Syntonic comma

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.

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