Music theory - basic, ground floor, absolute beginner

In another thread ksiegel was talking about how he prefers to play A7 as 2130 (rather than 0100)

So I thought I’d exercise my new Shipway book-learning and work it out. (Can you hear my brain creaking with the effort?)

A7 is a dominant 7th. It’s a four note chord made by putting together the 1st, 3rd, 5th, and 7th notes of the major scale, counting from A natural.

ok, I got that. But how do I know which major scale? Which key?

(crash and burn)

So now I’m gonna cheat. An inversion is playing the same notes somewhere else on the fretboard. (I understand inversion, at least!)

0 - open G
1 - C plus one fret = C#
0 - open E
0 - open A, and the root note of the chord

G, C#, E, A

2 - G plus 2 frets = A, and the root note of the chord
1 - C plus one fret = C#
3 - E plus three frets = G
0 - open A and the root of the chord again

A, C#, G, A

Wait! Where did the E go?

Now I’m really lost. How many notes are in a dominant 7th? Three or four? How do we know what major scale to count through to harvest our notes? Why is 2130 an inversion if it’s missing the E? Is this like the Hawaiian D7 that has a note “implied”?

Shipway has 0100 and 2130 as A7 on the chord shapes chart in the back of the book, but doesn’t explain why.

I need coffee….

Edit: oh nooooooooooooo! There are three different 7 chords: dominant, major, and minor. Now I need a beer…
2130 isn't a 7 chord, it's a substitute for an A7. 7 chords have 4 notes. The root, the 3rd, the 5th and 7th. By doubling up on the root (A) on the 4th and 1st string, and moving the dominant 7 interval (G), or flat 7, to the 2nd string, they took away the 5th interval.

I would think of the 7 chords as:
maj7
Dom7,
and bb7 (double flat7) which is used in the fully diminished chord shape 1212 as opposed to the diminished triad or the half diminished 7 chord, which uses a flat 7.
 
2130 isn't a 7 chord, it's a substitute for an A7. 7 chords have 4 notes. The root, the 3rd, the 5th and 7th. By doubling up on the root (A) on the 4th and 1st string, and moving the dominant 7 interval (G), or flat 7, to the 2nd string, they took away the 5th interval.

I would think of the 7 chords as:
maj7
Dom7,
and bb7 (double flat7) which is used in the fully diminished chord shape 1212 as opposed to the diminished triad or the half diminished 7 chord, which uses a flat 7.
Ok, so that explains why the 0100 and 2130 didn’t make sense to me. Thank you!

You also clicked the diminished triad/flatting the note bit for my brain. Slowly, slowly, I’m getting it.

I feel like a toddler on a beach wearing a little floatie and dabbling my toes in the wavelets, yearning to be out surfing with the big kids!
 
Since dom7 chords have 4 notes and the uke has 4 strings, there are 4 inversion shapes of the 7 chord, not accounting for substitutions. A7 can be 0100, 2434, 6757 and 999-10. You can find other ways by using open strings, but these are the basic moveable shapes you can use to find any dominant 7 chord you want to play.
 
I think that at times I can get lost in the "theory" of music theory and miss the ear training part. There are lots of things to name and at times multiple ways to name them, plus detailed instructions for how to put these things together and take them apart.

However, it is hearing these things in action that actually allows me to understand a 7 chord. Of course, it is recognizing the sound in the context of a song or progression, but beyond that, once I can hear the sound, it represents a chunk of understanding. I know which note (and which finger) is the money note (as my teacher calls it), that makes the chord a dominant 7 chord. I know if I move that note up a fret it is a major 7 chords with its own recognizable sound. Or if I move that note down a fret it becomes a 6 chord with its recognizable sound. Things become familiar as sound and finger shape chunks, and I am not thinking about diatonic triads and which notes are flatted or sharped, though I could go back and figure it out.

Back to @Plonky-tonk's observations, there are a bunch of different 7 chords. In my last lesson I ran into the horribly named but wonderful sounding Minor-Major 7 chord. Strictly speaking you need four notes to make a 7 chord as you point out, but particularly on the ukulele where we only have four strings, and especially in jazz and blues where you would like to throw some other notes (intervals) into your chords to extend them, it is not uncommon to drop the 5th. In the example you gave, 2130, we are not really taking advantage of dropping the 5th since we are duplicating A on the G-string and A-string. If instead you played 4130 so you had BC#GA, by leaving the 5th out you would be attempting a 9 chord (dominant 9th?, I am unsure of proper terminology). If you had more strings or were playing a piano, maybe you would keep the E note somehow and properly extend your 7 chord to a 9.

The thing I like about the analogy to "using my floaties" and paddling around, is that experience and comfort in the water is pretty critical for learning to swim or surf. What little I have learned of music theory has come largely because of playing my ukulele.
 
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