I was trying to do the technique described in this video on my acoustic guitar.
My hammer-ons sounded really dissonant once I got to around the 12th fret and above:How can I fix that?
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The "hammer-ons" technique causes vibrations on the string, due to the physics involved in striking it.
This means that the string vibrates in both parts: the part that goes from the bridge up to your finger, and that that goes from your finger to the end of the neck.
What happens in your case is that you're listening to both vibrations.
The fundamental difference between acoustic and electric guitars is what you (normally) listen from.
When listening from electric guitars, you usually get the electric signal of their pick-ups, and since pick-ups only capture the vibration of their string, you just get the vibration of that part only (just like in the first video).
In your video you're getting both of them, as you're recording the acoustic result of those sounds.
In fact, in the first video you're actually listening to the sound coming from the output of the guitar's jack; if the sound were recorded using a microphone close to the fingers, you'd listen the same as your second video.
As has been pointed out you are hearing the vibrations created in both segments of the string and they will not be in tune in general. However on an electric the dissonant part will not be amplified since there is no pickup under that part of the string.
Even on an acoustis guitar the dissonant part is not going to be picked up in the vibration of the bridge and body so it's not going to be heard. If you are playing a solid body electric guitar without an amp you will always hear both at equal volume. I've never seen a classical guitarist use a scrunchy or other mechanism to kill open string vibrations. They just play clean, and mute with the hands and fingers as needed.
Two questions here. 1. Why does my guitar do this? 2. How can I fix the disonnance?
As music amante states, a hammered string sounds on both sides - the note played between fret and nut being s a lot quieter, but still audible on acoustic guitar, and hardly picked up on an electric.
A scrunchie on the strings around fret 1 or 2 will solve the problem. If at all possible, playing as a pedal steel player would do is the other solution - muting that side of the hammer-on with the left palm or a finger from that hand.