Grade 7: Late Antique

The two thirds -- pythagorean (81/64) and just (5/4 = 80/64) -- may be approached intuitively as 5/4 is deemed sweeter. Then the pythagorean comma (81/80) is an arithmetically derived corrective factor. These two major thirds then suggest two full-tones (fat and thin) and two half-tones (thin semitome and fat hemitone).

Alternatively, one may follow up the ascending cycle of fifths, and compare 12 fifths up with 7 octave jumps down. See [Tame, p. 248].