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I have learned the concept of major and minor chords (as they are played on piano). However, when I look at how the chords are played on guitar, I see that 4 tones are involved instead of 3. The additional tone is a "bass" tone.

So, my question is what is the rule for determining the bass tone?

For example I play D chord from C-major key. So, I would use D, F and A tones and a bass tone? How can I determine what should it be? Is it just the root tone of the chord (D in this case) played one octave lower?

In basic triads, there are only three notes (tri) - on guitar, there's the possibility to play up to six notes, so using the note names, say your D minor, with D F and A notes, just using the top 3 strings will give that. Adding an open D will give you a root position chord - the root is the lowest note played. But you could also add 5th string ope, (another A), to produce D minor again, but with A (fifth note of the chord) at the bottom, giving a 2nd inversion. You could also play bottom string, 1st fret - another F note - which also fits the chord, giving a 1st inversion of Dm, although it's not as convincing and as strong as the other two options - and a bit awkward to play. But it's still Dm!

There is no rule for determining the lowest note, as long as it's one included in the chord, but the strongest sounding is usually with root at the bottom, which is how a lot of guitar sites would have you believe it's the only one. Not true!

I have learned the concept of major and minor chords (as they are played on piano). However, when I look at how the chords are played on guitar, I see that 4 tones are involved instead of 3.

Major and minor chords each contain 3 distinct notes (distinct letters). So a C major chord always contains C, E, and G.

However, the chord can be voiced as you like, with one or more instances of each of those notes spread over the range of the instrument. That's equally true on guitar and piano - there's no difference in what a 'chord' means on each of those instruments, although of course different voicings may sound better, or be easier to play, on different instruments.

So, my question is what is the rule for determining the bass tone?

Simplistically speaking, you choose one of the notes from the chord - which one you choose will affect what inversion of the chord you are playing, but it won't change the name of the chord. Usually you'll choose the bass note (and thereby inversion) for each chord such that the overall result sounds best to you, within the limits of what's practical to play.

If you want to be 'cleverer', you could play a pattern involving two or more of the tones in the chord, and you might even play a bassline that strays outside the notes in the chord, e.g. using a passing note.