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Three Finger Technique Across Strings

I think you should try it a variety of ways and find what works best for you.

Try everything, and if you find one way that works better than anything else, work that until it's second nature. At some point, that's what you want. To let your hands do the work without having to think about it. Then you can just concentrate on the music and trust your hands to go where they need to.

I don't even think in terms of patterns, but more which finger is free to make a smooth transition to the next note or string.
I've trained my hands to move in whatever manner keeps what I 'm playing moving comfortably.

Certainly, some things are considered "technically correct" and others are not. Who cares? We are all individuals, and we should all play as such, according to what works best for each of us.
 
" I'm having trouble moving from string to string. When you play say four quarter notes on one string & move to another string, do you "start over" with r-m-i-r or do you just use the next finger in the pattern? Like r-m-i-r on one string then go m-i-r-m on the next string?"

R M I R
M I R M ect

The way I use to teach myself to play it was by playing 16th notes slowly and building the tempo up on the same string and doing an adjacent on the beat. Which would always be the next finger (if doing it RMI) ect (R) m i r (M) i r m (I) r m i (R) m i r
So then when it came to climbing up strings in beats of 4, I was fine. I've tried it with 2 notes on the string and doing up and descending - and that's tough!

But in most patterns like that, just to be safe I switch to 2 fingers :)
 
I need that ring finger to play on the same string as the middle & index fingers.

Here's a fact of learning finger style.
Any finger follows the previous finger.

In the exercise video i posted earlier ( and the others in that series) they develop a players understanding of the relationship to counting and playing.

Once that concept has been internalised you will use it use and addapt it to suit when playing live. But to check you are practicing it correct you use all fingers you wish to use, so when you do play live you are removing fingers,...not adding them.
It is easier to remove ( not use ) a trained developed finger than it is to try and add one that is not a trained developed finger. This is why practice has to be more than the sum of the use, you push yourself in practice to make it easy to play.

Now take the idea of alternating fingers.
yes they alternate, but they alternate to a beat, not a pattern, tempo or specific length of time...but a beat.

With this in mind, you can at any time re-start any sequence of fingers to function the beat, the fingers do not truley need to alternate when playing live, they need to alternate to function the beat.
How can this be so?
Well a rest is a stop regardless if it is for a beat, a quarter,a half etc. is makes no difference, the fingers can re-set. If you have a whole bar rest, or say 10 bars, you do not need to follow it with the next finger in the sequence, you will start again. So if your sequence is R-M-I-R-M-I-etc if you stop on the Ring you can start again on the ring.
So long as the sequence flows when moving you service the beat.
Again why can this happen? Well if a rest is a stop, then the time taken in between songs in a set are stops, so they are rests....if you finished the last song on the middle finger, to stay in sequence you would have to start the new song on the middle and carry on from there.
But we do not, that is ridiculous assumtion to make, but as a fact it is true.
So if we can start again with every new song we play, we can certainly look at it as we are starting a new sequence again after every rest/stop we make.
The thing to learn is how fast do we re-set the fingers without tripping over our thinking.
That is the true purpose of that exercise series, to allow the fingers not to be tied up in the thinking, let the finger play what is being felt and leave the thinking to the sub concious.
 
To add to many useful things that have been said:

when playing two or more notes downward (pitch-wise), so the next note is on the next string (not on the same) - always try to use same finger. It saves you a lot of effort - the finger is already there, so all you have to do is continue the plucking motion, instead of engaging a different finger.

E.g. you play G(high, 5th fret on D string) - D (5th fret A string) - G (low, 3rd fret E string). Use same finger to play all three, because after playing each note your finger ends up resting on the next string you need to play.

Of course, this doesn't always work, so use your judgement. Disco-style non-stop octaves are easier to play with a "claw" of sorts (I always use I-R for that). But the idea is: where you can get away with this - do it.
 
To add to many useful things that have been said:

when playing two or more notes downward (pitch-wise), so the next note is on the next string (not on the same) - always try to use same finger. It saves you a lot of effort - the finger is already there, so all you have to do is continue the plucking motion, instead of engaging a different finger.

E.g. you play G(high, 5th fret on D string) - D (5th fret A string) - G (low, 3rd fret E string). Use same finger to play all three, because after playing each note your finger ends up resting on the next string you need to play.

They call it 'raking'.