I had a breakthrough while I was making a tuna sammich the other day. I had to drop everything and fill a page with arpeggios and scales and interval cycles. I think I found a sensible approach to outside playing, a context that justifies all the nasty, dissonant, wrong notes.

I started with a maj7 chord. To find good notes for improvising over a maj7 chord, you could just grab the Lydian scale or take the following approach. Create a maj7 arpeggio from the root: 1 3 5 7. Fill in the other notes using whole steps above the 1, 3, and 5 (a technique that I found yields the most consonant scale). You get 1 3 5 7 | 2 #4 6, all notes from the Lydian scale. The same process with a m7 chord yields 1 b3 5 b7 | 2 4 6, all notes from the Dorian scale.

A well-known jazz technique is to play arpeggios with higher extensions starting on the 3, 5, or 7. Over a maj7 chord, you get 3 5 7 9 (a m7 arpeggio), 5 7 9 #11 (maj7), and 7 9 #11 13 (m7). Now, what if I apply the process above to these arpeggios, working out a new scale in each case? You get 3 5 7 2 | #4 6 #1, 5 7 2 #4 | 6 #1 3, and 7 2 #4 6 | #1 3 #5.

Using the #1 and #5 over a may7 chord seems like a bizarre strategy. Calling anything #1 instead of b2 seems bizarre. (The technical name is probably #15, but this exploration will make more sense in one octave.) This is where the breakthrough makes sense. Most players arpeggiating past the 13 will loop back to 1 because they're sticking to notes in the scale: 9 #11 13 1, #11 13 1 3, etc. But if you're open to the possibility of a #1 note, a whole world of outside playing reveals itself. Besides, I noticed a while ago that raising the 1 and 5 in a Lydian scale over a maj7 chord sounds cool. Really weird, but not necessarily wrong or awkward. And the more I experiment with those notes, the better I understand their sound.

I'll continue the process further:

1 3 5 7 (maj7) | 2 #4 6
3 5 7 2 (m7) | #4 6 #1
5 7 2 #4 (maj7) | 6 #1 3
7 2 #4 6 (m7) | #1 3 #5
2 #4 6 #1 (maj7) | 3 #5 7
#4 6 #1 3 (m7) | #5 7 #2
6 #1 3 #5 (maj7) | 7 #2 #4
#1 3 #5 7 (m7) | #2 #4 #6
3 #5 7 #2 (maj7) | 4 #6 #1
#5 7 #2 #4 (m7) | #6 #1 #3
7 #2 #4 #6 (maj7) | #1 #3 #5
...

This goes on for a while before reaching #7 x2 x4 x6 (x is double sharp), the same as 1 3 5 7. I noticed a few things when I wrote that out. Notes shift to the left without changing. The arpeggios alternate between maj7 and m7. More specifically, the intervals between adjacent notes alternate between major third and minor third. I could write out the whole cycle in a single line, alternating major and minor thirds.

1 3 5 7 2 #4 6 #1 3 #5 7 #2 #4 #6 #1 #3 #5 #7 #2 x4 #6 x1 #3 x5 #7 x2 x4 x6 (those last 4 are 1 3 5 7)

You can make an arpeggio out of any 4 adjacent notes or a scale out of any 7 adjacent notes, but they'll sound curiouser and curiouser the farther you stray from the root.

Here it is over a m7 chord:

1 b3 5 b7 2 4 6 1 3 5 7 2 #4 6 #1 3 #5 7 #2 #4 #6 #1 #3 #5 #7 #2 x4 #6 (those last 4 are 1 b3 5 b7)

I worked it out for a dominant 7 chord as well. I used the same process, but the intervals alternate two major thirds with two minor thirds.

1 3 5 b7 2 #4 6 1 3 #5 7 2 #4 #6 #1 3 #5 #7 #2 #4 #6 x1 #3 #5 #7 x2 x4 #6 (those last 4 are 1 3 5 b7)

In that last cycle, the dominant 7 arpeggio alternates with m7b5, m/maj7, and aug7, so those can be found in there too. I won't bother writing them out.

Now, how to apply all this? I've already been fooling around with #1 and #5 over maj7 chords. Now I know how to use them in arpeggios. I guess the next step is to experiment with the first few "wrong" notes over the other chords. Natural 3 over a minor chord? They don't call it outside playing for nothing. Here goes.

Don't worry if your head hurts; mine did too. It'll go away in a few hours. Think about rainbows. Rum helps too.