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This effect is known as inharmonicity, and it is important for precision piano tuning.

Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force.

The solutions are of the form $\sin(kx - \omega t)$, where $\omega = kv$. Applying fixed boundary conditions, the allowed values of the wavenumber $k$ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. This predicts evenly spaced harmonics.

However, piano strings are made of thick wire. If you bend a thick wire, there's an extra restoring force in addition to the wire's tension, because the inside of the bend is compressed while the outside is stretched. One can show that this modifies the wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} - A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^2}.$$ Upon taking a Fourier transform, we have the nonlinear dispersion relation $$\omega = kv \sqrt{1 + (A/v^2)k^2}$$ which 'stretches' evenly spaced values of $k$ into nonuniformly spaced values of $\omega$. Higher harmonics are further apart. We can write this equation in terms of the harmonic frequencies $f_n$ as $$f_n \propto n \sqrt{1+Bn^2}$$ which should yield a good fit to your data.

Some remarks about this result:

• As you noted earlier, the frequencies don't change as the sound decays. This is because our modified wave equation is still linear in $y$, though the dispersion relation is not.
• The above effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. This produces stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect.

This effect is known as inharmonicity, and it is important for precision piano tuning.

Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force.

The solutions are of the form $\sin(kx - \omega t)$, where $\omega = kv$. Applying fixed boundary conditions, the allowed values of the wavenumber $k$ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. This predicts evenly spaced harmonics.

However, piano strings are made of thick wire. If you bend a thick wire, there's an extra restoring force in addition to the wire's tension, because the inside of the bend is compressed while the outside is stretched. One can show that this modifies the wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} - A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^2}.$$ Upon taking a Fourier transform, we have the nonlinear dispersion relation $$\omega = kv \sqrt{1 + (A/v^2)k^2}$$ which "stretches" evenly spaced values of $k$ into nonuniformly spaced values of $\omega$. Higher harmonics are further apart. We can write this equation in terms of the harmonic frequencies $f_n$ as $$f_n \propto n \sqrt{1+Bn^2}$$ which should yield a good fit to your data. Note that the frequencies have no dependence on the amplitude, as you noted, and this is because our modified wave equation is still linear in $y$.

This effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. This results in stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect. That is, a piano whose fundamental frequencies really were tuned to simple ratios would sound out of tune!

This effect is known as inharmonicity, and it is important for precision piano tuning.

Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force.

The solutions are of the form $\sin(kx - \omega t)$, where $\omega = kv$. Applying fixed boundary conditions, the allowed values of the wavenumber $k$ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. This predicts evenly spaced harmonics.

However, piano strings are made of thick wire. If you bend a thick wire, there's an extra restoring force in addition to the wire's tension, because the inside of the bend is compressed while the outside is stretched. One can show that this modifies the wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} + A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^2}.$$ Upon taking a Fourier transform, we have the nonlinear dispersion relation $$\omega = kv \sqrt{1 + (A/v^2)k^2}$$ which 'stretches' evenly spaced values of $k$ into nonuniformly spaced values of $\omega$. Higher harmonics are further apart. We can write this equation in terms of the harmonic frequencies $f_n$ as $$f_n \propto n \sqrt{1+Bn^2}$$ which should yield a good fit to your data.

Some remarks about this result:

• As you noted earlier, the frequencies don't change as the sound decays. This is because our modified wave equation is still linear in $y$, though the dispersion relation is not.
• The above effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. This produces stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect.

This effect is known as inharmonicity, and it is important for precision piano tuning.

Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force.

The solutions are of the form $\sin(kx - \omega t)$, where $\omega = kv$. Applying fixed boundary conditions, the allowed values of the wavenumber $k$ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. This predicts evenly spaced harmonics.

However, piano strings are made of thick wire. If you bend a thick wire, there's an extra restoring force in addition to the wire's tension, because the inside of the bend is compressed while the outside is stretched. One can show that this modifies the wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} - A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^2}.$$ Upon taking a Fourier transform, we have the nonlinear dispersion relation $$\omega = kv \sqrt{1 + (A/v^2)k^2}$$ which 'stretches' evenly spaced values of $k$ into nonuniformly spaced values of $\omega$. Higher harmonics are further apart. We can write this equation in terms of the harmonic frequencies $f_n$ as $$f_n \propto n \sqrt{1+Bn^2}$$ which should yield a good fit to your data.

Some remarks about this result:

• As you noted earlier, the frequencies don't change as the sound decays. This is because our modified wave equation is still linear in $y$, though the dispersion relation is not.
• The above effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. This produces stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect.

This effect is known as inharmonicity, and it is important for precision piano tuning.

Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force.

The solutions are of the form $\sin(kx - \omega t)$, where $\omega = kv$. Applying fixed boundary conditions, the allowed values of the wavenumber $k$ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. This predicts evenly spaced harmonics.

However, piano strings are made of thick wire. If you bend a thick wire, you'll see there's an extra restoring force in addition to the tension, because the inside of the bend is compressed while the outside of the bend is stretched. One can show that this modifies the wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} + A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^2}.$$ Upon taking a Fourier transform, we have the nonlinear dispersion relation $$\omega = kv \sqrt{1 + (A/v^2)k^2}$$ which 'stretches' evenly spaced values of $k$ into nonuniformly spaced values of $\omega$. Higher harmonics are further apart. We can write this equation in terms of the harmonic frequencies $f_n$ as $$f_n \propto n \sqrt{1+Bn^2}$$ which should yield a good fit to your data.

Some remarks about this result:

• As you noted earlier, the frequencies don't change as the sound decays. This is because our modified wave equation is still linear in $y$, though the dispersion relation is not.
• The above effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. This produces stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect.

This effect is known as inharmonicity, and it is important for precision piano tuning.

Ideally, waves on a string satisfy the wave equation $$v^2 \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial t^2}.$$ The left-hand side is from the tension in the string acting as a restoring force.

The solutions are of the form $\sin(kx - \omega t)$, where $\omega = kv$. Applying fixed boundary conditions, the allowed values of the wavenumber $k$ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. This predicts evenly spaced harmonics.

However, piano strings are made of thick wire. If you bend a thick wire, there's an extra restoring force in addition to the wire's tension, because the inside of the bend is compressed while the outside is stretched. One can show that this modifies the wave equation to $$v^2 \frac{\partial^2 y}{\partial x^2} + A \frac{\partial^4 y}{\partial x^4} = \frac{\partial^2 y}{\partial t^2}.$$ Upon taking a Fourier transform, we have the nonlinear dispersion relation $$\omega = kv \sqrt{1 + (A/v^2)k^2}$$ which 'stretches' evenly spaced values of $k$ into nonuniformly spaced values of $\omega$. Higher harmonics are further apart. We can write this equation in terms of the harmonic frequencies $f_n$ as $$f_n \propto n \sqrt{1+Bn^2}$$ which should yield a good fit to your data.

Some remarks about this result:

• As you noted earlier, the frequencies don't change as the sound decays. This is because our modified wave equation is still linear in $y$, though the dispersion relation is not.
• The above effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. This produces stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect.