Well, the reason I've been carrying on about barre chords is to prepare you for this song. Now that the Mission website is back up, we can carry on as usual. There are lots and lots of barre chords in Carry On-- all easy ones. Just the regular, old, barred E, Em, E7, A, Am, and A7. The only hard chord in the song is Cadd5, which is NOT a barre chord! (It's actually not that hard, either.)
You COULD play this song using only the non-barre versions of these chords. This would be OK if you were just strumming accompaniment for a singer, but for a guitar solo, it sounds terrible, as the melody would often be going down, when it's supposed to go up.
This piece does require a reasonably high-quality instrument, where the high notes are not grossly out of tune with the low notes on the same string. On nearly all guitars, as you play higher and higher up the neck, the the notes sound progressively more out of tune. This happens because you have to exert more pressure on the strings to fret them, stretching them and making them play sharper than intended.
If your high notes and chords seem excessively sharp, try using less pressure on the strings. If this doesn't help, change to high-tension strings. The extra pressure needed to fret the high notes is a smaller fraction of the total string tension, with high-tension strings. Besides, there's hardly a guitar in the world that can't be improved by putting new strings on it. If that's not good enough, have a professional luthier (guitar technician) look at your guitar. There are modifications that are relatively easy and cheap for a luthier to make, that can help a lot. Most good guitar stores have a luthier on staff. If he can't fix it, it's probably time to upgrade your guitar. This song is a good one to use when trying out new guitars.
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