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Inharmonicity

The inharmonicity of the piano string (wire) leads to a deviation of the frequencies of the overtones from perfect (integral) multiples of the fundamental, i.e. the partials do not coincide with the harmonics. Harmonics are integral multiples of the fundamental, thus the piano string is inharmonic. The discrepancy becomes larger with each overtone, so that the frequencies of the overtones are increasingly sharp relative to what is expected for an ideal harmonic oscillator. The following table shows the frequencies expected for a perfect harmonic oscillator with a fundamental at 440 Hz, i.e. the harmonics for A4 (calculated), along with the actual frequencies measured for a single string of A4 on a Steinway B piano.
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1
2
3
4
5
6
7
8
Measured
440.0
880.9
1323.9
1769.8
2219.7
2674.6
3135.4
3603.1
Calculated
440
880
1320
1760
2200
2460
3080
3520
Diff (cents)
0.0
1.8
5.1
9.6
15.4
22.5
30.9
40.4

The increasing sharpness of the overtones is perhaps best seen graphically (differences are in Hz):
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The nonlinear increase in frequencies of the overtones is very nicely described by the following equation:

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where n is the partial number (n = 1 for the fundamental) and B is the inharmonicity coefficient (see, for example, Rasch and Heetvelt (1985), and references therein). The following plot shows a nonlinear least squares fit of the dependence of the frequency of the overtones as a function of the partial number using the above equation to obtain the value of B (0.00075145) for A4.
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The ability of the above equation to describe the inharmonicity with a single coefficient (B) can be seen in a comparison of the measured overtone frequencies with those calculated from the equation with the fitted value for B for A3:
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1
2
3
4
5
6
7
8
Measured
219.79
439.83
660.00
880.66
1101.82
1323.99
1546.78
1770.32
Calculated
219.82
439.79
660.03
880.70
1101.93
1323.89
1546.65
1770.42
Difference
0.03
-0.04
0.03
0.04
0.11
-0.10
-0.13
0.10
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Inharmonicities vary from one end of the keyboard to the other

The inharmonicity depends on the string length and thickness (or more properly, the stiffness). The plot below shows the deviations of the overtones from ideal harmonics for A0 (black), A3 (green), and A4 (red). For this piano there is negligible inharmonicity in the bass, and increasing inharmonicity as we move up the keyboard (values from a Steinway B).
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The values of B obtained from fitting the original frequency data were:
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A0 0.000311692
A3 0.000213645
A4 0.00075145
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A characterization of the inharmonicity across the full range of the piano is essential for proper tuning using an electronic device. The variation of B across the keyboard for various pianos is shown on the B value page.


Inharmonicity varies with pitch on the same string

The inharmonicity can vary slightly for a single string as its pitch is changed. This can be a problem when tuning a piano which is far out of tune at the start. To see this, I show below the inharmonicity for a single A4 string as a function of its frequency.

Offset (in cents)
Fundamental
B value
0
439.955
0.000749032
-10
437.445
0.000758101
-20
434.805
0.000770484
-30
432.353
0.000775297
-40
429.717
0.000789170
-50
427.322
0.000801439


With a 50 cent starting offset: The B value changes from 0.000749032 to 0.000801439, which represents a change in the frequency of partial f4 from 1769.2 to 1769.9 Hz (or 0.7 cents), and for f6 from 2673.6 to 2675.9 (1.5 cents).

For a 20 cent deviation: The B value changes from 0.000749032 to 0.000770484, which changes f4 from 1769.2 to 1769.5 Hz (or 0.3 cents), and f6 from 2673.6 to 2674.6 (0.65 cents).