Wave

Two notes, one number ratio

The article Why the fifth sounds hollow explains why consonance is not a matter of taste: two notes sound the more fused the simpler the ratio of their frequencies. The octave is 2:1, the perfect fifth 3:2, the fourth 4:3. At 3:2 the shared pattern of the two waves repeats after two and three vibrations respectively, a very short common period that the ear takes as one sound.

Beats

The same page describes an experiment with the guitar: tune the fifth and fourth strings (A and D) exactly, play them together, then turn the fourth string about five cents higher. The sound begins to pulse. These are beats: sometimes the wave crests reinforce each other, sometimes they cancel. For two notes that are almost the same pitch, the beat frequency is the difference of the two frequencies: a gap of two hertz pulses twice a second. A (110 Hz) and D (about 147 Hz), however, are a fourth apart. Here the partials that almost coincide beat: the fourth partial of the A (4 × 110 Hz = 440 Hz) and the third of the D (3 × 147 Hz ≈ 440 Hz). Five cents more on the D shift that partial by about 1.3 Hz, so you hear roughly one beat per second.

Organ builders and piano tuners tune by this. According to the article, when setting equal temperament one tunes to a given number of beats per second: an equal-tempered fifth beats slowly, about once in a second and a half, a third clearly faster. How the compromises of tuning arise from this is in the article Equal, meantone, just — why your piano is slightly out of tune.

Waveforms and tone colour

The page Notation symbols explains under “Audio design of the listening examples” how its tones are produced: they are synthesised directly in the browser with the Web Audio API. Soft dynamics and notes use sine waves, loud dynamics and accents sawtooth waves, articulation such as staccato triangle waves with a short envelope.

The table summarises this assignment. Sine is soft and round there, sawtooth richer in overtones and stronger.