There are some related answers (like "What is the formal definition of 'transpose'?", and "What is the idea of "transposing by perfect unison"?"), but they didn't answer my question, unfortunately.
That's a tricky question, let's give it some context first. Every note has three pieces of information related to its pitch, — octave, letter, and alteration (we're ignoring frequency):
Similarly, every (conventional) interval has two pieces of information that define it completely, — type and kind (I'm not sure about the terms though):
- minor third (a third, minor)
- perfect unison (a unison, perfect)
- doubly-augmented fifth (a fifth, doubly-augmented), etc.
If I were to be asked to transpose any note to any interval, that won't be too hard:
A♮4transpose up by a minor third →
E♭2transpose up by a perfect unison →
G♯5transpose down by a doubly-augmented fifth →
I'm trying (and failing!) to pin-point the exact steps that I take to perform these transpositions, the way I'm manipulating these five pieces of information to produce the resulting note. Can somebody help me figure them out?