Most people learning to play from written music work on some etudes. There is theoretically nothing stopping people who play by ear from learning the same etudes.. except you'd either have to find a recording someone made of themselves playing them or ask someone who reads to play one for you a few times so you can catch it or maybe work it out from the written score yourself or punch it note by note into a midi player or something.
I propose something a bit simpler. Etudes are basically short musical pieces that are a useful exercises that frequently incorporate some opportunities to focus practice on one or more elements that we need to be able to play well. While there are some exceptions, they also are sometimes a bit dull. Most people who want to learn to play by ear want to play songs more than exercises (well, ti think that is probably true for everyone, actually). So if we take songs that are fairly short and easy to remember, but give an opportunity to practice some useful things, they can serve the same purpose as the written etudes in books or etc.
This also can be of interest to players who usually play from written music, but would like to work on developing their ear more. Or you would like to learn a piece or two that you can easily play without written score for trying out instruments in stores or jamming with friends, maybe even working up for small ensemble play.
One of the pieces I use is the Gloucester Wassail song, which is a Christmas/holiday song that a lot of people would recognize when they hear it, and happens to have some useful features. Firstly, it can be played on any one string, which allows for getting in some practice on any particular string that maybe gives you a little trouble. Secondly, if you use a few embellishments with it, it gives the opportunity for practising some slurs and some sorta fast finger drops to practice getting groups of short notes so you can play them clean and quick. Lastly, if you don't jump to the next string for the high note, it gets a reasonable amount of practice in for your 4th finger.
Now you may ask, "Who in the world would want to practice holiday music in the heat of summer???" Well, kiddies, this is exactly the time to practice it if you want to have it down to a reasonable state of effortless perfection by the time the Christmas season rolls around. Also it dates back to the medieval period, so it can be done with a medieval sort of feel or dressed up a bit to make it a bit more baroque if you like. It is definitely in the public domain, so if you work up a nice version of it and put it up on youtube, it shouldn't get taken down. And lastly, most people would not think of it as a "kiddie song".
For those who usually play from written music, one more thing to consider is that even though you probably have seen the piece written out before, it is old enough that it predates the use of modern music notation and most of the people that played it in the first centuries after it was written learned it by ear. So you may consider that it can be more traditional to learn this particular piece "by ear" and memorizing it rather than playing it from sheet music.
To use it as a sort of "etude", I suggest setting a timer to 5 minutes and playing it over and over at a tempo that is comfortable for you until the timer runs out. Then take a little break for a couple of minutes and do it on another string, until you have done them all. You could also use a metronome as well, if you want to work on your timing and steadiness of tempo. Don't have a timer?
You do now!
The song has a fair number of verses, so 5 minutes isn't really overplaying it. It amounts to 20 minutes of total playing time which is a good warmup for first position. Every time you use it as an "etude" for practice, you'll be just that much better when the holidays come around and you want to show off your musical genius. If you don't want to risk annoying the neighbors or room-mates or family by playing it so much all at once, play it for 5 minutes on one string and then do another string at a practice or play time later in the day. But I personally do really find it worth the bother to try and play it on each of the 4 strings for 5 minutes every day.
I've added a few embellishments and taken up the tempo maybe a bit in the few weeks I've been working on it. Simplify it and slow it down to suit your tastes and what makes it easy to play. You'll be doing it faster and fancier in no time.
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