The random singer can be a dangerous beast but sometimes its the only thing to do!
I have to transpose "in my head" because I play manual bass pedals which don't trasponse.
The relative key is not usually a problem; I seem to learn songs "serially" so once the key is established the rest of the song is played automatically; I appear to remember the chord intervals as the song progresses!
Nevertheless it is strange how you usually end up in Db... or F# Minor!
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John Allcock