inharmonicity. The highest strings must be under the greatest tension, yet must also be thin enough to allow for a low mass density. The limited strength of steel (i.e. a too-thin string will break under the tension) forces the piano designer to use very short and slightly thicker strings, whose short wavelengths thus generate inharmonicity. The natural inharmonicity of a piano is used by the tuner to make slight adjustments in the tuning of a piano. The tuner stretches the notes, slightly sharpening the high notes and flatting the low notes to make overtones of lower notes have the same frequency as the fundamentals of higher notes. See also Piano wire, Piano tuning, Psychoacoustics. The Railsback curve The Railsback curve, first measured by O.L. Railsback, expresses the difference between normal piano tuning and an equal-tempered scale (one in which the frequencies of successive notes are related by a constant ratio, equal to the twelfth root of two). For any given note on the piano, the deviation between the normal pitch of that note and its equal-tempered pitch is given in cents (hundredths of a semitone). As the Railsback curve shows, octaves are normally stretched on a well-tuned piano. That is, the high notes are higher, and the low notes lower, than they are in an equal-tempered scale. Railsback discovered that pianos were typically tuned in this manner not because of a lack of precision, but because of inharmonicity in the strings. Ideally, the overtone series of a note consists of frequencies that are integer multiples of the note's fundamental frequency. Inharmonicity as present in piano strings makes successive overtones higher than they "should" be. To tune an octave, a piano technician must reduce the speed of beating between the first overtone of a lower note and a higher note until it disappears. Because of inharmonicity, this first overtone is sharper than a harmonic octave (which has the ratio of 2/1), making either the lower note flatter, or the higher note sharper, depending on which one is tuned relative to the other. To produce octaves that reflect the temperament and accommodate the inharmonicity of the instrument, the technician begins the stretch from the middle of the piano so that, as the stretch accumulates from register to register, it results in the desired stretch at the top and bottom of the instrument. Shape of the curve Because string inharmonicity only makes harmonics