Piano Tuning - Stretched Octaves

Stretched Octaves

The tuning described by the above beating plan will give a good approximation of equal temperament across the range of the temperament octave. If it were extended further, however, the actual tuning of the instrument would become increasingly inaccurate. This is due to a factor known as inharmonicity, which is present in different amounts in all piano strings. Strings' harmonic series do not fall exactly into whole-number multiples of their fundamental frequency; instead each harmonic runs slightly sharp, the sharpness increasing as 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 music, but classical literature in particular, requires this deviation from the theoretical equal temperament. It is because 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. Subtle compromise between tonal brilliance, intonation and an awareness of gradation of tone through the compass of the instrument, gives the perceptive craftsman the necessary understanding to enable the appropriate single, double,triple & quadruple octaves are reasonably beatless, always ensuring that his M10s and especially M17s (octave doublings and triplings of the M3) are not "wild". 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. Some pianos—and some tuners—achieve this better than others.

It is commonly said that the octaves of a small piano need to be stretched more than those of a large piano. But from the concert tuner's perspective it is the opposite. Because smaller pianos' inharmonicity is so extreme, establishing octaves based on a triple octave makes the single octaves beat noticeably, and the wide, fast beating intervals in the upper treble—especially M17s—beat wildly. Of a necessity the tuner limits his stretch to what he deems acceptable. Concert grands' lesser inharmonicity allows a 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 ability contributes mightily to the response, brilliance and "singing" quality that concert grands offer, and concert artists require.

Stretched Fifths: A serendipitous 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 (about one beat every two seconds) would double each ascending octave. At the top of the keyboard, then, the theoretically (and ideally) pure fifth would be beating as many as eight times per second. Modern ears easily tolerate fast beating in non-just intervals (seconds and sevenths, thirds and sixths), but not in perfect octaves or fifths. Happily for pianists, the string stretch that accommodates inharmonicity on a concert grand also nearly exactly mitigates the accumulation of dissonance in the perfect fifth.

Other factors, physical and psychoacoustic, affect the tuner's ability to achieve a temperament. Among physical factors are inharmonic effects due to soundboard resonance in the bass strings, poorly manufactured strings, or peculiarities that can cause "false beats" (false because they are unrelated to the manipulation of beats during tuning). The principal psychoacoustic factor is that the human ear tends to perceive the higher notes as being flat when compared to those in the midrange. Stretching the tuning to account for string inharmonicity is often not sufficient to overcome this phenomenon, so piano tuners may stretch the top octave or so of the piano even more.