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I've been learning the violin now for a little over a year, taking it seriously since last July.
I still seem to struggle learning songs so they just come naturally and I can play them from beginning to end reliably. How would you recommend learning a song, like Londonderry Air for example, to performance quality, so that you can play it any time, anywhere with good intonation and rhythm?
My own experience is that I can learn a simple song in a couple of days or less--able to stumble through it without stopping (reading the music, I don't try to aurally-only learn a new tune). Then comes the hard part of polishing it so it sounds presentable. Depending on how hard the song is, and my present level of ability, that could take weeks or even months. If the song is at or maybe a little over my level of ability then even some parts may never be played properly, or may just be a random thing--one time would be played properly, another time not. I think you just have to keep progressing, getting better on the instrument. Eventually you can go back and play that same song and realize you're now playing it much better than, say, 3 months ago.
As an example, see my video submission for the St. Paddy's Day Virtual Party thread. I chose an Irish tune I'd never played before (John Ryan), last Sunday when Diane announced the Party, and attempted to learn it in time to submit a video. I decided I'd learned it well enough to do the video I submitted. You can clearly hear all the mistakes I made in the opening part. If I'd recorded the song 10 times one of those times I'd have the first part perfect, but some other part would stumble. So I just made one recording and decided that was good enough. Today I played that tune again, and played the whole thing almost perfectly, one time out of the 5 I played it today. So perhaps if I'd waited until today to do the video I'd have done better, perhaps not. So that particular song is near my present level of ability for learning in a relatively short time.