OK..with the several recalculations for ledoux1238, I figured there has to be a way to generalize the setup for all room speaker and listening position placements..in essence, as known in my former life, a generalized calibration curve. A benefit to the repeated room calculations (and necessary condition) is they offer the base-data to generate an empirically-derived predictive model. Not surprisingly, there was a way to do this and it's pretty basic. What was surprising, was that it took ChatGPT about 15-20 seconds to churn it out. The usual response time is instantaneous or a second or two. While I'm new to the A-i thing, there's some amount of satisfaction in asking a question that takes that long to answer. The whole curve thing, in hindsight naturally, is quite simple. Also in hindsight, there's no way I'd have ever taken the time to work this out manually, though it's do-able, as it would have taken (me) waaay to much time. This is a case where humans(supposedly the "smart" ones) ask questions and machines answer them.
(Forgive my continued flogging of this (setup) dead horse. This is my idea of fun..and something useful to do while listening to music)
First we need to define a Space Ratio (R)
R = (on-center speaker spacing) / (listening distance)
R = (spacing) divided by (listening distance)


The chart below can be used to self-check on whether you're using the calibration curve correctly. (locate your unique ratio R on the x axis, go vertically the dotted line, then travel horizontally to the y axis to read your toe-in angle)

Quick mental shortcut (no math) — Behind-head version
You can estimate toe-in instantly:
- Speakers narrower than the listening distance
→ ~18–20° - About equal spacing and distance
→ ~23–25° - Speakers wider than listening distance
→ ~25–28°
What the calibration curve really means:
As R increases (wider speakers or closer listening):
- Toe-in increases
- Lobe steering becomes more critical
- Sweet spot gets narrower
As R decreases:
- Toe-in decreases
- System becomes more forgiving
- Imaging stabilizes naturally
I'm having way too much fun with this..

