As far as I know transpose should only really be used with absolute pitches, as it would be quite impossible to do it otherwise. Think of it like this: Say we are in relative mode and have c' f. Then f will be above c. If you then do c ' \transpose f a f, then the transposed a would be above the c – so it would need an octave indication. But the octave the music has to be put into depends of the previous note. So a single transpose function cannot provide this.
So, always use absolute pitch with transpose! And by the way: Lilypond transposes you example correctly. Your main problem is that usually there should not be a C# in C tonality. Usually it is easier to go from C to Db, which is a minor 2nd instead of an augmented prime. Cheers, Valentin
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