Finding dominant sevenths from the triad

On the mandolin, my "guerrilla" learning style has thus far been focused on constructing four-string chords, both for a fuller voicing and (let's be honest) to allow the right hand to cruise sloppily.

Structurally, it's been about regular permutations deliberately limited to moveable forms. So, for triads, I carried over what I'd learned on the Guitar Craft-tuned guitar, from a California Guitar Trio exercise: one form per inversion, root, first (third in bass), and second (fifth in bass), with the other strings advancing diatonically up the neck as well. So, for A major, the three inversions with A, C#, and E in the bass would be:

This proved to be so helpful in my learning that I sweated out the same thing for the diatonic sevenths...again, one form per possible inversion, advancing up the neck, moveable forms. Translating directly from the guitar, for Amaj7 as an example, the forms for A, C#, E, and G# in bass, are: The really geeky may notice that in first inversion I altered the triad, flatting the root to the major seventh; this allowed me to use the root on the top string, keep all four component notes in the chord, and have each form truly advance the bass and treble strings up the neck each time. It was an arbitrary choice, but it helped me construct a box that really helped me to "see" the chord forms advancing up the neck.

On the mandolin, I ran into another problem: some of these forms are simply too tight to work well on the smaller instrument. (By contrast, stretches like the second-inversion maj7 voicing are very workable on the mandolin while being a real workout on the guitar.) I realized that I would eventually need to find other convenient voicings for the mando.

Somewhere in there I ran across Jethro Burns' outstanding book, which advocates the concept of the three-string chord (even for chords with more than three tones) on the basis that it can be altered more easily than a four-string chord. True, but that introduces an unintended problem for the beginner: how can you understand what chord tones you can omit, before you've got a good grip on where the possibles are? I ignored the three-string approach at the time, staying with my fours and developing some confidence (this was a good decision, at least for me), but how funny what comes back in due time. With a good grounding in where all those notes actually are, and watching Mike Marshall articulate the same advantage of alteration in one of his excellent video resources, I actually started to "get it". In working with a couple of David Grisman tunes, the three-string chord really started to show its value, and with that came another revelation: the three-string chord is much, much easier on the tightest fingerings.

Great. So I've started to explore the three-string chord concept, with an eye to building a library of chords for my own use. And, quickly enough, the "omit tones" problem rears its head, to wit: how does one intuitively know where to find a three-string chord that omits the root?

As with many things I've learned about music so far, at least one example turned out to be a "duh" moment. I was playing around with the four suggested inversions of a three-string dominant seventh, as noted by Burns. Going up the neck, this is A7 with G (7th), A (root), C# (3rd), and E (5th) on top:

Now you may notice that the note A only appears once in those four chords, as the top note of the seventh-in-bass inversion. So: how the heck do I quickly find an A7 that doesn't even have an A in it?

I found myself noodling a bit on this, and stumbled on creating the seventh out of each triad inversion, by flatting the root note by a whole tone. Whoa, there they are!


Then, what really bent me is that suddenly I also recognized that these root-omitted dominant seventh chords are enharmonically the same as the diminished triad build on the chord's third--in each inversion. That is, A7 with no root (C#-E-G) is also a C# diminished triad. This makes complete sense, of course, since a dominant seventh is nothing more than two minor thirds stacked on top of a major third, and if you omit the major third, what you're left with is two minor thirds--or a diminished fifth. Quo erat duhmonstratis. (Okay, maybe it takes a geek to appreciate that, but sue me, I love this stuff.)

Excepting the form with root on top, then, which kind of lives "in between" the third inversion and the first, I can now find any dominant seventh by finding the appropriate triad and altering it. This three-string chord thing may have some value yet!