I know that a C augmented triad is C, E, G♯ and an E augmented triad is E, G♯, B♯. However, I think we can replace the B♯ in the E augmented triad by its enharmonic equivalent, a C. Is it appropriate to treat a C inverted augmented triad (i.e. E, G♯, C) as an E augmented triad in a song, or should B♯ be used instead of C?
Specifically, if the we start with an E major triad and then augment the B, should we get a B♯?
The answer to your final question is yes. Correct enharmonic spellings aside, an augmented triad that is inverted (not “transposed” like you said) twice produces two other augmented triads. The means E+ G#+ and C+ are all augmented triads that use the same notes. When you throw correct enharmonic spelling into the mix, an augmented triad should have a 1,3 and +5. That means using every other letter for the chord tones. That gives you:
E+ = E G# B#
Ab+ = Ab C E (more practical than G#+ but G# can be used)
C+ = C E G#
When thinking of an augmented triad think of a major triad and raise the 5th a half step but keep the same letter. Flat becomes natural, natural becomes sharp, sharp becomes double sharp.
The inversions of an augmented triad are symmetrical, meaning the relative intervals of each inversion are the same, at least enharmonically.
C E G# is two major thirds stacked.
But, technically, E G# C is not two stacked major thirds. It is a major third and a diminished fourth, which enharmonically is the same as a major third.
E G# C is not an E augmented triad, E G# B# is an E augmented triad.
When you get into chord spellings and enharmonic equivalence, you should not just say it's all the same. The more obvious case is the German augmented sixth chord being enharmonically the same as a root position dominant seventh chord, but functionally the chord or very different. The former is a pre-dominant chord, while the later is a dominant chord.
We can make the same observation about augmented triads. Look at the harmonic context. If, for example, you were in the key of F major, and the chord was C E G#moving to an F major triad, you would call the first chord a C augmented triad, rooted on the dominant, a chord acting as a dominant. You don't then willy-nilly, say that chord is the same as an E augmented triad or G# augmented triad.
Specifically, if the we start with an E major triad and then augment the B, should we get a B♯?
Yes. An augmented triad has an augmented fifth above the root.
If you are thinking B# is enharmonically C, and then respell the chord to E G# C, and then call it an E augmented triad, instead of a C augmented triad, you're just misspelling chords.
It isn't really clear what the "problem" is. Is it a notation issue, an analysis issue? Part of the problem is asking about "sameness." That isn't a musical term. Enharmonic equivalency is the musical term. If the question is just general, I would say: spell chords correctly, and enharmonic equivalents are not necessarily the same in the sense of harmonic function.
Very similar to the diminished triads that use the same notes, but often with different names.
The augmented triad uses a root, major third and augmented fifth. Hence C aug = C E G♯, while E aug uses E G♯ B♯. C and B♯ being *enharmonic. But let's not forget the third use of those very same notes - G♯ aug - G♯ B♯ Dx. In fact each augmented triad has three incarnations.
So, why call the same pile of notes 3 different names? It will basically depend on its situation in the scheme of things. In key C, playing C>C+>F, it has to be C augmented, whereas in key E, playing E>E+>A it makes more sense to call it E+.
Not necessarily those triads in those keys, but that's where they are found most. And, as such, their roots will reflect their names, and just like root position of any triad sounds strongest, it's the same with augmenteds.
Augmented chords, in my experience, are used in the way described above far more than any other, so for that reason, this is my explanation.
Yes, an augmented triad is symmetrical, a pile of major 3rd intervals that add up to an octave. (In the same way that a dim7 is a symmetrical pile of minor 3rds.)
There’s no transposing involved in your question. Maybe a choice of enharmonic spellings. E+ is properly spelled E, G♯, B♯. In some circumstances it may be acceptable to spell it E, G♯, C, or even E, A♭, C. It may also be debatable whether it’s harmonic function is C+, G♯+ or C+. Symmetrical chords are often used specifically BECAUSE of their ambiguity.
A B♯ is only inequivalent to a C in theoretical contexts of analysis. For instance, in the context of C♯ minor, you probably want to think of the "C" note of the harmonic minor scale as a "B♯" because it fits better within the theory of scales and triads. Disregarding any of the contexts of harmonic analysis, C+, E+, and G♯+ (A♭+) augmented triads are all the same chord, just inversions of each other.
Interesting note: The augmented triad divides an octave into three even parts. Therefore there are 12 / 3 = 4 unique augmented triads. A fully diminished 7th chord divides an octave into four even parts. There are 12 / 4 = 3 unique fully diminished 7th chords.
Each augmented triad has three enharmonically equivalent versions. Taking the case of C augmented, one has
The importance of how the chord is spelled relates to the context in which it's used. For example, a common chord progression is I - I+ - vi, which in the key of E would be E - Eaug - C#m. By spelling the E augmented chord E - G# - B#, the performer is cued into the larger harmonic context. Spelling the augmented chord E - G# - C, would suggest, perhaps, a key change.
In harmonically ambiguous contexts, or sometimes just for readability, an augmented chord will be intentionally "misspelled", but these are special cases.
