If given a youtube video, where one part of the video is normal, and the other part is sped up so slightly that it's hard to notice... is there a technical way of proving that it is sped up or not?
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There is unfortunally (or maybe fortunally) no way to see if it is slowed down or sped up if done properly. What you have to look for is their movements. I'm not a guitar player, so I can't tell if it is real, but I think it sounded and looked real.
If you study the people there you can see that they move pretty natural. You can see on several occations that they blink in normal speed, but you also see that the main guitar player looks from place to place pretty rapidly. But from my experience with the piano you do everything a little bit faster when you play fast. Because you need to look at what you do, and when you are working hard with your brain and "slowing" everything down to keep up the pace you do things fast. The left hand finger movement also looked pretty natural. If it was sped down it would probably have been unnaturally fast and awquard movements.
Everything looked like it goes in normal speed, even their foots. Normally you have a constant speed on your foot when you tramp the beat. Depending on how slow you play you just increase the time you have your foot flat on the ground. Their foots didn't look like they went up and down too fast. If they had sped it up they would probably have had a small brake holding their foots flat on the ground before tramping the beat again really fast.
Secondly I don't see why a spanish TV programme would slow down their footage or why a good guitarist wouldn't be able to play that.
Technical reason's why I think this video is both not sped up but looks unnatural.
Using the camera crane tracking shots as a time consent, I am pretty certain the video was not sped when the music tempo stepped up as the camera crane tracking shots that follow this appear to be consistent with the earlier shots. So for my eyes, I do not see a sped up video.
The key thing I think that is going on here is that this was shot in a studio with studio lighting. Even with bright studio lights the stage area is large enough to demand the iris to be stopped down to get the most depth of field but the trade off is a slower shutter speed. When the music tempo picks up the close ups of the hands playing the lead guitar appear to be more slurred which looks unnatural. The longer shutter time explains why there is this slurring or blurring.
Another element to consider in favor of the video not having been sped up: this looks like a live show, in other words, there was no post production, everything was done live as for broadcast. You can't speed up video in real time and match it with the sound unless maybe you have a whole chain of effects and delay lines which does not seem practical for this.
There is nothing in this that looks or sounds even slightly sped up. As @leftaroundabout said, the original is paced a bit faster all the way through, but also has that tempo increase halfway through.
While being a fairly fast piece, it is not challenging enough that I would anticipate speeding it up for any reason.
Slowing down a video is a magic thing, but isn't cinema magical?
You should look for this(repeated or mixed frames), as a prove of the "slowdown" process, made by a computer. If you made the "slowdown" in the cam (shooting more frames per second) the process is smooth, and no prove can be extracted. Sure you should download the video to your computer to do this kind of observation.