This post provides an illustration of the syntonic comma in musical tuning.
The audio example (view score) starts at C and ascends in a series of four perfect fifths, touching G, then D, then A, then E. From this high E, the example descends by a major third, hitting an instance of C two octaves above the starting note. This high C is restated, and then played together with the original C in the bass.
In equal temperament, the high C that we reach at the end of the progression forms a pure octave with the starting C — you’ll notice that the first clip ends without a trace of dissonance.
However, if we use pure fifths and thirds as opposed to tempered intervals, the progression doesn’t bring us to a pitch that matches our starting point. At the end of the second clip, you can hear how the starting and ending “versions” of C create a mistuned compound octave that flutters or beats slighlty. In some timbres, particularly where the sound has a bit of vibrato, the mistuning at the end is imperceptible, but I tried to use a plain sound that doesn’t mask the clash.
Here’s a way of thinking about the progression. A pure perfect fifth is slightly wider than a tempered fifth; obviously, traversing a wider fifth will take us to pitch that’s sharper than the one we would reach via a narrower tempered fifth. As we traverse four of these pure fifths we end up at an E that’s sharper than a tempered E, which is itself sharper than a justly tuned E. When we then descend by a just major third from our sharp “Pythagorean” E, we reach a C’ that’s in beautiful agreement with the E; but since that E is so sharp with respect to the original C, descending by a just major third cannot bring us all the way back home. We land at a C’ that’s 21.51 cents higher than the starting point.
Syntonic Comma — Equal Temperament:
Syntonic Comma — Pure Intervals: