Hi Bob,

I have never heard of a C2 chord. The notes C,D, and G you are playing and which you call a C2 chord is very likely the first inversion of a Gsus4 chord. The notes for Gsus4 are G, C, and D with the G note being the root. Go to your keyboard and play G, C, and D first, then play C, D, and place the G note on top (the way you play it) and you will hear the sound of a Gsus4 in its first inversion.

What is the difference between a C2 chord and a Cadd9 chord? There is one significant difference. The Cadd9 chord is a major chord and your so-called C2 chord is a suspended chord. Let's see why.

You say you usually do not play the note E when you play your C2 chord. In a C chord, the note E is the third note as we discussed above. The third note of any scale determines whether the scale is major or minor. If you play a C scale and use the note E, you are playing a major scale. If you lower the E note a semi-tone (half tone) and play Eb as you play the scale, you are playing the C minor scale. So the tonality of a scale is affected by the treatment of the third note i.e., playing E natural or Eb.

Now, to play a suspended chord, you play the root note, the fourth note, and the fifth note of a scale. If you play Csus4 for example, the notes would be C, F, and G. Notice the third note (E) is not played. If you added the third note you would have a discordant sound because you are trying to mix a major chord and a suspended chord.
In the case of your C2 chord, you are in fact leaving out the note B of a G major chord and substituting the note D which makes it a Gsus4 chord - not a C2 chord.

Suspended chords have the same effect as dominant chords in that they create tension in music and our ears require that tension to be resolved, hence, as an example, G7 is resolved to C major. A suspended chord will also require resolution to another chord, though not necessarily a major chord.

So Bob, to summarize, your C2 chord should, I suspect, be renamed Gsus4 regardless of the inversion you use. And the primary difference between your C2 chord and Cadd9 is that your C2 chord is a suspended chord and Cadd9 is a major chord.

I hope I have offered clear answers to your questions. If not, let me know.

Most Sincerely, Chuck