#1
I have noticed something quite strange when dropping from E standard to Eb and vice versa. When I tune from standard to half step down, the strings that I tune first become slightly sharp as I tune the other strings.

For example, I would first detune the low E string first, then by the time I have tuned the high e, the low E would be slightly sharp. As would the rest of the strings, but less and less sharp as they reached the e string. The order would be the opposite if I first detuned the high e string. The same would happen when tuning up from Eb to E. Only this time they would go slightly flat.

I would have to keep retuning the strings (maybe once or twice) by small adjustments until they are all in tune. After that, they're all held perfectly in tune.

I'm wondering if anyone knows what the hell is going on here? I've last had my guitar set up last year August and there doesn't seem to be any problem with the neck. It's not a major problem, but it makes tuning tedious sometimes, especially if I want to play in drop tunings. I would have to retune some of the strings if I dropped the low E down a step.
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Epiphone Les Paul Standard
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Marshall BB-2 Boost/OD
Joyo JF-07 Classic Flanger
Joyo JF-37 Analog Chorus
#3
Could be the truss rod bowing backwards due to the decrease in tension when you down tune, and then bowing forward again when you tune back up
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#4
Actually, my guitar teacher told me this would happen a while ago, told me to always tune frpom the lower E up. I'm not sure how this helps because he was very vague describing it to me
#5
It's a lot more rough on a Strat with a floating tremolo! It just has to do with the slight increase/decrease in tension when you tune strings up or down.

Similar to Brian Mayden's suggestion, you could try tuning by alternating high and low strings. E-E-A-B-D-G to try to keep tension a little more even across the fretboard.
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#6
It's quite normal, actually. Same thing happens with my basses and it also happens when you have a floating bridge. Due to the change in tension on the neck it bends back (or forward) a little, so the other strings go out of tune a little bit.

When you have a floating bridge (like a Floyd Rose) the bridge moves backwards when you drop-tune your low E, causing the other strings to go sharp. So you have to keep tuning till you got it right!