increasingly higher tones in the harmonic series are reached. This problem is mitigated by "stretching" the octaves as one tunes above (and to an extent below) the temperament region. When octaves are stretched, they are tuned, not to the lowest coincidental overtone (second partial) of the note below, but to a higher one (often the 4th partial). This widens all intervals equally, thereby maintaining intervallic and tonal consistency. All western music, but western classical literature in particular, requires this deviation from the theoretical equal temperament because the music is rarely played within a single octave. A pianist constantly plays notes spread over three and four octaves, so it is critical that the mid and upper range of the treble be stretched to conform to the inharmonic overtones of the lower registers. Since the stretch of octaves is perceived and not measured, the tuner is aware of which octave needs "more" or "less" stretching. A good tuning requires compromise between tonal brilliance, intonation and an awareness of gradation of tone through the compass of the instrument. The amount of stretching necessary to achieve this is a function of string scaling, a complex determination based on the string's tension, length, and diameter. The Railsback curve is the result of measuring the fundamental frequencies of stretched tunings and plotting their deviations from unstretched equal temperament. In small pianos the inharmonicity is so extreme that establishing a stretch based on a triple octave makes the single octaves beat noticeably, and the wide, fast beating intervals in the upper treble—especially major 17ths—beat wildly. By necessity the tuner will attempt to limit the stretch. In large pianos like concert grands, less inharmonicity allows for a more complete string stretch without negatively affecting close octaves and other intervals. So while it may be true that the smaller piano receives a greater stretch relative to the fundamental pitch, only the concert grand's octaves can be fully widened so that triple octaves are beatless. This contributes to the response, brilliance and "singing" quality that concert grands offer. A benefit of stretching octaves is the correction of dissonance that equal temperament imparts to the perfect fifth. Without octave stretching, the slow, nearly imperceptible beating of fifths in the temperament region (ranging from little more than one beat every two seconds to about one per second) would double each ascending octave. At the top of the keyboard, then, the theoretically (and