Why is it C# D# E# F#, and not C# D# F F#?

Because a seven note scale uses each letter name exactly once. Those are the same seven keys on a piano, but written the second way there are two notes called F, no B at all, and the thirds are invisible. Spelling is what makes written music readable, and the two versions sound identical.

Where do the chords come from?

Each is a triad built from the scale itself: take a note, then every other note above it. Nothing comes from outside the scale, which is exactly why they sound like they belong together. That is what a key is, and it is why the same seven chords keep appearing in songs in the same key.

Why does a pentatonic have no key signature?

Because it is not a key. A key signature says which notes are sharp or flat throughout a piece, and that idea depends on the seven letter names each appearing once. A five note scale skips letters, so it is spelled by pitch here and the page says so rather than inventing a signature for it.

Scale and Key Finder

The notes and chords of a key.

Runs in your browserNothing you enter is uploaded, and it keeps working with the network off.

Root note

A letter and an optional sharp or flat.

Scale

The notes and chords of a key.

How to use it

Give the root. C, F#, Bb.
Pick the scale. Major, a mode, or a pentatonic.
Read the notes and chords. Spelled the way they are written.

When you would use this

A scale is a set of pitches, and writing it down is a separate problem from knowing which pitches they are. The rule for a seven note scale is that each letter name appears exactly once. C sharp major is C# D# E# F# G# A# B#, which looks strange until you see the alternative: the same seven keys written as C# D# F F# G# A# C has two notes called F, no B anywhere, and no way to see that the third is a third. Spelling is not decoration. It is what makes a stave readable, and the two versions produce identical sound. So the notes here are worked out by walking the letters and giving each one whatever accidental takes it to the right pitch, which is how the scale would actually be written. The key signature falls out of that: count the sharps or count the flats. The chords come from the scale and from nowhere else. Take any note, then every other note above it, and you get a triad using only notes already in the scale. Do that on all seven degrees and you have the chords of the key, which is why the same handful keep turning up in songs that share a key. Their qualities are not arbitrary either: major on the first, fourth and fifth of a major scale, minor on the second, third and sixth, and diminished on the seventh. Pentatonic and blues scales skip letters, so the one-letter-each rule cannot apply. Those are spelled by pitch and the page says which rule it used.

Questions

Why is it C# D# E# F#, and not C# D# F F#?: Because a seven note scale uses each letter name exactly once. Those are the same seven keys on a piano, but written the second way there are two notes called F, no B at all, and the thirds are invisible. Spelling is what makes written music readable, and the two versions sound identical.
Where do the chords come from?: Each is a triad built from the scale itself: take a note, then every other note above it. Nothing comes from outside the scale, which is exactly why they sound like they belong together. That is what a key is, and it is why the same seven chords keep appearing in songs in the same key.
Why does a pentatonic have no key signature?: Because it is not a key. A key signature says which notes are sharp or flat throughout a piece, and that idea depends on the seven letter names each appearing once. A five note scale skips letters, so it is spelled by pitch here and the page says so rather than inventing a signature for it.