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Outside of everything, Hibiki has more training than Ryuki. In an actual fight outside of any abilities or rider forms, I think Hibiki would win due to this.

With base Kamen Rider forms involved, Ryuki has the edge. Hibiki has a lot of abilities, but Ryuki's are stronger and slightly less situational. Ryuki has a delay by having to summon, but things like Guard Vent an Advent trump any of Hibiki's abilities in my eyes. Hibiki has an advantage in that he deals with fighting large monsters normally, but he fights ones he's trained for and they are vulnerable to his sound based techniques. I don't think Hibiki would fair very well against Dragredder as a result, which knocks down this advantage.

When Hibiki powers up into his red Kurenai form, he has more power and it isn't so tied into specialty for dealing with monsters, so he could potentially fight Dragredder better. As such, I think this makes him gain an edge in the fight since the boost in power seems like it makes up for his faults and the added damage can overwhelm the base form of Ryuki in most situations. If anything though, Kurenai form makes it more an even fight.

Going into their upgraded forms is a bit tougher. Hibiki's Armed form is a lot more armored and more powerful than the other two forms, plus the sword adds more versatility to his attacks. Survive Ryuki gains more versatility and power as well, though not a great leap overall compared to how different Armed is from the base form. This one is close too, but I think in the end Ryuki might gain the advantage more than not due to his cards and main weapon seeming to be longer in range.

The last thing I can think to touch on is bikes. They both really don't use bikes much despite being Kamen Riders, but Ryuki definitely edges this out in Survive form. Hibiki gains one and has a driving filler arch like Piccolo and Goku but I never felt he was any stronger for having it or really needed it.

Hibiki has training and strength, but Ryuki has his own strengths as well. Its a close one, but I think for me, it comes down to Ryuki having the ability to Advent overall. Having Dragredder really helps Ryuki gain the edge and having him as a bike in Survive Form gives him some bonus Kamen Rider points in my calculator that Hibiki doesn't get.
In the fight of Hibiki vs Ryuki, I have to go with Ryuki.

Hibiki's training helps him make up for not having as powerful abilities as Ryuki, but his training to me is more skilled towards Makamou(and mainly certain types of them) than towards fighting other riders or non-makamou. It would help in the fight, but I don't think he'd have a very large edge with his experience. Typing this up the idea hit me too: Hibiki rarely had to fight other riders while Ryuki had to deal with it quite often. End-show Ryuki had a lot of experience himself and the added rider fighting might give him an edge in being able to beat Hibiki.

The fire element argument I can see and wasn't something that crossed my mind. Hibiki would be quite a bit resistant to it where Ryuki probably isn't and would require guard vent or some other kind of defense against it. The plus to Ryuki here though is that all of his abilities aren't strictly fire-based so it doesn't null most his attacks. Hibiki definitely has an edge with fire though.

As far as Dragredder goes, Hibiki does have more of an advantage against him than some riders probably would. The thing with that to me though is that this is a very different monster. Not only one that isn't weak to the sound attacks Oni riders use, but one that is likely just as resistant to fire as Hibiki (Ryuki I can see not being so, but a fire-spitting dragon monster should be resistant at least a good bit.)

Another problem with Hibiki having to fight Dragredder to me is that during the end of the series, Hibiki fights less larger monsters. While I know he's stronger as Armed Hibiki, he barely fought large monsters by then so he didn't have much experience using those powers on larger monsters.
I think that, plus Ryuki attacking at the same time, would make it tough for Hibiki to compensate. His training and experience would help, but it feels the odds are against him with it being 2 vs 1 and his training being mainly for a specific type of threat.

If this was deadliest warrior, I would imagine the excel spreadsheet they use to calculate everything would make Hibiki win a good amount of the battles, but I just don't think he could win more often than Ryuki could.

