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I am studying non-diatonic 7th chords in chord progressions and was wondering if most non diatonic 7th chords would still sound good in their triad version? I guess the answer is yes but are there ever any cases where using a 7th note can yield some chord progressions that may not sound as good in their triad form?

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It depends on the style of music you're talking about, but the immediate answer that sticks out to me is the diminished sonority.

The fully diminished seventh chord often found in root position. Diminished triads, however, are almost never found in root position. The idea is that, historically, composers didn't want the interval of a tritone above the bass if they could help it. In a diminished triad, the way to get around this is to put the chord in first inversion; this puts the tritone between two of the upper voices. (The half-diminished seventh chord is also often found in first inversion for the same reason as the diminished triad.) In a fully diminished seventh chord, however, it's impossible to not have a tritone with the bass, so we can use whatever inversion we like, including root position.

This would thus be a clear instance that would answer your question in the negative: if you had a score that used a fully diminished seventh chord in root position, you wouldn't want to remove that chordal seventh without also changing the subsequent diminished triad to first inversion.

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If you are treating the harmony as voices with voice leading, then just dropping the sevenths from the chords may result in various parts with gaps in the lines.

With sevenths...

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Omit the sevenths and the alto and tenor have gaps (rests) in the lines...

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...that may be considered "good" or "bad."

Re-arrange parts for only three voices...

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...but strictly speaking that is a new progression, at least from a voice leading perspective.

Of course the premise here is the seventh resolves to the next chord's third. If you omit the seventh, then some other voice needs to move to the next chord's third.

Another way you might think about this is: the seventh is general considered a defining tone for chords. By omitting the seventh you redefine the chords (you are changing the chord qualities) and therefore you are changing the progressions. Those triadic progression may be OK, but they aren't exactly equivalent to the seventh chord progressions.

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    In many cases, the seventh of a chord is even more of a defining tone than the root. If one is writing two-part music in C major, for example, a resolution like F-B E-C will function more strongly like a V7 - I cadence than G-B E-C would. – supercat Dec 6 '19 at 21:38
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Here's a chord progression that doesn't work if you leave out the 7ths:

  • Dm - Dmmaj7 - Dm7 - Dm6

But then again, you could rewrite it as triad+bass

  • Dm - A+5/D - F/D - F-5/D

(the "F-5" isn't really a so-called triad, but it's a three-note chord)

So, the answer to the question is: no, not all 7th progressions sound good if you drop the 7ths, unless you re-think what the essential triads behind the chord progression are.

But is there a chord progression that cannot be reduced to four voices while keeping the essence of the original?

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A good example would be V7 of IV. Without the 7th, people will tend to hear it simply as the tonic. If it then moves to IV, it will just sound like a I IV progression, not the resolution of a dominant.

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One way to think about 7th's is that they're 'flavor' to a chord. If you think about it that way, it's basically, "Will removing this bit of flavor hurt?" - and that answer depends quite a bit on what you're removing it from.

Here's an example - a basic blues background:

C - C - C - C7 - F - F - C - C7 - G - F - C - G

... would the C7's missing a Bb in the background mean much? Eh, not really. Likewise, someone throwing an F into the G chord to make it G7 isn't going to make much of a difference.

But what about this common chord progression, from songs like 'Till there was you' / 'Man or Muppet' / 'You're just too good to be true':

C - CMaj7 - C7 - F

... that flavor is incredibly important. First off, it's separating those chords into something different each bar, giving the chord a progression instead of the same chord repeated for several bars. And it's got an important voice perspective: it's a C, descending to B, to Bb, and finally into the A that's part of the F chord. Try playing a C triad for the first three chords when singing along to one of those songs, and you'll quickly see that there's something missing.

Likewise, the 7th can provide valuable context information about the chord. The easiest way to see that is to take a diminished chord. Imagine playing the triad: C + Eb + Gb. The problem is, that actually could represent three wholly different things, depending on whether Ab, A, or Bb is added to the mix:

  • Adding an 'A' = CDim. C + Eb + Gb + Bbb (eh, I'll just call it 'A')
  • Adding an 'Ab' = Ab7. Ab + C + Eb + Gb.
  • Adding a 'Bb' = Maybe Ebm+6 or Ab9

Or how about the first bit of the very first, very recognizable Mario Brothers theme?

D + F# + C + E (D9) => G + F + B + G (G7)

... try plucking those chords out on a piano... and them simplifying it down to just the D and G triads. It'll go from sounding like an immediately recognizable start to a familiar song... down to a simple chord resolution.

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