In 60 yrs of playing, I've rarely played only a 5 string version of an open C.

If I am strumming, I usually just deaden the 6th (lowest) string with the finger that is holding the C on the 5th string (unless I want the low E to sound, of course). Partial strumming is possible, particularly for professional players, but not something I'm very good at. Someone who doesn't play guitar probably wouldn't notice.

@tim I've only been playing 40-some years but I never play an open (low) E string with my C chord. I mute it with my thumb or ring finger. Not that either way is 'wrong', it's just how I was taught and what I've seen in most music manuals.

You are altering the chord pretty fundamentally if you put an E in the bass, especially if the guitar is the lowest instrument in the ensemble. It matters somewhat less if you put a G in the bass, but the result is basically the same: You're no longer playing a root position chord. This doesn't always matter, but it's going to mess up your voice leading in the bass pretty good and that often does matter. Indeed, the only reason this mistake is often permitted is because a lot of people don't hit the strings particularly evenly so the bass E isn't very loud.

@Fugu Re: “it matters somewhat less if you put a G in the bass”; that is highly style-dependent. In most classical music, a second inversion chord is considered dissonant (because of the fourth on the bass), whereas a first inversion obviously sounds different but has the same function so could conceivably be substituted.

@Glorfindel I would suggest to replace "higher tones" with a more accurate wording, like "octaves" or "multiples (in frequency) of the tones in the chord". For someone learning the basics of western music, it might not be obvious.

@Tim well it depends on the style of music you play. I barely ever have 6 string C-Chords in written music, almost always its a 5 chord one. It's a question of style. There's no facts there.

@Fugu - there are some country type songs which require a 1-5- bass pattern with a chord on 2 and 4. The C with G under (2nd inversion) is ideally voiced for that. And as far as voice leading goes, playing the bottom string open on a C chord before going up a 4th (to F) sounds like pretty good leading to me.

Despite all the upvotes, you have portrayed the guitar notes played on the piano one octave higher than they actually sound. The fifth string 3rd fret C is actually one octave below 'middle C'. And, he could easily be playing all 6 strings - despite there being a huge movement saying it's not good.

@11684 Agreed; I was just making the general statement that a second inversion chord sounds more like a root position chord than a first inversion chord.

@Fugu Maybe in some styles this is true. However, my point was exactly the opposite; in classical music a second inversion chord is functionally completely unrelated to the root position. A second inversion chord would (most likely) have dominant function, whereas a root position could be anything.

@11684 Yes, I too took a functional harmony course. I'm not talking about function, I'm talking about how it sounds. Call it subjective if you want. Besides, you can't generally substitute a root position chord with its first inversion in classical harmony anyway. Considering the thesis of my comment was "playing a chord in an inverted position versus a root position matters" this whole thing feels like semantics to me.

@Tim: How do you play an open D chord? Do you also let 6 strings ring?

@EricDuminil - ha! On the odd occasions I play open D, It's going to be a 6 string 2nd inversion, which is with 6th string 2nd fret as well as the usual triangle. Why do you ask, since the question is about a C chord?

@EricDuminil - not sure where that info came from, but the open C I would use would be C/G