While I understand that it is very important to include the root note in a chord, is it required?
Let's say I'm in C Major, and I want to play a I chord. Do I need to include C in order for it to be considered a I chord? Or is the chord progression enough to imply that the chord I'm trying to play is a I chord?
Additionally, if I can have such a chord, what is it called, or how is it notated?
There does exist what we call "rootless" voicings in harmony. These are chords in which the root is implied by the upper harmonies. Typically, the 3rd and the 7th are the primary indicators of chord quality, and the 5th is secondary. Rootless voicings are most commonly used in settings where an instrument such as piano or guitar is providing harmonic support in an ensemble/group setting, and an instrument (usually bass) has already stated the root of the chord. In composition, rootless voicings are utilized to keep the tonal center ambiguous and allow for the possibility of shifting the root of the chord underneath the harmony to create interest. Here are some examples of rootless voicings:
If you were to play the rootless voicings in the above picture without their root, the placement and use of the chord tones would imply the root. Why does this work?
Implied Roots
As I said earlier, the 3rd and 7th give the strongest indicator of the chord quality, and the 5th & 9th are the usual secondary indicators. With that in mind, chord quality can be determined without a root when the following tones exist within the chord:
Major => Maj. 3rd & Maj. 7th
Minor => Min. 3rd & Min. 7th
Minor Major => Min. 3rd & Maj. 7th
Dominant => Maj. 3rd & Min. 7th
Diminished 7 => Min. 3rd, Min. 5th, Dim. (bb) 7
Half Diminished => Min. 3rd, Min. 5th, Min. 7th
Augmented Major => Maj. 3rd, Sharp 5th, Maj. 7th
Augmented Dominant => Maj. 3rd, Sharp 5th, Min. 7th
Experiment with playing these chords & intervals without the root first, and then with the root -- you will become more familiar with the idea of not relying upon the root itself to express the tonal center of the chord. Good luck!
If some other instrument is playing the root, such as a bass, then that root note is in the chord that's being played by the whole band. If you're talking about playing a chord on, say, guitar or keyboard, then without the root it will sound odd, as it's not really a chord, which traditionally has 3 notes minimum (guitarists will disagree and talk about a '5 chord').
Don't forget, though, that chords may have more than 3 notes : let's take Cmaj7. When it's played without the root C , we're left with E-B-G. This happens to constitute an E minor, which is not, obviously, the same chord as the C major. Let's take a G7 chord - G-B-D-F.Sounds, funnily enough, like a dominant 7th. Take the root G away, and we have B-D-F, which is a minor flat 5 chord - different. In context, they may or may not sound similar but the moment the bass puts in the expected root, it's G7.
To answer the question, it's used, and is called C (no root) I've not come across it often, but generally, someone else will be putting in the missing note.
It's important that a chord contain the root of the chord, but not necessarily the root of the underlying scale. A G Major chord played as part of the C Major scale will not contain the note C.
Occasionally, you don't need to include the root of the chord either. Particularly, if the note is covered by another instrument (or vocals) then the guitar part (or other harmony instrument, like keys) may omit the note.
A C Major chord with no root would be notated "C (no root)" or "C Maj (no root)".
Technically, what remains is a dyad (E and G) which some consider to be an incomplete chord. But depending on the context, this dyad can serve the function of the full chord, and so it's appropriate to designate it a chord.
Yes, it is possible to lop off notes. Practical : for Major chords, play the 3, 5, and 6 of the chord, alternative : play the 5, 6, and 1 (root). You will hear that these same notes "become" the 5, flat 7, and 1, - alternative : flat 7, 1, and 3 of the Relative Minor. This is not the only way to take off notes, but it is a practical way of experimenting.
As an example for rootless chords used in a "standard" situation, accordions with just 3 chord button rows (either because they only have 5 rows altogether, or because they have 3 instead of 2 bass rows) have a "joint" major seventh and diminuished chord row: instead of C-E-Bb and C-A-Eb (the seventh chords are generally missing the fifth, so this is c7 and cdim) they have just G-Bb-E which serves as both c7 and gdim (the octave of the chord notes is unspecific). So for a typical Oom-pah accompaniment of C-c-G-c-G-g7-D-g7 the "g7" will actually consist of D-F-B.
Most players of accordions with just three chord rows (typical for French and Russian accordions) are not aware that their seventh chords are missing the root note and are actually diminuished chords (without fifth). Particularly since the chords are usually spelled as c7 et al and are in the same location as a c7 would be when having four chord rows.
And the effect is effectively indistinguishable from "true" seventh chords since the Oom-pah pattern will provide for root and fifth presence regardless of which of the two may actually be missing in the chords.
You can certainly IMPLY a chord without including its root. Look at Bach's 2-part inventions. Plenty of harmony is clearly implied with just two notes. Context can make an implication pretty solid. If it looks like a duck, hangs around with other ducks and goes 'Quack', chances are it's a duck. But unless all notes are included, we can't definitely state it IS that chord.
(Let's not get tied up in special cases like the 3rd not existing in an 11th chord. Unless it's a #11 the chord's very likely a sus4 or a F/G shape anyway.)
