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I've gone through a bunch of ♭VII-I and ♭III-I cadences in rock music. In C major, they are B♭-C and E♭-C.

Is there a specific name for the two cadences? So far, I've been calling them authentic cadence variants, since both ♭VII and ♭III are substituting for V. Others had been saying so, as well.

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No. A widely cited scholarly paper calls them just what you do, ♭VII-I and ♭III-I. It offers a plethora of adjectives to distinguish different kinds of cadences, but none have the centuries of weight of the terms for the cadences that Mozart used.

Edit: One might call them variants of the authentic cadence, but that's an awfully broad name, not the specific name that you ask for.

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    @MaikaSakuranomiya I'm not sure where you got that information from. No theory books I'v ever seen mention any kind of III as a substitute for V and bVII will never be viewed in classical as authentic and from classical perspective will pull to bIII (the relative major). – Dom Jul 13 at 21:10
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    It isn't clear to me in what sense you could call either bIII-I or bVII-I even variants of an authentic cadence. Maybe you could expand on this idea.... – David Bowling Jul 14 at 4:09
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    OP said that they were variants was the idea of OP's teacher (answer and its disbelieving comments were deleted.) – Camille Goudeseune Jul 14 at 4:12
  • I've totally seen papers that call cadences of III-I and VII-I type authentic variants, and VI-I and II-I plagal variants, e.g. in analyzing Rachmaninoff and other late Romantic music. – Mirlan Jul 14 at 21:20
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    @Mirlan that's still broader than what the OP hopes for. If those are called variants of authentic and plagal, then they exhaust all possible cadences ending on I, and one might as well replace the archaic word "plagal" with a postmodern "non-authentic." – Camille Goudeseune Jul 14 at 22:48

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