I was just wondering if i were to play in any given key, say c to make it easy, what are my options for changing keys? I know i could play in A minor because it is it's relative and i understand all the theory behind that but i was wondering if there was a theoretical approach to finding all possible options of keys to change to.
You can switch keys whatever you like basically, but I suggest you start with switching keys that are close to the initial key in circles of fifths and fourths. These keys would be the keys that have very similar notes like the initial key.
For example if you have C major key, the closest two keys would be F major and G major. All 3 of these keys share the same notes, which are referred to as "common notes" with one difference in every key respectively. F major has Bb instead of B, and G major has F# instead of F, when compared to C.
One more important thing to mention is that these close keys share many triads as well, but in different functions, which can open doors for lot of improvising combinations and directions. For example you can find C major triad in all 3 of these keys, and so you can find A minor triad. Other triads are shared between two keys at-a-time as well.
One example of improvising exercise would then be to develop harmonic background content that is universal for all 3 keys (like simple triads that repeat in these keys). Then you can play scales/arpeggios/melodies over that harmony and try to experiment with various melodic functions over these chords. These sequences played on different positions will acquire a new meaning and dimension.
Later on, you can investigate some more complex key changes and analyze what sets of notes could be used, why, when and how. It's all about practice, and lots lots of exercises basically. The goal is clear - to learn to improvise over any given harmonic structure.
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