sharper (never flatter), the Railsback curve—which is functionally the integral of the inharmonicity at an octave—is monotonically increasing. A piano is tuned beginning in the center, so the Railsback curve has a shallow slope in this area. But as the piano tuner stretches octaves to compensate for inharmonicity, the stretch accumulates as tuned notes ascend and descend, so the curve becomes more pronounced at the ends. Inharmonicity in a string is caused primarily by stiffness. Decreased length and increased thickness both contribute to that stiffness. For the middle to high part of the piano range, string thickness slowly decreases as string length quickly decreases, contributing to greater inharmonicity in the higher notes. For the low range, string thickness drastically increases—especially in shorter pianos, which must accomplish the lower pitches by using heavier strings rather than longer ones—producing greater inharmonicity in this range as well. In the bass register, a second factor affecting inharmonicity is the resonance caused by the acoustic impedance of the piano sound board. These resonances exhibit positive feedback on the inharmonic effect: if a string vibrates at a frequency just below that of a soundboard resonance, the impedance makes it vibrate even lower, and if it vibrates just above a resonance, the impedance makes it vibrate higher. The sounding board has multiple resonant frequencies that are unique to any particular piano. This contributes to the greater variance in the empirically measured Railsback curve in the lower octaves. The actual tuning is not a smooth curve, but a jagged line with peaks and troughs. It has been suggested with Monte-Carlo simulation that such a shape comes from the way humans match pitch intervals. Multiple strings All but the lowest notes of a piano have multiple strings tuned to the same frequency. This allows the piano to have a loud attack with a fast decay but a long sustain in the attack-decay-sustain-release (ADSR) system. The three strings create a coupled oscillator with three normal modes (with two polarizations each). Since the strings are only weakly coupled, the normal modes have imperceptibly different frequencies. But they transfer their vibrational energy to the sounding board at significantly different rates. The normal mode in which the three strings oscillate together is most efficient at transferring energy since all three strings pull in the same direction at the same time. It sounds loud, but decays quickly. This normal mode is responsible for the