Just to confirm, when dropping a key on the guitar tune, would you subsequently change the chords as well? or just play the same chord?
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The answer depends on what exactly you are trying to accomplish by dropping the tuning.
I often drop my tuning from standard E A D G B E to half step flat tuning which is Eb Ab Db Gb Bb Eb (or D# B# C# etc. if your clip on digital tuner defaults to sharps instead of flats). I do this so I can play the songs I want to play with the usual chords for each of those songs but can sing the song a half step lower so it falls more comfortably within my vocal range.
So if you want to tune your guitar lower in order for the singer to be able to sing in a lower key while you play the easy chords say from the G major chord set (as an example) - then you can play those easy G C D chords - but the singer can sing the song as if you were playing in the key of F if you tuned your guitar a whole step flat.
I am not sure why else you might want to tune your guitar a whole step flat, but if your guitar is tuned a whole step flat, you would have to play different chords than written for the song in order to play the song in the key the chords were written for.
So if you are looking at a chord chart/lead sheet/fake book and the chords are written for a song in the key of G (using G and C and D major chords) and your guitar is tuned a whole step flat, you would not be able to play it in the key of G using the G and C and D major chord shapes. If you did, you would actually be playing the song in the key of F instead of G.
If you wanted to still play it in the key of G you would have to transpose the chords in your brain (or write down the new chords) so that you would know that you now must play as if it was in the key of A in order to be in the key of G. So whenever you saw a G chord you would play an A shape which would sound like a G (but look like an A) if you are tuned a whole step flat. To play a C chord (sounds like) on a guitar tuned a whole step flat you would play a D shaped chord.
One easy way to transpose on guitar without changing your guitar's tuning - is to use a simple tool called a capo. There are many different types of capos but they all do the same thing - allow you to instantly transpose a song on guitar to a new key while using the familiar and easy chords you like to use. Here is a picture in case you have not yet learned about the magical capo.
Here is a useful chart you can use to help you transpose to different keys using a capo.
Guitar is entirely relative, assuming you're keeping the tuning the same but dropping all the pitches. If you do this, the guitar will "play" the same way no matter how low or high you go. The important part to remember here is the relativity of tuning. If you're comparing the tuning to standard E (EADGBE) and tuning a whole step down to D (DGCFAD), then all your fingerings and chords will transpose a whole step down. If you play a C major chord in standard E tuning, it will turn into a Bb automatically if you tune everything down a whole step.
If you're playing a guitar that's down tuned a whole step to D and someone asked you to play a C major, you would really be playing a Bb major, so you would need to transpose that C major to your guitar and play a Dmajor shape.
Check out this resource for a more traditional explanation and how it relates to concert (orchestra) instruments.