Instead of using the somewhat random-looking piano-centric major scale as a basis for melody and improvisation, I prefer to add to a small pool of notes that is easy to memorise from the patterns they form on the clock. To be more specific,
The notes that make up the patterns of 'safe' notes are referred to as symmetric scales. The non-symmetric scales are extensions of the symmetric scales.
Some symmetric scales fit several chords. You might know that a C pentatonic scale can be used on both C major and A minor. In fact, it can be used on C7 as well. Thus, the pentatonic scale will fit any minor, major, or dominant chord provided you adjust the root note accordingly. Below is a summary of the relationship between the symmetric scales and the chords they can be used on.
The three scales - pentatonic, hexatonic, and pentadominant - are the building blocks of the method I am promoting for the M3, and you will find plenty of material on this site covering them. As an introduction a brief explanation of their names and properties follows.