Is there such thing as harmony independent timbre characteristic of a sound? I was wondering if any of you know about any technique to extract a harmony (e.g. harmony in a shape of a chromagram) independent timbre characteristic of a sound frame.

What I'm after is a version of spectrum that will make all the piano notes look almost the same despite the fact they occupy different parts of the actual frequency spectrum.

Googling that didn't give me much yet.

Thanks a lot in advance!

EDIT: Great folks in comments pointed out some ambiguity in my question. I'm interested in a metric that is independent of the notes themselves, but captures the whole set of timbral characteristics that could help describe the instrument independently of the note content. It could be a psychoacoustical metric. I'm really curious if anyone knows anything that could fit this task. Thanks a lot!
well, since timbre is a multi-dimensional perceptual phenomenon, there is no single metric.  so lotsa different metrics.

possibly the most common metric is obtained by high-pass filtering the note, measuring the average power, and normalizing that against the power of the unfiltered note and also normalizing against the pitch.  and it yields a "brightness" measure.  the HPF is a digital differentiator.

let your input note be $x[n]$.

differentiator output:
$$ x_1[n] = x[n] - x[n-1] $$

"sliding" average-power of HPF

$$ x_2[n] = \sum\limits_{m=0}^{M-1} (x_1[n-m])^2 \, w[m]$$

$x_2[n] \ge 0$ and $w[m]$ is a window function.

Normalized against the note power (so that louder notes having the same timbre do not get a larger measure of brightness:

$$ x_3[n] = \frac{x_2[n]}{\sum\limits_{m=0}^{M-1} (x[n-m])^2 \, w[m] + \epsilon} $$

$x_3[n] \ge 0$ and you can multiply $x[n]$ by nearly any value you want and get virtually the same $x_3[n]$.  $\epsilon&gt;0$ is necessary to add to the denominator to preclude division by zero.  but $\epsilon \ll |x[n]|^2$ for most non-zero $x[n]$.

now this is normalized against different note loudness, but is not normalized against the pitch of the note.  higher pitched notes will result in higher values of $x_3[n]$, even if the waveform is otherwise identical.

to normalize against the pitch of the note, you will need a pitch detector or a fundamental frequency detector.  that's another whole problem and i have discussed it here on SE.