The F7 Chord

If I hear "a moveable C shape," I think 5433. Others may say, "No, that's a moveable Bb shape," and they'd be right.
Perhaps we should think of another way to describe the shapes. I first thought "Root on the first string," but that could mean that a D chord could be either 7655 or 2225.
I agree that there should be a naming convention for moveable shapes. I ran into the exact example Jim mentioned when making a reference sheet of moveable shapes for my own use.
I organize my moveables by root. For every chord quality I have four shapes: one for the G string, the C string, the E string, and the A string. And I only count minor chords because major chords are just mutations of minor chords.

It isn't very epigrammatic, but in my head I reference things like m7 rooted on E string.
Naming conventions? Here's an example. I usually see it as Cm7.
It's no wonder we get so confused.

Cm(maj7) Chord Aka: Cm+7 CmM7 CmΔ7
The C minor major seventh Chord has the notes C Eb G B​

... This is true. And when it comes down to it, even less essential is the fifth, because in the vast majority of standard chords—major minor and extended—the interval to the fifth is a perfect fifth (which is also the second overtone of the root), in which case it provides no significant additional information...
This is interesting in that I love the natural drone of a root (or 1 va)+5th and will generally sacrifice something else to retain it. 99% of the time it's just me and my voice trying to do my best. I have less than a 2 octave (comfortable) range from low A to high D (on a good day, E) and I rely on 5ths (especially, low G - the note I cannot sing) told hold things together.

I love the feeling in my chest of a pure 5th.

<edit1> Chords that require 4 or more notes force me to choose which note(s) to not play. This I love to do, as I can make the chord fill in where my voice is lacking.

<edit2> My books are full of 4-digit "codes" (fret #'s on strings, arranged by string #4-3-2-1) expressing the voicing of what I think fits.
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I often forget that F major is a nice key to play in on ukulele. A turnaround like "F F7 Bb Bbm F C7 F" would kill my hand on guitar! Both versions of F7 have their uses.
I did not know the F7 without the high C before this video. For those who like bar chords it definitely is a moveable E7, but I also prefer to play with four fingers like in video and then it is its own shape.
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I often forget that F major is a nice key to play in on ukulele. A turnaround like "F F7 Bb Bbm F C7 F" would kill my hand on guitar! Both versions of F7 have their uses.
That's why there are so few songs in F for guitar players, but with capo on first fret the turnaround is easy peasy with guitar E key chords (which would be more challenging for some on uke).
I have to apologize in advance because I don't know the proper names for the shapes, but I like the other chords that are close to the F7 shape, like Dm7 which also happens to be F6, and the Dm6, and the Ddim. In C chord/melody arrangements, since F is the four chord, often the B section or bridge contains nice sequences of chords starting with a 6 or minor 6 or a diminished and progressing, for instance F (or F6), Fm6, Dm7, G7sus4, G7, C. In this case it works our nicely in first position holding the C on the A string with the pinkie, but there is also a version working with closed chord shapes only up around the fifth fret. The F6 (barre fifth fret) and G7 (F7 shape played two frets up) are kind of backbone chords in C major chord/melody arrangements, so the one or two note variants to those chords come in very handy. I guess that is what I mean by close.
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