Lennie Tristano’s Augmented 15th Arpeggio
Above is the “Classic’ Lennie Tristano superimposed #15 arpeggio for C Major7. The #15 sounds “Correct” because all of the 5ths in the arpeggio are PERFECT 5THS.
EXTENDING AN ARPEGGIO TO THE 23RD
TERTIAN HARMONY
If we keep stacking 3rds with superimposition and extension we create “Tertian” harmony. So, above a 15th we have a 17th, above that a 19th, above that a 21st and above that a 23rd. Hence a 23rd chord will use all 12 notes.
The 23rd chord example below has a major leaning
This example provides a 23rd chord with a minor leaning
Here we have a 23rd chord with an augmented leaning
Lastly we have a 23rd chord with a diminished  leaning
Going back to our old friend George Russell and the Lydian Chromatic Concept we become aware of the shifting of tonal gravity and what is available with a 23rd chord that shifts out and how we can manipulate it to create new ideas and avoid cliches. Also going back to our 1st example of Lennie Tritano’s #15th arpeggio we hear how we can exploit the perfect 5ths to make something not right “Sound” right but fresh and new.
How to create music and improvisation with the 23rd chord
Below is a great example of my personal favourite 23rd chord creating a lovely Tertian Stack. 
This example also reverts back to our 12 tone blog for improvisation and jazz heads/tunes. There are 3 sets of TETRACHORDS. Or 3 sets of 4 note groupings.
Next we can take the same 23rd chord pitches and make Hexatonic or 2 groups of 6 from the 23rd chord tertian superimposition.
Finally we can make 3 note cells to form a 12 tone line from the same tertian 23rd chord extension.
23rd chord Improvised line ideas:
Employing 23rd chord outlines in Major, minor, diminished augmented.
Line 1
Line 2
Line 3
Line 4
 