The error arising in any just intonation, due to the fact that the octave is incompatible with the simple frequency ratios of the intervals of the diatonic scale. If ƒ is the frequency of the tonic C, the first sixth produces A, with a frequency of ƒ. In going to D, an interval of a fourth is required, and this is a frequency ratio of , so its frequency will be of A, which is of ƒ, or ƒ, which is ƒ. The descending fifth gives G, at a frequency of ƒ = ƒ. The last fifth results in the tonic, C, with a frquency of ƒ = ƒ. This discrepancy is called the syntonic comma, and is equal to about one-fourth of a half-step. It results in the fact that, after the above simple five-chord progression, the tonic is no longer at the same frequency at which it started. Intervals of major thirds are not commensurate with a perfect fifth, the difference being the syntonic comma. The following integer equation must, therefore, be false for all integers:
where X, Y, n, and m are integers, and ƒ is the frequency of the tonic. The left-hand side represents successive steps of musical intervals, and the right-hand side represents octave transpositions. It can be shown that this equation can never be satisfied. See also the diatonic comma.