Word to denote a negative tone in the spoken voice

I think of a harsh tone, perhaps a snide tone or remark, or more casually a snarky tone.

Snarky, from Merriam-Webster, including example sentence:

crotchety, snappish

sarcastic, impertinent, or irreverent in tone or manner

Example of snarky in a sentence

The writer at No. 10, Fred Mustard Stewart, died last February at 74. His obituary in The Guardian contained this snarky observation: “Year in, year out, the 600-page mark did not daunt him.” —Dwight Garner, New York Times Book Review, 24 Feb. 2008

Patronizing could work for a slightly negative and condescending manner.

Patronizing - displaying or indicative of an offensively condescending manner (dictionary.com)

Since you said:

Snide is so close! (And much better than anything I found.) It's still a bit too strong. People who choose to do this are masters at it; they make it so subtle that even if someone overhears it, they will probably not tell them they are being rude. It's like a "subtle passive-aggressive rudeness".

I'm going to hazard arch, which is used like @Jon's applicable patronizing, but with a subtler tone of snarkiness.

From Wordnet:

arch (adjective): (used of behavior or attitude) characteristic of those who treat others with condescension

And from Collins:

arch: knowing or superior

An arch tone is not quite a cliche, but very common, and used when the social situation is recognized by both parties as licensing the superior attitude, such was when the person using it is from a higher social class (back in the days when those were more explicit and subscribed to), or today when the other party has screwed up in a visible manner.

In short, think old-school British upper class dismissively ordering about or correcting someone they think beneath themselves.

Alternatives include superior (not as snarky or witty) and condescending (more explicit and often more brutal, and sometimes felt by the other party to be unjustified).

pompous tone

Pompous: Overly formal, often contains passivity and jargon. Many businesspeople mistake the pompous tone for a professional one and use it regularly.

Words at Work: Business Writing in Half the Time with Twice the Power

You could get away with conceited tone as well, but it is less common as a manner than as a personal attribute. It also seems to be a bit dated.

We thank thee, Father Simon," said a voice, which strove to drown in an artificial squeak the pert conceited tone of Oliver Proudfute.

Sir Walter Scott, The Fair Maid of Perth

Antagonizing is another possibility and is closely associated with passive aggressiveness.

Crass is what came to my mind initially. (?)

Crass: adjective; lacking sensitivity, refinement, poise or intelligence. "the crass assumptions that men make about women"

A tone can also be “Whiny” as if the pitch remains high and moves up and down and all about the person expressing their irrelevant disappointment in something that may, or may not be, a big deal. Those who overreact to “stressful” experiences that bring out lack of emotional control. The high pitch is usually emphasized with certain pieces with the subject matter, and usually, this whining is an overreaction to a benign problem that usually gets centered on the person with said tone.

Furthermore selfishness and the inability to see past their own and try and see other perspectives are completely irrelevant. Age groups that normally express such a tone will be anywhere from 1 to 3 years old for two even teenagers. And then we have the whiny young adults who are either coddled or never taught strengths. These individuals were always given things without working for them and or saved from consequence that they never learned on their own lessons that should have been learned. These demographic of people are coddled as children and “baby“ only resulting in a poor weak ignorant adult. No offense just facts.
-the witty writings of Stephanie Joanne Lewis from the sunshine state of Florida.

contemptuous (or "contempt")

Collins defines it as:

contemptuous (kəntemptʃuəs ) adjective [usually verb-link ADJECTIVE] If you are contemptuous of someone or something, you do not like or respect them at all.

Synonyms: scornful, insulting, arrogant, withering

He spoke to her, his voice dripping with contempt. His contemptuous tone was backed up by his contemptuous body language.