Seventh chords

I've watched a few YouTube videos that purport to talk about seventh chords; normally they show the most frequent chords and ignore the more exotic varieties. I am going to put my point of view forward. First, some definitions: a seventh chord has four notes (root, third, fifth, seventh), and so has three intervals, where each interval can be either a minor or a major third. Thus there are 2³ = 8 different seventh chords. That's actually not true, as there are a few really exotic chords that have a flat 5 (i.e. the interval between the third and fifth is a diminished third).

Name	Third	Fifth	Seventh	Notes in C
Dominant 7	Major	Minor	Minor	C E G Bb
Major 7	Major	Minor	Major	C E G B
Minor 7	Minor	Major	Minor	C Eb G Bb
Minor Major 7	Minor	Major	Major	C Eb G B
Half diminished 7	Minor	Minor	Major	C Eb Gb Bb
Fully diminished 7	Minor	Minor	Minor	C Eb Gb Bbb
7#5	Major	Major	Diminished	C E G# Bb
7b5	Major	Diminished	Minor	C E Gb Bb
Major7b5	Major	Diminished	Major	C E Gb B
Major7#5	Major	Major	Minor	C E G# B

I've ranked the chords in approximately their frequency of use. I have used on occasion the 7#5 that sounds like an enhanced dominant seventh and I have used here and there the Major7b5, but I have never used or even seen the Major7#5 - this sounds like a fully diminished seventh chord, but it isn't.

Some of these chords, when inverted, give other chords; for example, C minor 7 in the first inversion is the same as Eb6 (Eb G Bb C), and C half diminished 7 in the first inversion is Ebm6 (Eb Gb Bb C). There are only three fully diminished chords, as the same notes in different inversions give differently name diminished sevenths - C Eb Gb Bbb, Eb Gb Bbb C, Gb Bbb C Eb and Bbb (A) C Eb Gb - are all the same.

I suppose that theoretically there are even more exotic chords, such as C Eb G# Bb, which might be called Cmin7#5. This sounds remarkably consonant when playing it on the piano, as the G# makes the chord sound like Eb6sus4, especially if it is played in first inversion with Eb in the bass.

There's a very easy chord shape on the guitar for FMajor7b5: first play an open position E chord (x22100), then move the fingers up a fret, leaving the top two strings to ring out (x33200). These notes are C,F,A,B,E. The first three notes are an F major chord, the fourth note is the flat fifth and the fifth note is the major seventh. 

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