My former piano student, u/physicsstudent137, has created a brilliant and simple method for identifying the tonic of a given key signature:
Step 1: Count the number of sharps or flats in the key signature.
Step 2: If an even number of sharps or flats, start on C. If an odd number, start on F♯ (F♯ is the 'odder' note, after all).
Step 3: For flats you go left from there that many half-steps. For sharps you go right from there that many half-steps.
Step 4: (If necessary) disambiguate between enharmonic equivalent note labels (e.g., F♯/G♭). For sharps, the note is the sharp note name. For flats, the note is the flat note name.
Whatever note you land on is the major tonic of the key signature.
I numbered the notes of the octave 0-11 and found that the key signatures can be identified based on whether the note number is even or odd. It's as if there are two sub-cycles within the circle of fifths: one for even key signatures, and one for the odd key signatures. Also, the number of accidentals in a key signature has the same parity as the tonic note. That is, if the note number is odd, then the number of sharps/flats in that key signature is also odd. Same for evens.
The even notes start at the index of 0 accidentals on the circle of fifths and go clockwise in the order 0, 2, 4, 6, 8, 10.
The odd notes start at the index of -5 (aka +7) accidentals and go clockwise in the order 1, 3, 5, 7, 9, 11.
Assuming sharps to be positive numbers and flats to be negative numbers, I found that the tonic note of a given (major) key signature can be determined by adding the number of accidentals in the key signature to either the note 0 (note C) for even key signatures or to the note 6 (note F♯/G♭) for odd key signatures. I sometimes call these two "landmark notes" in this context since they are fixed reference points.
I made a visualizer to hopefully make the pattern clearer (see link at bottom of post). I suggest trying the following:
Step 1: Set the note labels to "Numbers"
Step 2: Click on all of the odd-numbered notes in the circle of fifths in ascending order and observe the pattern.
Step 3: Click on all of the even-numbered notes in the circle of fifths in ascending order and observe the pattern.
(I used G♭ instead of F♯ for the key signature of note number 6 for simplicity, but the pattern still holds for F♯ and can be extended to enharmonic and theoretical key signatures.)
To go the in the reverse direction and answer the question "how many accidentals are in the key signature of this tonic note" is a little trickier, but I just visually reverse the procedure above, either while looking at a keyboard or by visualizing a keyboard in my mind. After a long time of staring at the numbered circle of fifths, I ended up memorizing the note numbers and whether each note is even or odd, so I just ask myself "how many semitones away is this note from its corresponding landmark note?"
TL;DR: Given a number of flats or sharps in a key signature... start on C for even and F# for odd... go left that many half-steps for flats and right for sharps... you land on major tonic. For any mode, start with the mode's prototype on the white notes (eg D for dorian) plus its respective tritone and do the same. At first glance you might hate, at second you will love.
Step 1: Count the number of sharps or flats in the key signature.
Step 2: If an even number of sharps or flats, start on C. If an odd number, start on F♯ (F♯ is the 'odder' note, after all). Step 3: For flats you go left from there that many half-steps. For sharps you go right from there that many half-steps.
Step 4: (If necessary) disambiguate between enharmonic equivalent note labels (e.g., F♯/G♭). For sharps, the note is the sharp note name. For flats, the note is the flat note name.
Whatever note you land on is the major tonic of the key signature.
Try it yourself!: https://editor.p5js.org/remote-files/full/-n651WXmu
Here is his description below:
I numbered the notes of the octave 0-11 and found that the key signatures can be identified based on whether the note number is even or odd. It's as if there are two sub-cycles within the circle of fifths: one for even key signatures, and one for the odd key signatures. Also, the number of accidentals in a key signature has the same parity as the tonic note. That is, if the note number is odd, then the number of sharps/flats in that key signature is also odd. Same for evens.
The even notes start at the index of 0 accidentals on the circle of fifths and go clockwise in the order 0, 2, 4, 6, 8, 10.
The odd notes start at the index of -5 (aka +7) accidentals and go clockwise in the order 1, 3, 5, 7, 9, 11.
Assuming sharps to be positive numbers and flats to be negative numbers, I found that the tonic note of a given (major) key signature can be determined by adding the number of accidentals in the key signature to either the note 0 (note C) for even key signatures or to the note 6 (note F♯/G♭) for odd key signatures. I sometimes call these two "landmark notes" in this context since they are fixed reference points.
I made a visualizer to hopefully make the pattern clearer (see link at bottom of post). I suggest trying the following: Step 1: Set the note labels to "Numbers" Step 2: Click on all of the odd-numbered notes in the circle of fifths in ascending order and observe the pattern. Step 3: Click on all of the even-numbered notes in the circle of fifths in ascending order and observe the pattern. (I used G♭ instead of F♯ for the key signature of note number 6 for simplicity, but the pattern still holds for F♯ and can be extended to enharmonic and theoretical key signatures.)
To go the in the reverse direction and answer the question "how many accidentals are in the key signature of this tonic note" is a little trickier, but I just visually reverse the procedure above, either while looking at a keyboard or by visualizing a keyboard in my mind. After a long time of staring at the numbered circle of fifths, I ended up memorizing the note numbers and whether each note is even or odd, so I just ask myself "how many semitones away is this note from its corresponding landmark note?"
For those who are curious, pedantic, or masochistic, I wrote a much lengthier and more detailed write up, see the link below. Derivation write-up: https://drive.google.com/file/d/1Oyi2s9HowefMtI3-I5A7z10VlDj... Visualizer: https://editor.p5js.org/remote-files/full/-n651WXmu
All credit goes to u/physicsstudent137.
TL;DR: Given a number of flats or sharps in a key signature... start on C for even and F# for odd... go left that many half-steps for flats and right for sharps... you land on major tonic. For any mode, start with the mode's prototype on the white notes (eg D for dorian) plus its respective tritone and do the same. At first glance you might hate, at second you will love.