Here’s an interesting way to get around the circle of 5ths using 7th chords on the C6th tuning. You need to have a lever that raises your C’s to C#, and another that raises your A’s to Bb.
A7 D7 G7 C7 F7 Bb7 Eb7 Ab7 Db7 Gb7 B7 E7 D____________________________________________________________ E____________________________________________________________ C__12#__11#__10#__9#___8#___7#___6#___5#___4#___3#___2#___1#_ A__12___11#__10___9#___8____7#___6____5#___4____3#___2____1#_ G__12___11___10___9____8____7____6____5____4____3____2____1__ E__12___11b__10___9b___8____7b___6____5b___4____3b___2____1b_ C__12#__11#__10#__9#___8#___7#___6#___5#___4#___3#___2#___1#_ A____________________________________________________________ F____________________________________________________________ C____________________________________________________________ even frets: C>C# lever odd frets: C>C# lever + A>Bb lever + p6
The first position (C# lever only) makes a 7th chord 3 frets above the root position for that chord. For example, F root position is 5th fret, so F7 with the C# lever is at the 8th fret.
The other position (C# + Bb + p6) makes a 7th chords 3 frets below the root position for that chord. For example, F root position is 5th fret, so F7 with this combination is at the 2nd fret.
You can also get a 7th chord at the root position with just the Bb lever. There are lots of 7th chords on the C6th neck.