Many melodies are chord notes played in sequence with occasional scale notes added. So, many melodies can be played by placing fingers over chord patterns. The thumb (finger 1) becomes useful for playing the lowest notes in a chord.
A song that demonstrates an arpeggiated melody is Clarinet Polka.
["The Clarinet Polka" or "A Hupfata"[1] (Polish "Polka Dziadek", Estonian "Vanaisa polka", Russian "Полька Дедушка" – Grandpa Polka)]
https://www.google.com/search?q=clarinet+polka&safe=off&espv=210&es_sm=93&tbm=isch&tbo=u&source=univ&sa=X&ei=W237UofrEIrJ0gGV_YDACw&ved=0CCkQsAQ&biw=932&bih=687
http://www.youtube.com/watch?v=0WCMzRLov58
http://en.wikipedia.org/wiki/Clarinet_Polka
http://www.youtube.com/watch?v=FqKxcNVtm9k
Perfect Practice Makes Perfect Performance
At the beginning of each practice session, reestablish and reinforce perfect performance of the basics before introducing work on a new skill.
Scale and chord patterns are based on using three vertical rows of buttons.
Finger 2 in the Left-most Vertical Row of Three Rows
The primary pattern begins with finger 2 in the left-most vertical row of buttons.
Begin with two-octave chromatic scales up and down. When you have played them several times correctly, evenly, and crisply, move on to the next skill.
Play major scale two octaves up and down. After several correct performances, move to the next skill.
Play natural minor scale two octaves up and down. After several correct performances, move to the next skill.
Play major and minor scale segments of three to six notes up and down.
Finger 3 in the Center Vertical Row of Three Rows
Move the hand to the left one row and perform the above sequence beginning with finger 3 in the center of the three vertical rows.
Finger 4 in the Right Vertical Row of Three Rows
Move the hand to the left one row and perform the above sequence beginning with finger 4 in the right of the three vertical rows.
Chord Notes as Melody
As you play more music, adapt the basic playing techniques to the requirements of the music.
For example, you start with a finger 2 pattern, but a certain section fits more easily using a finger 3 pattern. Experiment with blending various fingering patterns to best fit the music requirements.
Chord Patterns
The primary pattern begins with finger 2 in the left-most vertical row of buttons.
Perform a major chord arpeggio two octaves up and down.
Then perform a minor chord arpeggio two octaves up and down.
Then perform a dominant seventh arpeggio two octaves up and down.