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In Scott Joplin's Gladiolus Rag near the end there is a brief modulation from D♭ to A then back to D♭. I know a composer can modulate to any key at any time as he likes, but I suspect this is a common pattern in ragtime and perhaps even some classical pieces. Is this modulation some kind of identified type?

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By definition, a modulation must reach a cadence in the new key. Since there is no cadence on B♭♭ here, it's technically incorrect to call this a modulation.

Instead, he's simply using a B♭♭ chord right here. We understand this from a theory standpoint by calling this a "borrowed" chord or a "mode mixture" chord. What we mean by this is that Joplin is "borrowing" a chord found not in D♭ major, but in D♭ minor; in order to create a colorful harmony, he's mixing these modes (hence "mode mixture"). In Roman numerals, this would be a ♭VI chord in D♭.

But he does make this even more interesting by prolonging this B♭♭ chord by means of its own V, F♭ major. So not only is he using mode mixture, he's actually tonicizing this mode mixture! Great example.

PS: The cadential six-four that comes after this chord is, from a textbook standpoint, "supposed" to be on the downbeat of the measure. Interesting that it isn't!

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    Thank you. Part of my confusion stemmed from the fact that D♭ minor isn't a key signature (because it would have double flats), but I learned here that D♭ minor can still transiently exist and be borrowed from. I also assumed all the double flats used here were just a notational preference over other accidental changes. But its really great to learn that the double flats are actually communicating useful information. – J. Lenthe Aug 22 at 10:57
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    I was also assuming incorrectly that the F♭ was enough to make a cadence for the B♭♭ and therefore it was a modulation, but I suppose it has to come at the end of a phrase or section to be a cadence. – J. Lenthe Aug 22 at 11:04
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    @J.Lenthe The presence of a modulation tends to be considered in terms of duration: are we in the new key area long enough for the ear to adopt it as the new tonal center. Here it's short enough that, in context, it's not really heard as a change of key; just a change of color. Richard and Laurence are correct in this; my answer should have avoided the term. – Aaron Aug 22 at 20:49
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    As a gratuitous sidenote, I played in a ragtime quartet for years, and this was one of our favorite numbers. Joplin pulled out all the harmonic stops in this piece, but so elegantly that it's all coherent. It's one of his lovliest rags. – Scott Wallace Aug 25 at 11:26
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You're enharmonically correct that Joplin modulates to A, but that's misleading. Joplin is very careful to notate the passage as Bbb, which is the bVI of Db major. This is a reasonably common borrow from the parallel minor, leading to the dominant, which it does (Ab, in this case).

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There's no modulation here. Just a chromatic chord, the very common ♭VI. As we're in D♭major, that's correctly spelled as B♭♭.

(Now, if we respell it as A, the theory DOES get confusing! It's B♭♭.)

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    To be Bbb7, an Abb is required, and to be an Aug6, a G is required. – Aaron Aug 21 at 23:03
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I always try to explain a progression- or like in this example - a borrowed chord by transposing the music to C major key:

When the tonic C is extending to Ab, we have a lowered bVI degree: Ab-C-Eb, borrowed from Cm, the parallel key of the C.

As you see all other answers are correct! But I find it is more obvious or insightful explained the situation in C major or even in the language of solfege - for those who know it: I = Do Mi So (tonic) and bVI = Lu Do Ma, the VI of Cm.

Now in Db - this makes it more complicate to understand, the parallel is Dbm and the VI of Dbm is Bbb-Db-Fb (and not its enharmonic equivalent A-C#-E). But modern pop music notation isn't as 'careful' handling harmonic relations.

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    Thank you. I agree that it is easier to understand something like this in C major and that is a helpful technique I will use in the future. – J. Lenthe Aug 22 at 11:05
  • Our professor told us it isn't very good to think in C major for two reasons: (a)students aren't flexible enough when it comes to writing a, what we called, dictation, (b) people with an absolute sense of hearing, even though there are very few of them in a generation, can't help but feel what is written and that is C major. Just her words, not mine. – Invisible Aug 22 at 21:05
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    I wonder what your teacher thinks about the movable do ... my experience is that we become more flexible and movable ourselves ;) – Albrecht Hügli Aug 22 at 21:59
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This just seems like a temporary modulation to Cb Major. The only scale with both a Cb and a Fb

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