Cello pressed at half length higher than harmonic?

Question

I was playing around with harmonics on my e-cello, and noticed that when pressing at the first one (half-length), the pitch was higher than when merely touching for the harmonics itself. I thought it was something to do with my bridge set up and only managed to break the A string in the process of adjusting it (was old anyway and the whole set needed changing). Then I tried on my acoustic cello, and noticed the same.

So my question is: is there something quite wrong with my set-up, and I should bring even the acoustic cello to a luthier for adjustment, or is natural because pressing on the string bends it and so the pitch has to be higher than when merely touching it lightly? (only logical explanation that I could come up with)

Thanks in advance!

Answer 1

This happens on all stringed instruments. There are two reasons for that.

• As you already noticed yourself, pressing down the string does require bending it a little, i.e. stretching, which increases the tension. At least with steel strings, this is enough to audibly sharpen the pitch.
• Though an idealised model of the string is as a perfectly flexible, purely tensile, one-dimensional waveguide, in reality it does of course have a certain stiffness to itself. That's the reason even the flageolett harmonics are a little bit sharp compared to integer multiples of the fundamental, because the shorter wavelengths require more deformation at the same amplitude, thus the stiffness has more of an influence as a restoring force. The effect, called inharmonicity, is most studied for piano; for bowed strings it actually doesn't matter as much because the bow action creates a phase-locked loop, which forces every tone by itself to more or less exactly periodic, i.e. to have integer harmonics.
  But the stiffness also means that both the bridge and the nut or finger-stop aren't perfect vibration nodes, but add some Neumann boundary stiffness as well. In case of a flageolett note, this does not happen at the internal nodes because the vibration at both sides is in opposite phase. So effectively, this contribution is twice as strong for a fingered octave note as it is for the same note as a harmonic.

For thin steel strings and high-action instruments, the first effect probably dominates. For low-action instruments with nylon or gut strings, the second one might be stronger.

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Note on terminology: I write flageolett when I mean an “extracted harmonic” as you get by lightly touching the string at a vibration node. Contrast with harmonic partials that ring along with the fundamental in an unstopped note. (For piano and guitar the distinction doesn't matter – but in bowed strings, only flageoletts are subject to inharmonicity, whereas harmonic partials are enforced to be integer multiples by the phase-locking effect.)

Answer 2

Put in simple terms, when a string is touched at exactly its half-way point along the length, and made to vibrate, the note's pitch is exactly one octave higher than the original, open string. The string is mechanically effectively split into two equal halves, and by touching there, it introduces a node - a point along the string which doesn't vibrate.

When that same string is pressed against the fingerboard, at exactly the same point, it's actually under more tension - only slightly, but still more. So the pitch then is higher. The more the string has to be pressed down due to a high action, the more stretched it becomes. So to have that point produce a harmonic and a stopped note the string would have to be virtually touching the fingerboard, so there's no stretching or tension involved. That woud mean the string wouldn't vibrate freely when played, so the whole thing is a little bit compromised.

Answer 3

One factor that nobody has mentioned is finger thickness. When you touch a string lightly to produce a harmonic, there is a single point of contact; but when you press the string onto the fingerboard, keeping your finger in the same position on the string, the length of string that is free to vibrate is reduced by half the width of your finger. This will produce a slightly higher note.