The simple answer is in a demonstration. Go to a piano. Play a C. Knowing that a flat lowers the affected note by a half-step, play a C flat. Now play a B. Since the B is a half step below the C, C-flat and B are the same note. Any time this happens, it is an enharmonic spelling. 2048 Infinite uses random enharmonic spellings for the three notes that are the most useful to learn to spell enharmonically: B-Cb, F#-Gb, and Db-C#. We have the game switching between these spellings to make you learn them! They are some of the hardest to learn because they are so rarely used. But without them, you will not be able to think your way around the bottom of the circle of fifths.
In the image below, we show all of the most common enharmonic spellings. The beamed notes are enharmonic spellings of each other:
But it gets a little more complicated than that. (This post gets pretty heady at this point. You’ve been warned!) If you are playing an instrument on which you can make small modifications to your pitch, you should bend the note slightly in the direction that the accidental (the pitch modifier) is already going. This is because different intervals tune differently depending of where they are in the chord. For example, let’s say you are in the key of C and you see a B-flat. Most likely, that note is going to be part of a dominant seven chord (borrowed from the key of F), and your pitch should be very low. Do this, and you’ll be in tune. The degree to which you raise or lower your pitch depends on the harmonic context and really just comes down to listening. This concept is called just intonation. Use it!
And now for the nitty gritty stuff that theoreticians and composers like to argue about. The fights I’ve seen are really quite amusing. Is a C-flat equal to a B? Theoreticians (who tend to be more traditional) would say no. Composers (this one, anyway) would say it depends on the time period on harmonic context. Back in the day (1890’s and earlier) nearly everything was tonal. The purpose of equal temperament was for all keys to sound great on instruments that could not be tuned easily (piano, mallet percussion, and others). Composers bent over backwards to make sure they were always in a key. Then Schoenberg happened. When we play atonal music we strictly use equal temperament. Why? Because there’s no key to tune to! Sharps and flats are simply a way to tell the performer what note to play and nothing beyond that. Now, the circle of fifths is technically a tone row. So really it should be interpreted by modern theoreticians as such when no other context is given. The circle of fifths is a very broad tool and should not be limited to the tonal system when using it for analysis.
The circle of fifths is about more than key signatures. It’s a method of understanding sound in general. When teaching the circle of fifths without context, an F-sharp is the same as a G-flat.