# The Well-Tempered Tune, LLC

Friendly service to optimize the sound of your piano Inharmonicity Coefficients (B)

The values of B depend on the string length and diameter, and therefore B for each string varies significantly from A0 to C8. The range of values for various pianos is shown below, from A0 (27.5 Hz) to A6 (1760 Hz) and slightly beyond. An accurate measurement of the inharmonicity above C6 is difficult due to the absence of significant partials beyond the fundamental in these high notes. The cluster of values from 220 to 440 Hz covers the "temperament octave" from A3 to A4. The smooth curves through the points are defined by an arbitrary function that allows for interpolation. (See below for details.) Note that the Veritune measures the inharmonicity of each note individually. It does not assume an idealized smooth curve.

Steinway B |
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Kawai K-5 |

Baldwin F |
Kohler and Campbell KFM-700 |
Fitting of B vs frequency:

The curves through the B values for each piano were obtained by a nonlinear fit to the following equation: |
The equation was obtained empirically and is justified simply by the success of the fit. It allows estimation (interpolation) of the inharmonicity coefficient at any frequency x. For the Steinway B, the coefficients from the fit were:
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a = 5.22964 x 10^-6
b = 1.21012 x 10^-6
c = 8.3666 x 10^-10
d = -0.007927
e = 0.429601