The original Hanon exercises for the most part follow the shape of the (right) hand and place the principal 'skip' (third) between the thumb and the index finger.
Melodies, however, and the general run of piano playing, have no reason to show such consideration.
Varying the Hanon No.1 pattern by moving the skip to between RH fingers 2 and 3, or 3 and 4 (and so on) soon shows how spoiled we have been, and offers a useful opportunity to prepare the hand for a greater variety of demands.
Here's the original Hanon No.1 exercise.
We can be more independent and 'get away from the dots' by internalising the pattern and playing from a shorthand version like this.
It is understood that, between the dashed barlines, four repetitions of the pattern will creep up the keyboard note by note.
The 'gap' in the rising first half of each bar is between the lowest and second note up. If we move the gap up one note, we get this exercise (which you will learn from the shorthand).
The training for a different spread of the fingers leaves a new feeling in the hand.
The exercises in this series can be practiced with a held (or held and repeated) thumb. This balances the hand and develops independence of the fingers.
Caution! Approach these exercises with care!
Be sensitive to any build-up of strain in your hands, wrists and forearms, and rest and shake out tension regularly. Never 'work through the pain' - rather develop your stamina cautiously.
Here is how you would approach exercise MTG1.
The thumb plays at the start of the right-hand pattern and with the little finger (RH5) both times it plays.
Running up and down between fingers RH2 and RH5 twice gives you the odd time signature of 7-8 (seven quavers in a bar). You don't have to count this - or think about it at all. Just play the notes equally until you've got through the bar quota and that's it - you've played a bar of 7-8.
(Out of interest, the beat introduction divides the seven beats 1 2 1 2 3 1 2 |1 2 1 2 3 1 2. When you're proficient, you can count yourself in and try toplay along.)
The left hand does not need to do this exercise - especially not ascending, from the little finger side! - so it simply has a seven-note 'ambling around' pattern which is rather Hanon-like, too.
We can again devise a shorthand version to play these held-note intensive exercises from.
You're shown two patterns to 'get the picture', and are expected to supply the missing four bars between the dashed bar lines. The last pattern is shown in full out of courtesy.
Play exercise MTG2A straight from the shorthand version MS. The left hand has an even simpler 'ambling around' accompaniment.
Now here is the third possibility. The gap ion the pattern moves up one more finger, to between RH3 and RH4, and between LH3 and LH2.
This example is presented with just the shorthand version, straight then held-note intensive.
Now go on to Mind the Gap Page Two for the last of the possible simple patterns and some more advanced ways of using them.