Fake fingerings are an effective way to slow down harmonic rhythm without losing momentum. Subtle timbral shifts means the ear feels eighth notes, but you haven’t yet moved anywhere melodically. Michael Brecker and John Coltrane used fake fingerings extensively, especially in high-energy modal settings. The timbral and slight tuning differences also add sonic detail to your phrases with wonderfully folk-ish imperfections. See my fake fingering suggestions, but come up with your own, too.
Why did I start in E? Because that A was the lowest fake fingering I could find on the horn. Why did I choose to transpose each measure up a whole step? Because it sounded best to my ears. (copy that sentence into your diary or whatever and read it every day) Practice this pattern up a half step to hit every key.