# Cello pressed at half length higher than harmonic?

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)

• “because pressing on the string bends it” – you've answered your own question. Though I'm not sure whether this is actually the main effect. Commented Jul 19, 2021 at 8:41

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.

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.)

• "shorter wavelengths require more deformation at the same amplitude": there isn't any reason to think that the shorter wavelengths have the same amplitude, is there? Commented Jul 19, 2021 at 12:15
• @phoog indeed. The actual point is that the proportionality factor of amplitude to deformation is higher for the shorter wavelengths. That's what causes the frequencies to go higher than integer-multiple, regardless of the concrete amplitude. Commented Jul 19, 2021 at 12:34
• Why do you say it doesn't matter for guitar? Many styles of guitar playing use the equivalent of that technique to play harmonics. Commented Jul 19, 2021 at 16:55
• @DaveTweed on guitar, a flageolett harmonic has the same frequency as the corresponding overtone in the open-string tone. Thus “harmonic” describes both accurately. On bowed strings, this is not the case. Commented Jul 19, 2021 at 18:33

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.

• Oh! We were talking about a different "effectively"! I had intended to confirm that there was only one before I posted my comment, but then I forgot to do it. I was referring to the sentence "when that same string is pressed against the fingerboard, ... it's effectively under more tension." I had to pause and ask myself "why 'effectively'?" I would just say that when the string is pressed down, it's under more tension. "Effectively split" is less problematic, because you're not actually dividing the string when you touch it lightly. Commented Jul 19, 2021 at 14:10
• @phoog - I have effectively edited it... (I hope).
– Tim
Commented Jul 19, 2021 at 14:47
• I like it. Thanks for humoring my nitpicking. :-) Commented Jul 19, 2021 at 15:41

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.

• And this is more relevant to a cello (or violin, viola, etc.) since they're fretless, and you can move your finger slightly up to compensate. On a guitar, it's the fret rather than the finger which stops the string, so it stays a single point of contact even if you press it down. There's still variation possible by moving the string sideways (thus stretching it more) but finger thickness no longer factors in. Commented Jul 20, 2021 at 15:41
• As the question was for cello, it's totally relevant to the issue and something I had overlooked. Thanks! Commented Jul 21, 2021 at 16:01