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On Facebook XAII, Aidin Foroughi asked:
"Quick newbie question again.. Why does the augmented chord in 12TET sound so bad? I mean, the diminished is obvious but the intervals in an augmented don't theoretically sound as bad as they do in reality .. where am I wrong?"

People have disagreed with him, mentioned 3HE, etc; and asked him, "what about C-E-G#-B", etc.
I'd like to ask "what about D-F#-C-E-G#" (but I'm not on Facebook).

Steve M.

Hi Steve,

I'm assuming that with "bad" you / Aidin mean dissonant.

The augmented chord is the most enharmonically remote chord one can indicate with 3 notes without further context.
Meaning, that if one holds C-E-G# on the piano, there is no enharmonically equivalent simpler way to interpret it, C-E-Ab and C-Fb-Ab are all the same chord only in inversion.
The most remote interval in the chord is the diminished fourth / augmented fifth. It is an interval 8 fifths wide, augmented prime is 7 fifths wide, augmented fourth is 6 fifths wide, major seventh is 5 fifths wide, major third is 4 fifths wide, major sixth is 3 fifths wide, major second is 2, and perfect fifth is 1 of course.

I've found that the way we hear chords is quite a complex balance of things. And what I write below will only work if you follow Pythagorean / chain of fifths logic. (but you're talking about 12tet anyhow so this shouldn't be too much trouble I hope)
One of the things that makes this balance is the number of tones making the chord (octave equivalent tones not counting). Where a higher number of tones makes remote intervals more acceptable, we get a sort of averaging. (perhaps as easy as dividing the combined remoteness of all intervals by the number on tones of the chord, though I have not tested any actual formulas)
One of the other things that makes the color of a chord for a large part is its enharmonically equivalent simplest interpretation, meaning that while for instance a diminished fourth dyad has a strong dissonance, it still shares a large part of its color from a major third dyad (and when tuned just in Pythagorean they are very close in tuning this is the reason). In the case of the augmented triad its simplest enharmonically equivalent interpretation is still an augmented triad and it is therefore the most enharmonically remote "strong" colored chord for 3 tones there is.
If we go for example one enharmonic step further and play a 3 tone chord with an augmented second / diminished seventh, lets say C - Eb - F# then there is the simpler enharmonic equivalent chord of C - Eb - Gb. One needs 4 tones of for instance a diminished seventh chord to indicate the "strong color" of diminished seventh without the possibility of an enharmonically equivalent simpler interpretation.
This is the main thing at play for the augmented triad.

There is also the thing that above the true root of any chord there is always a perfect fifth implied, so in the case of an augmented triad this gives either C-E-G-G# or C-E-G-Ab or C-Fb-G-Ab (last one is very hard to indicate as such though) if one of the tones of augmented chord is indicated as the root. So there is also an element of roughness based dissonance here (even though the perfect fifths is not played, it is still there in our brain and still generating roughness).

The combination of these things, enharmonic remoteness, number of tones, simpler enharmonic equivalences, virtual perfect fifth above root, and roughness (I use a modified and simpler version than the one coming from sinetests) gives the total dissonance of the chord if it is played without any further context.
In context more things come into play (for a large part because the dissonance then becomes relevant to the the other chords and its place in the meter etc).

My description may seem a bit odd at first, but it works for all chords without exception.
It is by far the best model for consonance/dissonance I've ever come across (and the only practical one I'm aware of).
I've been busy with other things but I'll soon put it in formula form and make a web interface for it.

Kind regards,

Marcel de Velde

> On Facebook XAII, Aidin Foroughi asked:
> "Quick newbie question again.. Why does the augmented chord in 12TET > sound so bad? I mean, the diminished is obvious but the intervals in > an augmented don't theoretically sound as bad as they do in reality .. > where am I wrong?"
>
> People have disagreed with him, mentioned 3HE, etc; and asked him, > "what about C-E-G#-B", etc.
> I'd like to ask "what about D-F#-C-E-G#" (but I'm not on Facebook).
>
> Steve M.
>