4 ms·

> The modern piano mitigates this by having the hammers strike strings at a position that avoids exciting the 5th harmonic This is fascinating! I always wonder

> The modern piano mitigates this by having the hammers strike strings at a position that avoids exciting the 5th harmonic This is fascinating! I always wondered why a piano could work so well despite in reality having slightly ill-tuned thirds. Thanks for sharing the tidbit.

yesenadam 6y ago
But isn't everything in equal temperament "ill-tuned"? The fifths aren't real (3/2x frequency) fifths, etc. Everything except octaves - but piano tuners tell me that octaves that are "too big" (>2x) sound better![0] [0] https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and_inharmonicity https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and...
- vnorilo 6y ago
  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 6y 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 6y 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 6y ago
        In the case of the piano, it is definitely stiffness.
  - yesenadam 6y ago
    > a rough stuttering Well, I think of and hear a major third as the loveliest interval on a piano, not "rough stuttering"!
- elihu 6y ago
  Yeah, pianos deal with both the tuning inaccuracies that are inherent to 12-tone equal temperament in addition to piano-specific oddities like having to stretch the octave. Basically, the harmonics that rise off of piano strings aren't exact multiples of the fundamental -- they're a little bit off, because piano strings don't behave entirely like ideal strings, they behave like metal cylinders. In the mid-range they're pretty pretty close to plain equal temperament, but in the high treble the strings get shorted but the string gauge stays almost the same, which means the ratio of diameter to length increases and they act less like strings and more like cylinders. The bass has similar issues with single and double wound strings. So, the fix is to just stretch the octave enough so that the harmonics of low notes line up better with the fundamentals of higher notes, and so on. (What we perceive as "out of tune-ness" is the wobbly sound of two frequencies played together that almost but don't quite line up, creating a beat frequency.)
weinzierl 6y ago
Piano tuning is a science in its own. Google for Railsback Curve if you want to fall down a rabbit hole.

yesenadam 6y ago

But isn't everything in equal temperament "ill-tuned"? The fifths aren't real (3/2x frequency) fifths, etc. Everything except octaves - but piano tuners tell me that octaves that are "too big" (>2x) sound better![0] [0] https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and_inharmonicity https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_and...

vnorilo 6y ago
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 6y 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 6y 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 6y ago
      In the case of the piano, it is definitely stiffness.
- yesenadam 6y ago
  > a rough stuttering Well, I think of and hear a major third as the loveliest interval on a piano, not "rough stuttering"!
elihu 6y ago
Yeah, pianos deal with both the tuning inaccuracies that are inherent to 12-tone equal temperament in addition to piano-specific oddities like having to stretch the octave. Basically, the harmonics that rise off of piano strings aren't exact multiples of the fundamental -- they're a little bit off, because piano strings don't behave entirely like ideal strings, they behave like metal cylinders. In the mid-range they're pretty pretty close to plain equal temperament, but in the high treble the strings get shorted but the string gauge stays almost the same, which means the ratio of diameter to length increases and they act less like strings and more like cylinders. The bass has similar issues with single and double wound strings. So, the fix is to just stretch the octave enough so that the harmonics of low notes line up better with the fundamentals of higher notes, and so on. (What we perceive as "out of tune-ness" is the wobbly sound of two frequencies played together that almost but don't quite line up, creating a beat frequency.)

weinzierl 6y ago

Piano tuning is a science in its own. Google for Railsback Curve if you want to fall down a rabbit hole.