpreted as (scaled) frequencies, the arcs as the corresponding intervals, so that larger numbers give higher pitches. In historical sources, however, the numbers usually refer to string lengths on a (virtual) monochord, the reciprocals of frequencies. Ratios of small positive integers were applied to general countable quantities.

Click the nodes or the labels of the arcs to make the sounds and their intervals audible! The clavichord sounds are tuned to a 53-EDO approximation of the Pythagorean tuning, so that semitones measure 4 (limma) or 5 (apothome) and whole tone 9 units. The halved octave with 26 and 27 instead of 26.5 EDO-units are about 11 cent (half a Pythagorean comma) out of tune.

Top left: classical tetraktys, defining the perfect consonances of the Pythagoreans.

Top right: halved tetraktys, as described by Nicole Oresme (14th c.).
The halved octave with 26 and 27 instead of 26.5 EDO-units are about 11 cent (half a Pythagorean comma) out of tune.

Bottom left: definition of the Pythagorean whole tone through perfect consonances.

Bottom right: zooming into the whole tone reveals a similar symmetric tetraktys on a microscopic pitch level.