The Harmonium is one of the Factions that contest for the minds & souls of Sigilites in the Planescape setting for Dungeons & Dragons. Like all of the Factions, they are ruled by a guiding philosophy; in contrast to the Xaositects, who can be defined as the "Chaotic Stupid Faction", the Harmonium is essentially the Lawful Stupid Faction.

Nicknamed "the Hardheads", the Harmonium's philosophy can be summarized as a combination of "order is the key to goodness", "evil must be stamped out", and "I know best". They're essentially an entire Faction dedicated to the idea that law and harmony equates directly to goodness, and that when everything is working in perfect harmony, then goodness is achieved - but, if folks don't want to surrender to "the Greater Good", then it's justified to crack skulls until they see things your way.

The Harmonium began as an adventuring party of Lawful types - apparently, Lawful Good types - on some pissant Prime world called "Ortho". This party had a lot of success in clearing up evil and getting shit done, so they just kept getting bigger, and bigger, until eventually Ortho was swept by an Anti-Chaos Crusade. In dedicated pursuit to this, they wiped out all chaos on their own world - read, they committed genocide against everyone that was Neutral or Chaotic in alignment. Including Neutral Good and Chaotic Good types. In Sigil, those aware of Rajaat's Champions and their own crusade on Athas like to mock the Harmonium by pointing out the similarities, which the Harmonium never has been able to convincingly argue against.

Anyway, once they wiped out all non-Lawfuls on their own world, they found that, for some reason, Harmony still wasn't in their grasp. Studying their cosmology, they decided that the source had to be coming from the outer planes, so they managed to open gates to Sigil and launch another crusade.

...That went about as well as you'd think, given the much bigger fish on the Outer Planes. Beaten but defiant, the Harmonium has decided to try a more long-term approach, setting out to win hearts and minds from Sigil. By the time of the "present" (ie, when Planescape was originally set), they'd been working for some 500 years. Alongside their fellow "Law-Focused" Factions, the Fraternity of Order and the Mercykillers, they've made themselves integral to Sigil's justice system; Harmonium acts like the cops, the Fraternity of Order are the judges, and the Mercykillers are the gaolers and executioners.

At the same time, they're bitter enemies with several other Factions; not just the expected Xaositects (who worship chaos) and the Revolutionary League (who just want to tear down all of the stablished governments of Sigil), but also the Free League due to their independent mindsets and willingness to tolerate others. It says a lot about the Harmonium that, despite being led by a freaking Paladin with the required Lawful Good alignment of the time, if they learn about the evil Harmonium agents beating up, falsely imprisoning and even murdering Free Leaguers, those agents will be punished lightly and only be punished because the order to do that shit hasn't been officially given yet.

Despite the Harmonium's long history of failures and sordid "well-intentioned extremism", it's somehow still very popular with Lawful Good types, who keep focusing on the Factions ideology of "the greatest good possible for the greatest number of people possible".

Though it has no true patron gods, definitely the most popular god is Saint Cuthbert, whose general alignment (Lawful Neutral with some Lawful Good leanings) and philosophy ("serve the good of society by doing what you're told, or I'll thump you upside the head") are pretty close matches to the Harmonium's own.

The Harmonium may be an inspiration to the Hellknights of Pathfinder, as both are groups whose dedication to law and order in the face of a seemingly anarchic multiverse are strong enough and rigid enough to forge unity between members across the ethical (Good-Evil) spectrum. Unlike the Hellknights, who believe law is simply essential, the Harmonium does believe that its actions serve the greater good.

The Factions of Sigil
Believers of the Source
Bleak Cabal
Fraternity of Order
Free League
Revolutionary League
Sign of One
Society of Sensation
Transcendent Order
Clueless - Incanterium