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.
You get e from the primes by averaging the set size of only growing gaps (2,3,5)(7,11)(13,17)... or only non-shrinking gaps (2,3,5,7,11)(13,17)(19,23,29)...
Not a property of primes per se, just a property of growth. It works better with a set of randoms.
Chess.com seems to have excellent opening instruction. It actually tells you the opening name as you play. After you play, it gives you more. It shows where you screwed up, gives you a chance to find a better move, etc. If I was starting over (I'm over 2k now) I'd start with chess.com's interactive organic approach. Chessable is just rote memorization which is in essence trash.
After reading the complete Shakespeare, it is a natural continuation to go on to read the complete Milton. Outside the core of Paradise Lost/Regained, Samson Agonistes is another gem, and Lycidas is of course a classic poem.
The way Milton uses English the language exalts it into the profundity of Latin. One feels Milton's words are of sanctified origins; it feels as though they are coming from a higher power.
One random connection: In the same way Milton dictated his writing, Debussy dictated his music. To some extent this method may free the mind's work from the tedium of transcribing thought, and allow for more free-flowing expression in creation.
Robert Wilson theorizes that each character in Sideways symbolizes a type of wine.
Miles = Pinot noir (thin-skinned, great potential).
Jack = Cabernet Sauvignon (adaptable, survivor).
Maya = 1961 Cheval Blanc (Miles' prized bottle, actress born 1961).
Stephanie = Cabernet Franc (hollow, flabby, overripe, shallow).
https://www.youtube.com/watch?v=L0mO0UhBim4&ab_channel=Rober...
It was a divine move in the sense that he created a surprise tsumego that crashed AG. Lee Sedol called it his 'trick move' in the post game interview I believe. Move 78 shouldn't have damaged AlphaGo's moyo if it had responded correctly. See diagram 9 here for an explanation: chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/viewer.html?pdfurl=https%3A%2F%2Fdeepmind-media.storage.googleapis.com%2Falphago%2Fpdf-files%2Fenglish%2Fls-vs-ag4%2FLS%2520vs%2520AG%2520-%2520G4%2520-%2520English.pdf&clen=4258493&chunk=true
Move 78 was humanity's last great stand against AI in a board game. Lee Sedol, tired and inspired, reddens AlphaGo's ears with a move plucked from a higher dimension.
It now seems humorous that Kasparov once accused people of helping computers behind the scenes. Now chess masters have been caught huddled in bathroom stalls with their smart phones. Chess commentators choose to willfully ignore chess engines in their presentations, in order to enable our understanding of the analysis. The torch has been passed.
The OEIS can also be considered a sort of number encyclopedia, especially for large numbers.
It can also be interesting to link geometric observations with unrelated OEIS sequences. For example, the number of rectangles on a square grid turned out to be the octagonal pyramidal numbers.
If they don't cap the apm at 300 or so, this means nothing. I am not impressed with super speed, only decision making matters. Deepmind, you can do better.
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.