Tuning by Stretching
The tuner adjusts the widths (thus “stretching”) of the octaves across the keyboard by optimizing the coincidence of the partials which are most likely to be heard as poor tuning becuase they would lead to audible beating or oscillations due to interference. The table below shows the overtone frequencies for selected notes on a Steinway B piano (tuned with Verituner). The number of overtones is much larger for the lower notes. As we progress upward to C8, the major contributions shift sincreasingly towards the fundamental, with fewer overtones. For A5 and A6 the dominant partials are the fundamental and the first overtone, while for A7 only the fundamental is present to any significant extent.
If you look closely at the values of the overtones of a single note, you will notice that the frequencies are slightly sharp (higher) than expected. For example, the second overtone of A0 would ideally be at 54.34 Hz, while the actually overtone is slightly higer at 54.4 Hz. The deviation is greater for each succesive overtone, i.e. the overtones become sharper, with f12 being at 333.1 rather than 325.44. This is inharmonicity, i.e. the overtones do not coincide with ideal harmonics, and the deviation is nonlinear. The degree of inharmonicity is dependent on the metal used for the strings.
To optimize the coincidence of the partials between A0 and A1, we might choose the 6th partial of A0 overlapping with the 3rd partial of A1, referred to as a 6:3 partial for A0. Another would be 10:5. These emphasize the higher overtones in A0 to define the stretch. Color coding of partial pairs (stretch points) indicate the location of stretch points used in this tuning.
The stretch is adjusted outwards from the center octave (typically A3-A4), sometimes referred to as the temperament octave. The notes below the temperament (the tenor and bass) are stretched down (flat) so that their partials will coincide with the fundamentals closer to the center of the keyboard; and the notes above the temperament are stretched upwards (sharp) so that the overtones of the notes in the middle range (alto) will coincide with the fundamentals of the higher (treble) notes. In this way, the temperament is trasferred acroos the piano using octaves stretched relative to the center.