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Soloing in C

When the harmony dictates that the chord should be C, then the most important notes of resolution and repose are C, E and G. These are the notes of the C Major chord.

C D E F G A B C

So we think about ways to come to rest.

We can simply stay at rest, by just playing chord tones (arpeggios)
C E G
C G E
E C G
E G C
G C E
G E C

We can connect the chord tones:
C D E
E D C
E F G
G F E
G A B C
C B A G

We can alternate from chord tone to non-chord tone
C B C
C D C
E D E
E F E
G F G
G A G

Any of the atomic approaches above can be combined in an endless variety ways.

Also Note that the non-chord tones are B, D, F and A, which for the B minor b5 chord. It is logical, and helpful to think of the resolutions for the non-C (bm7b5) chord tones to the C chord tones.