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On my older question Are the diminished chords not that common in modern popular music?, I got this answer:

The half-diminished (or m7b5) is usually suggested as the diatonic chord on the 7th degree and is fairly common in most styles of music.


That's right. Half-diminished sevenths are pretty common in popular music. However, if such chord is used, it would not be usually labeled as [Nico]m7b5, but rather [Yazawa]m6. (Believe me, I always have been labeling them as m7b5.)

For example, in a circle of fifths progression in C major, they would usually write:

C - F - Dm6 - Em - Am - Dm - G - C

What's the reason here?

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    The question as it stands needs some editing. [Nico] and [Yazawa] won't make sense to many English speakers. – Tim May 25 at 7:10
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    Don't see the relevance of your last comment. A lot of people use keyboards insead of pianos - all my 88 note ones have A at the bottom, but my 76 has E, and my 61s all have C. Also - to follow the circle, C-F-B-E etc. is considered up in 4ths (or down 5ths). Count the letter names. – Tim May 25 at 11:11
  • @user45266 // Yes. [Nico] [Yazawa] is a character from an anime called Love Live. (not Hate Live! Love the sinner, hate the sin...) And how did you find out my profile pic and username is shared by a character from another anime? – Maika Sakuranomiya Jun 4 at 9:44
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Depends on the bass note. If the bass note of that third chord is in fact D, it functions as a mi6 chord. But if you put a B in the bass, it's a mi7(b5). Alternately, if you put a G in the bass it could also be a G9 chord. All about context. All those chords have somewhat different sounds, so choosing a different bass note for one chord can change the tone of the progression.

Edit:

Even though Dmi6 and Bmi7b5 have the same notes, they're different chords with different functions so I wouldn't recommend writing "mi6" to indicate "m7b5 a minor 3rd down" unless you truly don't care about the bass note. You could always write Dmi6/B if you want that half-diminished sound but aren't confident others will understand mi7b5 symbols.

Also, shouldn't Bmi7b5 usually lead to E7 or Ema rather than Emi if it's going to E in the bass, since usually its function is the ii of A?

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    If you put a different, extra note in the bass - or anywhere - it changes the chord and its name entirely! – Tim May 25 at 7:21
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If I understood the question correctly, you seem confused about the chords Dm6 (D F A B) and Bm7b5 (B D F A), because they have the same notes.

These two chords have different harmonic function, so you would distinguish the two of them like that. The vii usually acts like a dominant, so you'd find it followed by some kind of tonic, whereas the ii isn't a dominant chord, so you can follow it up with any other chord you want.

You can see that in the progression you posted:

C - F - Dm6 - Em - Am - Dm - G - C

The Dm6 is used here as ii-iii, which is acceptable. If you had a Bm7b5, it would be something like:

C - Bm7b5 - C

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    I'd also point out that OP's example is a circle of fifths progression, which usually uses the "Bm7♭5" notation to show the fifth between B and F, and also between E and B. – user45266 May 27 at 22:06
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D minor6 (Dm6) is spelled D F A B.

D half-diminished (Dm7♭5)is spelled D F A♭ C.

D diminished (Do) is spelled D F A♭ C♭.

So, diatonically, Dm6 works in key C. Dm7♭5 cannot. It may be the weapon of choice in pop songs, but it's not diatonic to the key.

They are not the same chord, don't sound the same, so what's the question?

EDIT - thinking that now, the OP is considering Bm7♭5 and Dm6 both with reference to key C, another thought comes along.

Yes, both contain the same notes (as do C6 and Am7), so context would probably be a deciding factor. Often I find Bm7♭5 followed by Em, whereas Dm6 is followed by G (circle of 4ths/5ths idea). In root position, those sound believable. Using inversions always plays tricks with the ears, so it makes it difficut to pinpoit a 'correct' name. As already stated, it's also the best part of G9 - made up from G B D F A - but missing the important G note can't actually make it G anything!

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    I think you're misunderstanding. The OP is saying that Bm7b5 contains the same notes as Dm6. In the given example, Dm6 is substituting for Bm7b5. – Peter May 25 at 6:55

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