I am quite interested in the topic of playing by ear. I don't often learn songs by ear, but I do play "by ear" in the sense of not using music when I play. I don't like to think of the process as memorizing a song, in the sense of only repeating what is written, but rather it's a process of understanding the song at a deeper level, and being able to play it in a way that I enjoy even if it may not always be what someone else (including the composer) might have envisioned.
At any rate, I thought I would respond to the request to play "Amazing Grace" starting on G, and play it "by ear" in the sense I'm using that term. I have just opened a soundcloud account where I posted "Amazing Grace." I'm hoping the following will lead you to that site, but I'm not certain that I properly inserted the link.
[soundcloud]https://soundcloud.com/richard-noel-7/a ... perimental
If that doesn't work, this might. https://soundcloud.com/richard-noel-7/a ... perimental
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I probably know something like 800 to 1000 songs that I can play without music. Once I properly learn a song I can usually play it without music and in various styles. In order to "own" the song, I typically remove the music immediately, which I find changes my orientation from that of a visual learner to a focus on the sound and the structure of what I am playing. I have no doubt that I think differently and play differently once the music is removed, and I equally have no doubt that this is a more effective way for me to learn songs than what I used to do, which was mostly to memorize notes and fingering. In fact, it was only when I was pushed to get away from the lead sheets I was using that I discovered a whole new world of music experiences when I focus on hearing rather than just reading the notes in front of me.
Anyway, here is an example of what works for me, and that is to play a song I'm learning in a variety of ways. I may change the rhythm, tempo, improvisations, chords, key, and voicings (I play v-accordions). The example is not meant to be a finished performance, just a demonstration of how I learn songs. I don't even like some of the versions I created, but even "bad" versions help me to learn the song and to decide how I like to play it. On a gig, I will often vary my approach to songs depending on the audience, and it also keeps my interest up so that I usually do not tire of playing a song.
I don't know if this will be of interest to anyone else, and I realize there are many different approaches, but that is what works for me.
In the song I posted, I am playing all versions using an accordion bass and musette treble. I would not use just these options on a gig, but it made the process simpler.
I begin with a 3/4 beat, playing a typical accordion bass. Then I add some chords, change the bass slightly. Then move to a more gospel style. Try the song in 4/4 rather than 3/4, with traditional accordion bass followed by a walking bass. I return to 3/4 at a slower speed, with chords and sustained bass, then finish with single notes in the treble, sustained basses, and more feeling.