A piano is routinely tuned to less than one cent by ear. Each note is actually composed of three strings struck together that are tuned in unison. For the highest note (4186 Hz), one cent amounts to 2.4 Hz which would be the beat frequency if two of the strings were tuned that far apart and would sound awful.

Don

 

So, can you elaborate on exactly what you are trying to do so that we have the complete picture?

 

Measuring the inharmonicity of the partials.

Don

 

Measuring the inharmonicity of the partials.

Don

Nice. Very helpful indeed.

 

Nice. Very helpful indeed.

Maybe this is more helpful:

https://en.m.wikipedia.org/wiki/Inharmonicity

I still think my way would work.

 

Maybe this is more helpful:

https://en.m.wikipedia.org/wiki/Inharmonicity

I still think my way would work.

Since this is how you tune by ear, I’m inclined to agree. Beat the signal against a precise clock and look for where low frequency beats occur.

 

Hi,

There was some work done sometime in the 1940's so perhaps you can look for that work and try to follow it.

This is a complicated measuring problem. Strings are very non ideal and so present unique problems. The possible variations include not only the string or strings involved but also the instrument itself.

The variation of a single string is very interesting however. Even the very basic tone will change a little based on how hard it is struck or plucked. The first moment after activation the tone will be slightly higher, and then progressively diminish in frequency as it gradually comes to rest. This at least can be measured quite easily. One way is to measure the audio tone itself as picked up from an audio microphone, and another interesting way is to use an LED as a strobe. The LED can be controlled very precisely using a microcontroller. You can actually tune a stringed instrument this way. You may even be able to pick up harmonics this way too even if not related to the fundamental tone.
However, you must realize that the fundamental will change slightly as the string gradually comes to rest. That means you'll get a range of frequencies related to time as well where the time is the time after the strike until the string is within reason no longer heard.

The change in base frequency is due to the variation of the stretching of the string after being struck, and because of the other variations it may be more worthwhile to approach this problem from a purely theoretical standpoint. You could study the physics of a string in motion and given various factors and parasitics you could come up with some reasonable answers to the possible frequencies and their amplitudes and how they change over time. Would be very interesting.

 

The strobe idea is interesting. I have a lab strobe and have used it to visually see the string vibrate. Maybe I can use it to "freeze" a harmonic visually, and monitor the strobe frequency with a phototransistor and my counter. The highest notes would be hard to see, since their amplitude is almost microscopic, but this could work for the rest of the strings.

Don

 

The strobe idea is interesting. I have a lab strobe and have used it to visually see the string vibrate. Maybe I can use it to "freeze" a harmonic visually, and monitor the strobe frequency with a phototransistor and my counter. The highest notes would be hard to see, since their amplitude is almost microscopic, but this could work for the rest of the strings.

Don

This is similar to my original concept. Use a PLL to lock to the fundamental, and multiply by n for the desired overtone. Use the output to drive the strobe.

 

The problem with getting the accuracy you want is mathematically the same problem that gives rise to the Heisenberg uncertainty principle. Using the Fourier transform, you can get any accuracy you want, but more accuracy requires a longer sample. I agree with wayneh that you will need multiple samples averaged. And this is if the harmonic content was not changing with time, which it is, significantly, with a piano tone.

Bob

 

And this is if the harmonic content was not changing with time, which it is, significantly, with a piano tone.

You might be able to use the phenomenon described by @MrAl to determine the time axis you use to overlay multiple sample runs. You could construct a normalized time axis by using the frequency error versus the fundamental as a measure of time. The error would start high and slowly disappear as the amplitude of the vibrations fades.

But the more I think about this, the more I think you could end up chasing ghosts. The behavior of nearby strings and resonance of the instrument will also have an impact on the string vibrations. If you really want to study string vibration, you need a single string in an anechoic chamber.

 

The strobe idea is interesting. I have a lab strobe and have used it to visually see the string vibrate. Maybe I can use it to "freeze" a harmonic visually, and monitor the strobe frequency with a phototransistor and my counter. The highest notes would be hard to see, since their amplitude is almost microscopic, but this could work for the rest of the strings.

Don

Hi,

Yes the lower amplitude frequencies would be harder to see. But since this has been done in the past, i would recommend that you look for that work and see hwo they did it. Do a quick search on the web maybe something will turn up.

Using the strobe would be hard to do yes, because the string changes frequency gradually and the strobe can only 'freeze' one frequency at a time. Since the lower amplitude frequencies are so small, you may have to use some kind of magnification to see the string moving/still. Possibly even a microscope.
I can only guess that the frequency change is exponential because the string damping would probably make it exponential, so with a strobe this would have to be measured little by little, one frequency at at a time, and plot the frequencies and amplitudes maybe.
Another idea might be an auto focusing circuit, that works like a camera focus. It would vary the frequency slightly as the string is vibrating.

But probably the best bet is to look in the past and see how it has been done and try to follow that, possibly improving on the method used.