Thanks T for that info, I had understood that I was for major and ii is for minor but never ever saw an I6 before, but that is good to learn just in case i do ever run across that, how popular is that notation now and days?
You're probably only encounter this notation in a music theory class. However, it's good to know from a historical perspective, seeing where the number system came from to what it is now. I'm glad that my posts are helping you become a better musician man.
Okay, good question dwest. Now that we know the "what" and the "when"... I need to know the "why". Why in I6 is 6 used to name the first inversion? There are no "6's" in the chord you named so it will be hard to remember w/o knowing why a 6 means the first inversion. Also, although you said it's the first inversion, you gave three different chords. I noticed the "E" is in the root of all those, so does the root/"E" make it the first inversion or is it the chord you're playing in right hand?
To answer the first question, the 6 is being used because there is the interval of a 6th between the bottom note (3rd) and the root. In the case of the C major first inversion chord, the notes E-C form the interval of a 6th.
A more formal name you would also see is I6/3 to include the interval between the bottom note (3rd) and the middle note (5th). In the C major first inversion chord, the notes E-G form the interval of a 3rd.
If you were looking at a piece of sheet music, it would make more sense visually. Keep in mind that all of this came about because in order to study something, you need a system. The system they used involved Roman numerals.
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To answer the second question, the root makes the chord a particular inversion no matter what other notes are being played. So, for the C major first inversion chord, having the E in the bass put the chord in first inversion, even though I was switching chords in the RH. You can break it up LH and RH in order to teach the voicing of the chord to someone, but ultimately the bass is gonna name the inversion.
This is where chord symbols come in handy. If you see a chord symbol like this C/E, the only information that you have is it's a C major chord and the E is in the bass. There is no clue as to how to voice it, just that the E has to be the lowest note. This gives you flexibility to play the chord however you like, but just keep the E in the bass.