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Why is chord labelled as IVb7?

In Hooktheory II, there is an example where you are in the minor mode, but borrowing the dominant 7th IV chord from the dorian mode. This chord is labelled as IVᵇ⁷ in the hookpad example (picture attached), but in the text it’s labelled as IV⁷.

I’m not sure about a couple of things:

  1. Which of IVᵇ⁷ and IV⁷ is correct? Or are both correct and referring to different things?
  2. If IVᵇ⁷ is correct, why is that the case?
    2.1. Is it because the b7 in IVᵇ⁷ refers to the b7 relative to the major scale? This doesn’t make much sense to me because both minor and dorian have a b7 relative to major.
    2.2. Is it because the dominant 7th IV chord has degrees 4, #6, 1, 3 relative to the minor scale, and the #6 is converted to b7 to make it more obvious that it’s a type of 7 chord? But then that’s still weird because usually the 7 in IV⁷ refers to the fact that we’re adding the 7th note from the bass (which is 3 in this case), not to the 7th degree from the root.

If anyone could clarify this, I’d be very grateful!

OK so I managed to figure it out - it turned out to be something completely different. I’ll explain below if anybody else is interested.

If you go to hookpad and select a minor key, and then choose to borrow from the major key, and then go through all the chord extensions from triad to 13th, you’ll see the following patterns emerge:

Completely ignore the accidental before the roman numeral, it’s not important to the superscript numbers. The superscript accidentals have nothing to do with which key you’re in or where you’re borrowing from, and only to do with the intervals within the chord itself.

You’ll see that 7 unique 13th chords are generated with different intervals in between them. What you’ll also see is that the rest are defined in terms of the first I¹³ and ii¹³ - this is true for ALL the chords you’ll see (e.g. even if you’re borrowing from harmonic minor or phrygian dominant).

For example, notice that the IV chord is the same as the I chord except for the #11. This is because you take the I¹³ chord, move all the notes up 5 semitones, and then add a #11. On the keyboard, if you make a C¹³ chord (which will be all white keys) and move them all up 5 semitones, you’ll be left with a chord that has a single black key, which happens to be the 11 of this chord starting on F. If you #11, you’ll have all white keys again, and you get F¹³, which is a different flavour of ¹³ chord to C¹³.

This is true for all the rest as well:

  • The V chord looks the same as I but that’s only because the b7 is implied in V⁷.
  • The vii half-diminished b13 chord is the ii¹³ minor chord but with a b5 (from it being a half diminished) and b9 + b13 from the diagram.
  • Something like #iiiᵇ¹³ means: start with ii, move it up one semitone, and flatten the 9 and 13.

The only exceptions are when you need to disambiguate whether something is a major/dominant 7 (thanks to the above convention that V7 is technically Vb7):

  • When borrowing from dorian, you get IVb7 to specify that it’s a dominant 7
  • When borrowing from lydian, you’ll get V#7 to specify that it’s a major 7
  • When borrowing from locrian, you’ll get bV#7 to specify that it’s a major 7
  • When borrowing from mixolydian, you’ll get Ib7 to specify that it’s a dominant 7

Works even for very weird things, like when borrowing from phrygian dominant you get a iv#7b13 chord - applying similar logic to before gives the correct chord.

Hope this helps somebody else! This wasn’t at all obvious to me and took me a while to figure out with some false steps along the way.