Different ways to improvise when playing a piece?

[spenstar]

 I often practice things quite far below my level. (Things such as hymns for my church or coming across a simple piece when practicing sight reading.) I've taken this as a change to start practicing improvising, but it's never something I've been taught. I've tried throwing chords in and little trills but that's about it. With the chords it works sometimes, but other times it won't work because it's a part that's supposed to be 'little' and the chord ruins the effect. As for trills, I can't throw that in a hymn or that would just ruin it. So what are some other things I can do when improvising on a piece of music?

[quantum]

Try approaching from a perspective of complete freedom.  No rules, no theory, no pre-determined theme, no such thing as "wrong notes".  Just start playing and make music with what you are doing.  This approach has been advocated by several on PS (Ted has written much about this).  The idea is that for one new to improvising, one needs to first develop a flow of ideas.  This flow is important as it is what also allows you to implement more established musical ideas in a fluid manner. 


You can also work on something specific like chords.  Do something like secondary dominants.  Good thing about them is that they can be used both inside and outside of a tonal framework.  You can have something normal sounding, or go off on come crazy adventure.