Is an inversion a different chord, or a different voicing?
If I play C4, E4, G4, it's a C-major chord. First inversion would be E4, G4, C5: is this considered a different chord, or just a different voicing of the same chord?
I'm thinking specifically for finding chords on a fretboard: do I need to make sure the root is the base note of the chord, or is it the same chord either way?
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It's not a different chord, and in terms of guitar I wouldn't even think of it as a different voicing. It's an inversion! That's what the term is for, to describe taking a particular chord structure and changing the order of notes so that a different note is on the bottom. Guitar voicings tend to have specific structures, like closed triad, spread triad, drop2, etc. Just think of it as an inversion, which is OK to use because you're probably not playing much of a bass line on the guitar anyways.
Unless you're going to make a clear and distinct bass line on the guitar, your role as a guitar player is not necessarily to lay down the bottom end. Most guitar players in an ensemble would consider a C, C/E or C/G to be equivalent for ensemble purposes AND avoid a voicing which falls beneath around fret VII of the sixth string or above around fret IX of the 1st string, because the function of the guitar in an ensemble is normally to play the meat of the chord and the bass player would play the bass. So if a bass player plays the note "C", then it doesn't matter what 'inversion' you play, the whole thing is still a "C" chord with "C" in the bass.
For just a single guitar accompanying, inversions are pretty much equivalent to the written chords, and playing a different inversion of a chord than the one written would not normally be considered a big deal. Especially in huge 6-string open chords, it won't affect the harmony that much. The real question is if your accompaniment has a clear bass supporting it or not, whether played by the guitar player or someone else. If it does, then you start to think more about what the low note is for a particular chord.
For examples of guitar accompaniment styles that have this feature, check out
Most singer songwriters who are any good at playing guitar
Bossa Nova guitar accompaniment pattern
Walking bass + chord pattern for guitar
The root-5 bass pattern ubiquitous on any Chet Atkins recording
These examples may be a bit dated, but they should give you some idea of the bass+accompaniment style.
Dr Mayhame is right, but I'll explain a little more. If you have a C, E, G, and no other notes it is a C major chord no matter what order it is on, how far apart the notes are, or if notes are doubled. The only thing that changes is the voicing of the chord which can make the same chords sound totally different.
The function of a chord changes depending on the inversion of the chord, but since you are playing guitar most of the time there will be an instrument under you (typically bass guitar). Because of this you don't have to worry too much about what note is the lowest and you can focus more on what voicing you like.
The first inversion of a chord is considered to have the same function as the chord in root position. This is esp. true for the primary triads: tonic, subdominant and dominant.
That said, the first inversion can help affirm the function of a secondary triad (substitution chord).
For instance, in C major, both Am and Em are substitution chords of C, so both Am and Em can have tonic function. But Am is also a substitution chord of the subdominant F, so Am can also have subdominant function. The first inversion of Am, c-a-e has the C in the root, and because the c is in the bass, it will be more effective as tonic than it will be as subdominant.
The second inversion of a chord is not considered to have the same function. That's because the second inversion has a (dissonant) fourth interval that needs a specific resolution.
The most abundant second inversion chord is on the first degree, and it will typically work as a suspension leading to the dominant chord. So in C major, the 2nd inversion would be g-c-e and it will typically go to G (g-b-d). Even though the 2nd inversion g-c-e is a chord of C, it does not have tonic function. You will notice immediately if you try to substitute the final chord of a song for its second inversion - it doesn't sound finished, because the second inversion does not have tonic function.
With regard to voicing: voicing is the exact placement of the notes, including any omissions and doublings of notes. The inversion is a more general concept that only says something about which chord tone appears as bass tone. Any inversion of a chord can be voiced in many different ways. But once the inversion is set, the voicing does not change the function of the chord - it merely colors it.
It is not considered a different chord. The name is still the same, the notes are still the same, they are just in a different order - so they are effectively a different voicing.
They will sound different, which is why inversions are used - you can impart a number of different flavours of sound to a piece of music.
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