Hyphenation of frequency-dependently

To start with, there is nothing wrong with the following:

… a dependently ^(adverb) amplified ^{(past-tense verb)} tone ^(noun)

… a dependently amplified tone

But what it lacks is a description of what the act of amplification was dependent on.

There is also nothing wrong with this:

​… a frequency ^{(attributive noun)} dependency ^noun

… a frequency dependency

The problem comes when you want to combine those two concepts.

We can parse the construction of the sentence in the question:

✘ … a frequency-dependently ^{(noun-modified adverb)} amplified ^{(past-tense verb)} tone ^(noun).

✘ … a frequency-dependently amplified tone.

At the very least, this is so unidiomatic that even if some kind of defence of its syntactical integrity could be made, it wouldn't matter. It's not a construction that can be reasonably accepted.

The problem seems to come from using a noun-modified adverb.

We can have a noun-modified adjective that acts adverbially and which comes after a verb:

She smiled Gorgon-like, freezing me in fear.

More traditionally, we can have an adverb-modified adverb before a verb:

She very quickly ran down the stairs.

But neither of these are noun-modified adverbs coming before a verb, nor do they help with the sentence in the question.

Although I can't definitively prove you can't have a noun-modified adverb, it seems unlikely, and I will drop that line of investigation.

However, consider this:

… a frequency-dependent tone amplification

Although amplification is a noun, it describes an action:

[Merriam-Webster]
1 a : an act, example, or product of amplifying

Using that as a basis, we can reconstruct the problematic phrase in a way that, while slightly awkward, is nonetheless grammatical and still preserves the sense of the original:

✘ … a frequency-dependently amplified tone

✔ … a tone produced from frequency-dependent amplification

This reconstructed version can be dropped into any sentence where the original, ungrammatical version had been previously attempted.

As suggested in a comment, a complete rephrasing of the sentence could also express the same thing:

… a tone that was amplified based on its frequency

It's simpler, but also uses a different style.

Which essential version to choose, with or without further tweaks, is subjective.