Stretch

Sampled notes

What is octave stretch?

Real piano strings are stiff, so their partials are inharmonic: the nth partial sits a little sharp of n×f₀, following fₙ = n·f₀·√(1 + B·n²), where B is the inharmonicity coefficient.

When tuning an octave by ear, a tuner matches a partial of the lower note to a partial of the upper note (e.g. the 2nd partial of the lower note against the 1st of the upper for a 2:1 octave). Because both partials are stretched sharp, the upper note must be tuned slightly sharp of its theoretical equal-tempered pitch. Tuning outward from A4 accumulates this: the treble ends up progressively sharp and the bass progressively flat — the classic Railsback curve.

This is a model. A single representative B never matches a real instrument, where B varies note-by-note (and grows toward the extremes). Wider partial matches (4:2, 6:3) probe stiffer, higher partials and yield more stretch. Toggle “B grows at the ends” for a rough sense of real-piano behavior.