Agreed, re: it's just right-hand lead., but I get why people say "closed" to differentiate it from "open" left-handed. That said, no one *has* to cross over to play right-hand lead on a righty kit. I picked up my technique from a combination of marching band technique (not having the stick be a straight-line extension of the forearm, but more of an angled, German palm-down grip), and from Bill Berry's intro in the video for "The One I Love". That was a revelation for me; "Holy- I *don't* have to cross my arms...!?!" I never did it again. In recent times, seeing the video, I've noticed that he may not have been using a traditional hi-hat stand, but maybe a cable hat. But, it works for me with a regular stand. BD and HH pedals symmetrically placed on either side of my snare, with my knees about an inch & a half from the drum, places the HH cymbals in the perfect place to be able to just move my right forearm forward & to the left a touch, and I can play the hats with my right hand and leave more clearance for playing the snare with my left than Berry shows in the video.
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Aronoff said in Modern Drummer (I'm paraphrasing from memory here) that he started playing open-handed when he needed a sloppier, Replacements/Stones-y vibe, which worked because he wasn't as proficient this way, but the problem ensued that because he did it so much, he got very proficient at that also and it lost its effect, so he had to stop.
Inspired by his statement though, I began practicing it a long time ago, to force some increased efficiency and speed in my left hand. 20-25 years later, I'm nearly equally adept at it also now, except I can't *quite* play as fast (usually as fast as I need to for my gigs, though), nor can I play patterns quite as syncopated, when playing lefty. But certain songs that aren't super fast or syncopated (Jumpin' Jack Flash, What I Like About You, etc), I'm equally confident playing the hats with left or right lead.
I played open handed for years, toured, gigged and recorded with it.
Let me let you all in on a little secret:
There is no "world of possibilities" presented to you just because you put your left hand on the hihat.
It's possibly the most over-rated drum technique thing that people advocate. It won't make you Simon Phillips, Billy Cobham or Claus Hessler and it won't advance your actual playing of music. Those great players got great by practicing and playing their asses off, not because of their hand position.
What it WILL do is waste your time relearning how to play stuff you can already do now except it will now be a little sloppier.
And all that stuff about how it will make your left hand so much better is also an exaggeration. Your hands will get better by diligent pad work not because you rode with your left hand.
My 2 cents, let the flames begin.
No flames here, but, I'm actual proof that it's not overrated; it's very useful. No, simply doing it won't make you Phillips, Cobham, Beaufort, etc, but if it wasn't viable, they wouldn't do it either. It's only a waste of time for those who think things will instantly improve from using it. Quite the opposite, actually; things will feel wobbly & hinky for a while, until you get used to it, and even then, it takes a long time for it to feel totally "normal". Also, the left hand WILL get better, as mine did- again, not simply from switching, because it takes time invested, but from by diligent *playing* more with the left hand. Pads not required (but encouraged when unable to actually play the hats).