Just finished a bottle of wine while watching Rudolph, so pardon any typos.
I write about arpeggios on a regular basis. I think they're important. I use them primarily as a framework for finding consonant tones over a chord. They're also good for rapidly moving over a large pitch range. If you want to transition quickly to a note in a higher register than where you're currently playing, ascending by arpeggio will sound better than straight up a scale.
Until this week, I've practiced only 7th arpeggios. I've done a lot of work on 1-3-5-7 and 3-5-7-9 arpeggios, and I'm sure they'll always be the foundation of my approach to improvisation, but I can't ignore the appeal of the monster 1-3-5-7-9-11-13 arpeggio. It's a cycle of 3rds, major or minor, based on the applicable chord or scale. Any 7th arpeggio ascends by 3rds until the 7-1 interval, a 2nd. The advantage of the 13th arpeggio is that every interval between adjacent notes is a 3rd, even 13-1.
The easiest way to apply a 13th arpeggio is to play every other note in a diatonic scale fingering. It works over any 7-note scale: any mode of the major, melodic minor, or harmonic minor scales. But watch out for awkward extensions. The 11 and 13 might give you trouble if you're not familiar with the differences between 11 and #11 or 13 and b13 over major, minor, and dominant tonalities. I use the most consonant scales (traditionally) as a starting point.
This week, I've been using these arpeggios over major 7 (1-3-5-7-9-#11-13), dominant 7 (1-3-5-b7-9-#11-13), and minor 7 (1-b3-5-b7-9-11-13) chords. Respectively, I use fingerings from the Lydian, Lydian Dominant, and Dorian scales to construct these arpeggios; I just play every other note. I'll soon explore alterations on the 5, 9, and 13 over dominant and Melodic Minor over minor. Just getting them under my fingers for now.