Why there is a single-flat key signature for a F major triad?

Question

I am reading a book for beginner about building the chords, basically on triads so far. It is said that the major triads is formed by a root, third and a fifth. From that instruction, it is easily to setup a C major triad by making the root as C; to get the third, I add 4 half steps to C so to get E; to get the fifth, I add 7 half steps to C so to get G. The C major triad is shown on the left of the following stave

Similarly, I am getting the F major triad by using a F note as the root, A as the third and C as the fifth, so F-A-C be the F major triad, correct?

In the book, the author introduces anohter way to construct major triads by taking the first, major third and fifth notes from a major scale. It then gives me F major triad as shown in the right side of above stave. The book does not tell why there is a single-flat key signature added. I am trying to reason it by listing the C major scale, C-D-E-F-G-A-B, from that, the first, major third and fifth notes is just C-E-G but this does not give me F-A-C. However, I find that if I lower the F by half step, I get the E, which may get E-A-C by adding one-flat signature on F. This still does not show me why the RHS staff is connected to the F major triad.

Also, is it any noticeable difference between the F major triads with and without the flat key signature? I tried to repeat the following pattern in a virtual keyboard and listen very carefully, I can't tell the difference honestly

(I am trying to search the related questions on generating F mjaor triad but it comes up with few threads with some terms I don't understand yet).

Answer 1

The key signature shown there doesn't affect the notes in the triad, as you've correctly worked out. It's just there because it's the signature for the key of F major. The author has just chosen to show the one flat of F major because that's the key they're talking about: it might be confusing to see the F major triad in a C major key signature. It doesn't affect the way you play the triad at all.

It would make more difference for B♭ major. The key signature for that has two flats, on B and E. Obviously the root is on B♭. So if you write a B♭ major triad, you have to either write out the key signature or use an accidental.

About the connection between the triad and the key signature: each major scale has the same sequence of steps and half-steps: TTSTTTS (where T is a whole tone and S is a semi-tone). This is what links the root note of a major scale to its key signature, and it's probably described in more detail later in your book. In C major, this sequence lines up with the pattern of "white notes" and "black notes", so there are no sharps or flats. The F major scale contains a B♭ (as its fourth), so the key signature has a B♭. It just so happens that in this case, the notes of the triad (F, A, and C) don't have any sharps or flats. In some scales (such as B♭ major or D major), the notes of the triad do contain some that are sharp or flat.

Answer 2

Triads are indeed formed from 1, 3 and 5 of a key, or the scale from that key. So in key C, 1 3 and 5 are C E and G, as you state.

On to the F triad. Exactly the same idea - but now we count from F. 1 3 5 is F A C.

Do the same for G, if you like - G B D. It all works as simply as that.

However (there's often a however!) sharps and flats in certain keys can affect which notes are actually played. Take A major triad. Count up A C E. Wrong! Because there's the key signature of 3♯ to consider, making that C into C♯.

On to your example. In key F, there's the key signature of one flat - B♭. That goes at the start of each line of music. Another however - that B♭ isn't played in an F triad, so is a red herring. I guess it's put in that music to show it's in key F. Confusing! But some would say it's technically correct. I wouldn't as we get an F chord in key C, so don't necessarily need to show that key sig. of one flat.

EDIT: noticing the last para. - don't know why the B♭ has changed to A♭, but now you have F minor and F major triads. They sound very different to most people, and ought to to you. If not, there's a serious problem with your discerning the difference!