The monochord and the tetractys
The Pythagoreans tested consonance on a single string with a movable bridge. Stopping the string at half its length sounds the octave, at two thirds the fifth, at three quarters the fourth. These ratios, 2:1, 3:2 and 4:3, are the ratios of neighbouring rows of the tetractys, the triangle of 1 + 2 + 3 + 4 = 10 points. The Pentametric Cycle produces exactly these ratios: the residue H picks rows H and H − 1.
The tetractys
With the silence read as 5 the triangle has a fifth row: 1 + 2 + 3 + 4 + 5 = 15 points. The historical tetractys stops at 4 (10 points); the fifth row is Lluis's reading, and it brings the one consonance the four rows cannot make, the pure major third 5:4 of just intonation (Zarlino's senario goes on to 6).
The sounding length
A string half as long vibrates twice as fast: the pitch ratio is the inverse of the length ratio. The tetractys has four rows, so neighbouring rows give exactly three ratios, 2:1, 3:2 and 4:3; with the unison 1:1 these are the four consonances of the cycle.
| H | rows | ratio | length | Greek |
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