I'm reading about dominant seventh chords and this is a summary of the notation that my book has introduced:


here


After a few moments' thought, I can deduce the convention for denoting the different positions in which a dominant 7th chord may appear -- the intervals from the bottom note of the chord are listed to the right of the V.

I'm a bit baffled, however, by the way they are abbreviated. Why do we choose to omit the 3 when abbreviating the first inversion, but omit the 6 for the second inversion?

My only guess is that the notes which form a 2nd are the only ones listed, as that is enough to reconstruct the correct inversion of the original chord (the 3rd inversion being an exception to this as the bottom note itself is one of those two notes).

Is this what is going on, or is there some other reason? Or is it simply arbitrarily decided upon and I ought to memorize and/or get used to it?

A big part of it has to do with the abbreviations for triads, which are traditionally taught before the figured bass for seventh chords.

(By the way: this whole process is called "figured bass," and for the exact reason that you discovered: the figures measure the intervals above the bass!)

First-inversion triads are listed as 6/3, or just 6 for short. Second inversion is 6/4, and root position is 5/3 (but the abbreviated form is just blank).

As such, the first-inversion seventh chord, or 6/5/3, can really only be abbreviated to 6/5. Abbreviating to 6/3 would confuse it with the first inversion triad, and abbreviating it to 5/3 would confuse it with a root position triad, so 6/5 is really the only alternative left.

The same is true for the second-inversion seventh chord. 6/4 and 6/3 are already taken, so 4/3 is the only viable option.

Although 6/2 and 4/2 are both available abbreviations for the third-inversion seventh chord (6/4 is taken up by the second-inversion triad), 4/2 is probably more clear because it doesn't "skip" a pitch in the figures: 6/2 might seem to omit the 4.

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