Perhaps Hibiki's standard attacks wouldn't hurt Ryuki as much as they would a Makamou. But I think in Kurenai form they would. I can't think of the name of it right now but the flaming ongeki form seems like it would hurt a person just as much as a makamou. And watching Hibiki take out hordes of Summer makmou makes me think he could fight any human one on one and win. Seeing how makamou are stronger than humans. That's why to me it comes down to skill and abilities. Hibiki has more skill and I don't feel any of Ryuki's vents are all that threatening. If Ryuki had more long ranged attacks I think it would be different.

The main problem would most likely be Dragedder. But, I think he could take him out in armed form. I think that because he beat Orochi in the Hibiki Film. I know that was a different Hibiki but I'm assuming they had close to the same abilities.

I agree with Kurenai form having the advantage over the standard Ryuki form. Since Ryuki has no in-between, Hibiki can overtake Ryuki in his base form by powering up. This would help against Dragredder too but not enough to make me think he could handle two non-fodder attackers every single instance of them fighting. He'd gain more than any other instance to me this way though.

His Armed form might work against the basic form of Dragredder but I think the fact that it goes into a Survive mode with Ryuki makes it more buffed up itself and able to stand up to the increased Hibiki power, just like Ryuki can. It comes back to the numbers issue and since this isn't Hibiki Riders vs Ryuki Riders, but the main riders themselves, Ryuki gains the advantage due to the type of rider he is I think.

Ryuki has upgraded forms of all his base cards and a new one(technically two) outside of just a different final vent. He can ride it as a cycle but just because he likes to for the final vent doesn't mean he has to during a battle and it can attack outside of just the fireballs.

Hibiki may have fought bigger enemies, but again its a specialized thing he fought and when he gets the Armed form he has only the one instance you brought up that I recall of him fighting anything large. If anything, ranged attacks would be harder to hit a speeding target. Granted your not as agile on a bike as you are on foot in terms of three-dimensional movement but being faster and on a cycle gives you quite a bit of dodging ability that someone on foot might not have.

I still think it comes down to Hibiki being powerful against the makamou due to the nature of how they fight with sound and given the descriptions of things early in Hibiki. For all I know, a large fire attack from Hibiki could hit anything else and barely burn it. I'm giving him the benefit of the idea that he is on a more equal footing against Ryuki, than having Ryuki be on par with simple monsters and being defeated like cardboard gundam fighting a flamethrower.

I recall them fighting together a bit, not all the time and maybe not as a pincer attack but its something in his arsenal and that I remember him using in battles. I'd rather go off of what they can do myself, so I'm fine with Hibiki using the guitar and trumpet. He's proven he can use them. The key thing Hibiki gets from that I think is the trumpet shooting bullets, but I don't think the guitar could help him much in this fight. Maybe due to not being a fire element, but it seems like he'd be best with what he's most trained on in terms of those two weapons.

My point with the sound stuff is that his is a highly specialized skill. Granted, a very strong makamou could possibly beat Dragredder, but what my problem with Hibiki's style is that he uses a type of music. I'm not 100% sure if his fire and other abilities draw the same, but if they damage makamou which are only really hurt by the purity of Oni music style, then it seems like it could be something that might not hurt people as much as other abilities might if the person isn't damaged by said pure sound.
Ryukis fire attack we know is fire-fire. Maybe its best against the mirror monsters from his show, but I've never seen or heard anything in the series say otherwise. Where with Hibiki it was underlined to me that makamou are hurt by Oni sound which they train to do.
Now I'm not saying I knock down Hibiki for this, just saying that because he goes SSJ3 by the end of the series doesn't mean that makes him able to defeat non-makamou things with little effort.

Obviously there is no convincing you otherwise and that's fine lol. I just wanted to clear up what I think a bit. Since it feels like we'll just be looping here between ourselves, I'll leave this debate with this post for now.
I think it will be a close fight but Ryuki has an edge to me and I can't see that changing due to how the Hibiki riders were set up versus Ryuki riders.

Alright. We have a winner with a vote of 2 to 1 (Based on poll and debaters)
RYUKI

If you wish to continue the debate, please do so in the Post Match Thread.