3 ms·

Yes, but while fifths are off by 2/100, major seconds by 4/100, thirds are by 14/100 or 16/100. The former create gentle swirling interference, the latter a ro

Yes, but while fifths are off by 2/100, major seconds by 4/100, thirds are by 14/100 or 16/100. The former create gentle swirling interference, the latter a rough stuttering. If you hit a fifth on a piano and listen carefully, you can hear the slow cycle in the sound. The stretched octaves are due to high string tension causing some inharmonicity, ie. sharpening higher harmonics.

HPsquared 5y ago
The inharmonicity is caused by the strings not being 'ideal' mathematical strings: they have stiffness, due to having some thickness - and require some additional force to bend them back and forth, beyond that of the string tension. This bending stiffness is more evident in higher harmonics than in the fundamental (tighter radius of curvature), manifesting in increased 'apparent stiffness' and therefore a higher frequency, at the higher harmonics, compared to the lower harmonics. Not sure if I've explained it well, but vibrations of an ideal string (no bending stiffness) can be described by a second-order partial differential equation [0], whereas a real string with nonzero bending stiffness is actually more of a 'vibrating beam' problem, which is a fourth-order PDE [1]. EDIT: this is why low notes have greater inharmonicity (thicker strings). [0] https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.pdf https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.... [1] http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation.pdf http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation...
- salty_biscuits 5y ago
  I don't think bending modes are the usual culprit, it is more often the frequency mismatch between axial and transverse modes that cause inharmonic tones. I'd need to get a real piano and make some spectra and do some maths to show it though.
  - vnorilo 5y ago
    In the case of the piano, it is definitely stiffness.
yesenadam 5y ago
> a rough stuttering Well, I think of and hear a major third as the loveliest interval on a piano, not "rough stuttering"!

HPsquared 5y ago

The inharmonicity is caused by the strings not being 'ideal' mathematical strings: they have stiffness, due to having some thickness - and require some additional force to bend them back and forth, beyond that of the string tension. This bending stiffness is more evident in higher harmonics than in the fundamental (tighter radius of curvature), manifesting in increased 'apparent stiffness' and therefore a higher frequency, at the higher harmonics, compared to the lower harmonics. Not sure if I've explained it well, but vibrations of an ideal string (no bending stiffness) can be described by a second-order partial differential equation [0], whereas a real string with nonzero bending stiffness is actually more of a 'vibrating beam' problem, which is a fourth-order PDE [1]. EDIT: this is why low notes have greater inharmonicity (thicker strings). [0] https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.pdf https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/string.... [1] http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation.pdf http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20Equation...

salty_biscuits 5y ago
I don't think bending modes are the usual culprit, it is more often the frequency mismatch between axial and transverse modes that cause inharmonic tones. I'd need to get a real piano and make some spectra and do some maths to show it though.
- vnorilo 5y ago
  In the case of the piano, it is definitely stiffness.

yesenadam 5y ago

> a rough stuttering Well, I think of and hear a major third as the loveliest interval on a piano, not "rough stuttering"!