ok you may not play guitar
so this may not help you too much
but going to your tri tones.
you can think of your left hand like your guitar player in a jazz
band.
he is playing the same basic harmony but with a different rhythmic twist some times.
often in order for the guitar player to really blend in with a big band
he has to play tri tones with maybe one extension
which is all we need our left hand to do.
there are several principles that come to mind to get you started
first of all you only need to play the 3rd and seventh of each chord in your left hand.
major or minor third flatted or major seventh depending on the quality of your chord.
second
any chord can be made dominant
thus the tritone
third
you can approach any dominant chord with its relative two chord
so if you are about to play a C7 your left hand would be playing an e and Bb
to insert the relative two chord
which is Gmin7
play FBb
principle four
any chord can be approached by its dominant
so going back to our C7
you can also precede it with
BF
which is the tritone for G7
principle 5
any chord can be preceded by its
dominants tri tone substitution
so the tritone substitution of G7 is Db
so the same FB but your bass foot can now play the Db
but even if the bass didnt play the G7
if you threw the Db on the end of your FB tritone
it would be like your guitar player was getting fancy and it would create an altered G7
either way it would sound interesting.
principle 6
each tritone sub
can be approached by its relative two
so back to our g7
its tritone sub was Db7
the Db7 relative two is Abmin7
so you would play
GbB - FB = Abmin7-Db7
then EBb = C7
I think that is enough for now.
find a jazz blues and mess around with preceding each two four and five chord
with its relative two
dominant and dominants tri tone sub
and see how much more lively and independent your left hand will start to become,
