Consonance and dissonance

The perfect fifth has a frequency ratio of 3:2, and on a piano it is tuned about 2 cents narrower than that.

Stable and unstable

Play C and G together, then C and D together. The first pair sounds settled; the second seems to rub and want to move. Musicians call the stable kind consonant and the restless kind dissonant. Open Music Theory describes consonant intervals as ones that sound as if they do not need to resolve, and dissonant ones as sounding as if they do. Dissonance is not a mistake. It is the tension that makes music move, and much of the pleasure in a piece comes from tension resolving into rest.

The traditional classification

In the common-practice tradition taught in most theory courses, harmonic intervals fall into three groups. The perfect consonances are the unison, the octave and the perfect fifth. The imperfect consonances are major and minor thirds and sixths. The dissonances are seconds, sevenths, every augmented or diminished interval including the tritone, and, when sounded against the lowest note, the perfect fourth. That last one surprises people, since the fourth sounds fairly smooth. Its treatment comes from the rules of counterpoint, and Open Music Theory stresses that these categories have varied between periods and places.

Simple ratios

Why do some pairs blend better? A very old explanation points to frequency ratios. An octave is 2:1, a fifth 3:2, a fourth 4:3 and a major third 5:4. The simpler the ratio, the more often the two waves line up, and the smoother the result tends to sound. Tradition credits this discovery to Pythagoras, but historians point out that tuning by pure fifths is older and that the story about him is largely legend. Either way, the link between simple ratios and consonance has been a starting point for theorists for more than two thousand years.

Modern pianos are tuned in equal temperament, where every half step is the same size: 100 cents, with 1,200 cents to the octave. That makes every key equally usable, at a small price. A pure 3:2 fifth is about 702 cents, while the piano's fifth is exactly 700, about 2 cents narrow, which almost nobody notices. A pure 5:4 major third is about 386 cents, while the piano's is 400, about 14 cents wide, which trained ears can hear as a slight shimmer.

You can hear tension and release on the keyboard. Play B and F together, a tritone, and let the two notes move inward by a half step each: B rises to C and F falls to E. The restless tritone closes into the major third C to E, one of the most common resolutions in Western harmony, found at the end of countless hymns and classical pieces. Try it slowly, listening to the moment the sound settles.