The matrix arithmetic for ancient harmonic theory is presented here for two tuning systems : “Spiral fifths” as presented by Nicomachus, a Syrian Neo-Pythagorean of the second century A.D., and Plato’s “Just tuning” as reconstructed by the ethnomusicologist, Ernest McClain, from clues preserved by Nicomachus and Boethius (6th c. AD). These tables lie behind the system of architectural proportions used during the Renaissance, and their basic ratios now pervade modern theories of music. Calculation employs an early form of log table governed by vectors of 2- 3-4 in the first, and by 3-4-5 in the second. These systems govern 12-tone theory from the perspective of four primes-- 2, 3, and 5 generate all ratios under the overview of 7—as disciplined “self-limitation” within a “balance of perfect opposites. “