If you "flip" all of the notes (so that a half step up becomes a half step down), then the G-B-D-F chord (which sounds nice) turns into the B-D-F-A chord.
Well, I suppose I should say the A-F-D-B chord, since the G "flips" to the A, the B "flips" to the F, etc.
What are these chords called? But, more importantly, how come the first one sounds nice but the second one doesn't? They have all the same intervals, so shouldn't they sound equally pleasing?
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This is tough to answer because everyone has a different interpretation of "nice." So whereas you might not think B D F A sounds "nice," plenty of others do!
What we can say is that they'll certainly sound different, since you're inverting the makeup of the chord. While G B D F is built (from the bottom up) with a major third and two minor thirds, B D F A is built (from the bottom up) with two minor thirds and a major third. Much like how tangrams create different images using the same pieces,
so too will these chords be different, despite being made of the same intervals.
But these chords are called inversionally symmetrical. In your case, all of the pitches invert (or "rotate") around the pitch axis of G♯:
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Chords are frequency components. A seventh chord (with a "blue" seventh from the overtone series having a frequency ratio of 7/4 rather than the nominal 9/5) has the frequency relations 1 : 5/4 : 3/2 : 7/4 with a greatest common divider of 1/4, two octaves below the fundamental: if you indeed use these frequency ratios, the resulting signal has a periodicity of the note two octaves below the fundamental note. Using a "proper" seventh and/or equal-tempered tuning breaks this mathematical relation but the sound is close enough to still hint at it.
Reverting the relations yields frequency ratios 1 : 4/5 : 2/3 : 4/7 with a greatest common divisor of 1/105. Even if you use the last (lowest) note as reference, you get 7/4 : 7/5 : 7/6 : 1 with a common divisor of 1/60. Neither ratio results in a periodicity that has a reasonably straightforward relation to the original pitch.
So chords that are to some degree modeled after overtone series result "almost" in a periodic signal with a period in reasonably straightforward relation to the chord fundamental. Inverting the frequencies does not result in a similarly straightforward relation since the relation of "notes with overtones" cannot just be swapped to "notes with undertones". For starters, the various notes could not have overtones themselves for that concept to work: you'd have to work with pure sines. Those do not actually form chords as nicely as one would think as the blending of overtones is missing. And the concept of "undertones" just does not have a similar mathematical connection to periodicity as "overtones" have.
Chords and scales don't exist in a vacuum, and the tradeoff between their symbolic musical meaning and the underlying physics falls apart when inverting the frequency relations: in that case, only the symbolic meaning remains.
And that, on its own, is more arbitrary than we are used to.
I agree that to the normal ear, B D F A does not sound particularly nice, however all chords such as these sound nice in their rightful place - the example you gave of flipping a Gmaj7 gives you a Bmaj7 diminished which is commonly used in jazz.
The reason simply inverting 'nice' chords doesn't always give you another 'nice' chord is because of which notes sound prominent. The bass note and the highest note tend to stand out.
In the case of the G chord, the bass note G stabilises 3rd, 5th and 7th so they do not sound like a diminished chord. If you have heard the rule that in orchestration you should not 'double the third' then you may understand this further. Putting the 3rd (B) at the top - as you have done in your flipping - is a bit like doubling the 3rd (B) in an orchestration. It makes it more prominent and takes over the rest of the chord, making it sound diminished.
The reason inverting intervals does not work in the way you want is because of the way degrees and intervals of a scale are found. For example: take the key of C. The 4th is F and the 5th is G. This is working up from C. However, F to the C an octave above is a 5th, similarly G to that same C is only a 4th. Just as velocity changes when direction changes even when the speed is the same; intervals change when the direction changes even if the notes used are the same.
