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A7 = A Dominant 7 = A C# E G = R 3 5 b7
Am7 = A Minor 7 = A C E G (You will also see it notated as A-7) = R b3 5 b7
Am△7 - A minor, major7 = A C E G# = R b3 5 7
You can call the interval between the Root and the 7th in a dominant chord either a minor 7th or a flat 7.
I find it is easier and simpler to use flat 7 because it removes the confusion between a minor 7th chord and a minor 7th interval.
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Half diminished 7th chord consists of R - m3 - d5 - m7 and its symbol is a circle with a line through it
Fully diminished 7th chord consists of R - m3 - d5 - d7 ie. the 7th is a diminished 7th, a semitone lower than a minor 7th and enharmonically equivalent to the major 6th. The symbol for a (fully) diminished 7th chord is a small circle with no line through it.
There's a bunch of alternate symbols for chord notation, for example a triangle can be used for a major chord, etc
Its inaccurate to describe a 7th (or any other note in a chord or scale) as "flat" because it depends what its flatter than. For example in G natural minor, the 7th is F natural but is a minor 7th. And if the scale had a diminished 7th in, that would be F flat (not double flat). Of course, a diminished 7th is always (by definition) a semitone flatter than a minor 7th, and a minor 7th is always (by definition) a semitone flatter than a major 7th. In theory the interval of an augmented 7th exists too, but it would be rarely seen and possibly considered confusing/wrong, even though its technically possible!
@octatonic has mentioned b3 and b7, while @paul_c2 has referred to them as m3 and m7 . Is there a consensus?
From what I can gather it would appear that "minor 7th" would be most appropriate, "flattened" or "b" being a function of notes rather than scale degrees (and by extension diminished 7th instead of double-flattened 7th?
Is that about right?
Some interesting things that can crop up.
For instance Abm△7 has the notes Ab Cb Eb G. Cb is the minor 3rd (or flat 3rd), G is the major 7 (or just the 7th).
Ab7 (Ab dominant 7) has the notes Ab Cb Eb Gb. Cb is the minor 3rd (or flat 3rd), Gb is the flat 7 (or minor 7th interval).
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I'd not use "double flattened" but use "diminished" instead, for similar reasons.
I guess, talking amongst guitarists 99% would know what you mean by b7 but amongst pianists or other instruments, they might interpret it as whatever note letter it is, but with a flat (especially a pianist who obviously sees the notes with accidentals easily, as the black notes). Also bear in mind that another musician might be "thinking" in another key signature, eg you could be in Bb and regard the b7 in that key as Ab, where an alto saxophone player would be thinking in the key of G and the minor7 would be F natural for him/her.