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There are so many ways to mute
1. palm
2. slight finger lift off string and just muting those strings.
3. fingernail slap and variations.
4. finger across strings and plucking those strings, I’m sure there are a lot more variations.
For this song I do curl the fingers and place the fretting hand pinky across the strings and slap the strings fairly lightly and then slide my muting down in a slight 3 finger strumming stroke. This is one continued motion. I also achieve different tones by where my bent finger nails strike the string, close to the bridge or over the sound hole produce different muted tones.
Another method is the middle finger up stroke instead of the finger nail approach.
Pinky over the strings and use an middle finger brush stroke for a different muted tone and then your fingers are in position to pick the last 2 notes of this mute. For me this is easier and I vary the 2 approach’s for different tones.
Each has different mute tones and also where on the strings you strike them also affects the sound – close to the bridge or closer to the fret board end. To me, all are valid ways to to mute.