What is the logic behind a chord consisting of the 3rd, 5th, and 7th of a chord with the root of the chord in the bass?
With my answer I’ll try to make understandable your question for other users. It is a translation of an abstract/thesis:
Opening lecture: The logic of chords Prof. Dr. Ingolf Max (Leipzig)
I have adapted the example to your question:
A logic can be understood as the specification of a set of constitutive rules for a formal game. That these rules set the internal meaning of all involved in this game.
Harmony can be understood as the internal relationship between chords and thus as the internal meaning of music.
The constitutive rules include the specification of the logical space. This is the room of the whole game. This also means that anything that can not be related to this space can not be grasped with this logic / game.
The logical space of the representation of chords and the harmonic relations between them is the chromatic scale (the logical tone space).
Constituent determinations include the indication of the smallest units to which all further operations should refer. In the logic of chords, I assume that not the sound, but the chord is the smallest unit. Chords are considered as two or more directly related intervals. That is, the smallest units of our logic are, so to speak, specific molecules.
A consequence of this representation is that more complex chords (e.g., "consisting" of 5 or more tones) may contain multiple subpatterns which are identical in their interval structure to certain chords.
The chord c'-e’-g'-b 'contains the contiguous subpatterns C-major c'-e’-g', E-minor e’-g'-b 'and the non-contiguous subpatterns c'-g'-b 'and c'-e’-b' (both certain "incomplete" seventh chords).
Another consequence of this representation is that even between two chords there are always harmonic relationships due to their internal structures. Harmonic relationships are not created in addition, but are always present with the sound of several chords.
That's a major 7th chord in its first inversion. According Jimmy Bruno's approach to studying "jazz" or extended chords on the guitar, if a chord has 4 notes, know that it has 4 inversions, all being;
That's it. Unless if you are looking for a different "logic behind".