Is there such thing as harmony independent timbre characteristic of a sound?
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.