It would take too long lay this out for you guys… but in a MAJOR KEY, there ARE 3 MAJOR chords, 3 Minor chords and 1 Diminished Chord
Major is written in Roman Numeral CAPS
Minor is written in Roman Numeral lower case
Diminished Chords use a degree “⁰” symbol
Augmented Chords use a plus “+” symbol
Key of F Major has 1 flat (Bb) & uses these notes:
F,G,A,Bb,C,D,E (F is Root because it is #1)
Major Chords in key of F Major:
I (F,A,C) because F is the Root of F Major scale
IV (Bb, D, F) as Bb is the 4th of F Major scale
V (C,E,G) as C is the 5th of F Major scale
Minor Chords in key of F Major:
ii (G,Bb,D) as G is the 2nd of F Major scale
iii (A,C,E) as A is the 3rd of F Major scale
vi (D,F,A) as D is the 6th of F Major scale
Diminished Chord in key of F Major:
vii⁰ (E,G,Bb) as E is the 7th of F Major scale
So… what makes something sound “sad” isn’t necessarily a minor chord per se (because Major Keys have Minor chords) but where the chord progression leads or ends will make it sound that way.
If the key is minor, the root (1 chord) is minor and the positions and Major chords and Minor chords shift.
To make it easier on myself, the Key of F Major has a relative Minor key which is always 3 semi-tones lower than the Major key: D minor - because the notes are the exact same but shifted: D,E,F,G,A,Bb,C (D is Root as it is #1)
So…
Major Chords in key of D minor:
III (F,A,C) as F is 3rd tone of D minor scale
VI (Bb, D, F) as Bb is 6th tone of D minor scale
VII (C,E,G) as C is 7th tone of D minor scale
Minor Chords in key of D minor:
iv (G,Bb,D) as G is 4th tone of D minor scale
v (A,C,E) as C is 7th tone of D minor scale
i (D,F,A) because D is Root of D minor scale
Diminished Chord in key of D minor:
ii⁰ (E,G,Bb) because E is 2nd tone of D minor scale
So… The “sad” sound is due to the shift as most minor keys would resolve to their root chord - which in this structure is minor.