In the third picture, however, the points where the curve crosses the axis come in two regularly spaced families. And as the pictures on the facing page indicate, for any curve like Sin[x] + Sin[α x] the relative arrangements of these crossing points turn out to be related to the output of a generalized substitution system in which the rule at each step is obtained from a term in the continued fraction representation of (α – 1)/(α + 1).

When α is a square root, then as discussed in the previous section, the continued fraction representation is purely repetitive,

Curves obtained by adding together various sine functions. In the first two cases, the curves are ultimately repetitive; in the second two cases they are not. If viewed as waveforms for sounds, then these curves correspond to chords. The first curve yields a perfect fifth, while the third curve yields a diminished fifth (or tritone) in an equal temperament scale.