Mean free path
In molecular physics, mean free path is the mean distance a molecule, atom, or other gas unit can travel before collision with other particles. It determines the largest frequency that can possibly propagate in a gas. Any larger frequencies will not propagate, owing to the fact that molecules will not transfer the pressure wave.
In air at STP, mean free path is about 68 nm. Together with the sound speed of 340 m/s, the maximum physically possible frequency is about 5 GHz.
Definition
Mean free path is defined by
- [math]\ell = \frac{\mu}{p} \sqrt{\frac{\pi k_\text{B} T}{2 m}},[/math]
or alternately,
- [math]\ell = \frac{\mu}{p} \sqrt{\frac{\pi RT}{2 M}},[/math]
where:
- [math]\ell[/math] is the mean free path,
- [math]m[/math] is the mass of the particle,
- [math]M[/math] is the molar mass of the substance,
- [math]\mu[/math] is the gas's viscosity,
- [math]R[/math] is the universal gas constant,
- kB is the Boltzmann constant,
- T is the temperature,
- P is the pressure.
Analysis with respect to equal-step tunings
There exists a point in frequency which marks the transition between rhythm and pitch. The beat rate of an interval is their frequency difference. If the beat rate is lower than the the said lowest frequency, beats are distinct, and the interval sounds like a "wahwah" instead of a true sound pair. It can be argued, by such criteria, that the smallest interval useful as a pitch material has the same beat rate as the lowest frequency.
If an interval whose base frequency is played at the mean free path limit, and beats at the lowest frequency, given as 16 Hz for human beings[citation needed], the interval would have a size of about 5.5 × 10-6 cents, corresponding to a step of about 200-million-edo. Playing in a tuning system which makes a distinction for intervals this small is physically impossible without beating interference on all notes.
However, higher-pitched sounds will get heavily attenuated the closer their wavelength gets to free path. When the sounds attain levels of just several MHz, some estimate they will fail to travel through the air longer than a few centimeters[citation needed].
In small-atom solids such as dense metals, the sound can travel at much higher frequencies due to both higher sound speed and the periodicity and strength of the lattice (→ Wikipedia: Speed of sound). Theoretically, an alien civilization with their "ears" fit for hearing through solids rather than gases would be able to make music with these higher frequencies, and perceive intervals as distinct instead of "wahwah".