How can I modulate the distance of a tritone using 3 secondary dominant chords?

Question

So, I have been composing this "Requiem for String Quartet" on and off for a few weeks now. I call it a Requiem because of its dramatic arc through lamenting melancholy, painful harshness, solemn peace, and a majestic feel. And to reinforce the fact that it is a Requiem, I have been using the Gregorian chant "Dies Irae" as a melodic basis for quite a bit of the piece. I have however been treating it differently in different parts, some parts more contrapuntal and others more solo. Some parts, I wrote the melody as close as possible to the chant and others I took some liberties with, but kept the overall direction the same. And I'm at the point where the chant goes from the Tuba Mirum to the Rex Tremendae. I intend to treat the melody in yet another way here in the Rex Tremendae, having the strings in a rhythmic unison, giving a sort of chorale harmonization of the melody.

I'm currently in the key of Ab major and I'm wanting to modulate to D major. I figured, since I have 2 question phrases followed by an answer phrase, I could do 3 little tonicizations instead of 1 big modulation without tonicizations. So like I would go from Ab major to a second key, keeping things quieter and maybe having the last chord of the phrase, the dominant seventh be staccato. Then I would go from that second key to a third key, again quiet and possibly ending staccato. Then I would go from that third key to D major, with a confirmatory crescendo.

So basically, I would be after something like this harmonically:

Pre-Modulatory Phrase |    Phrase 1   |   Phrase 2   |       Phrase 3      | Post-Modulatory Phrase
Ab:I -> chords -> V7 -> I -> Sec. Dom.
                           1: V7   ->  I -> Sec. Dom.
                                             2: V7  ->  I -> chords -> V7/D
                                                                      D: V7 -> I -> chords

And I put numbers 1 and 2 for the keys that I don't know yet because I couldn't really use letters, or it would look like I decided on the keys when I haven't. Anyway, the distance that I have to modulate is the largest distance I could possibly modulate, the tritone axis. This makes things a bit tricky if I don't want to use the very quick diminished seventh modulation(which I'm not wanting to do here) or the Tritone Substitution(I think it will sound too abrupt to use Eb7 as a Tritone Substitution after all the material that has come before in F minor, C minor, and Ab major).

Is it even possible to string 3 dominant seventh resolutions across a tritone axis to get this smooth modulation via 3 tonicizations? If so, is there more than one possibility or is there just 1?

Answer 1

We can go considerable modulatory distances by using secondary dominants of closely related what would be minor keys as secondary dominants of major keys instead.

One such example that gets you from A flat major to D major with 2 intermediate keys would be this:

A♭: I -> V7/iii = C: V7 -> I -> V7/vi = A: V7 -> I -> V7/IV = D: V7 -> I

As you can probably tell by the order of the use of V7/iii and V7/vi, this is obviously not the only such way, and you can pick another way if you like.

Answer 2

We can re-write the question as how one makes x+y+z=6 mod 12 then reinterpret the numbers as scale steps. A quick answer is -2-2-2 or 2+2+2 moving the harmony by whole steps. Another is 1+2+3 (in some order) making moves of a minor second, a major second, and a minor third; this can be done as -1-2-3.

Other possibilities are +4+5-3 or 4-3+5, etc.

Answer 3

Consider that a commonly used key change, or probably modulation here, is up a semitone. That's easily achieved, and used often, as the pivot V of the new 'key' contains one of the old tonic notes - the tonic itself.

That would take us to key A major. That happens to be the V of the target key - D, so it's quite simple to move into that. Thus, two simple, easily listened to changes, and we're there - aren't we?

So - starting in A♭, V(7) of next key would be E(7), taking us into key A. Which could then go ii-V-I, meaning Em, A, D, into the target of D major. I think the two changes would work, instead of the requested three, but there are so many different ways to get from A♭ to D, in so many different stages, I felt this is what would work well.

And, of course, there's always the diminished chords which always ease any 'strange' modulations.

Answer 4

There are many ways!

Quite stereotypical is using any tone as dim7/V to I46-V-I. With this we can go directly from Ab to G#dim7 (=vii°7 of A) instead of A you take A46 - A7 - D. (Note that the V46 chord is the same as the 2nd inversion of the tonic.)

But with this trick you can try any other way like e.g.

1. Tonization of Ab: Ab-Fm-Ddim7-Eb46-Eb7-Ab

2. Modulation to F: Ab-Bdim7-C46-C7-F

3. Modulation to D: F-G#dim7-A46-A7-D

another proposition would be using ii7-V7 (iv-V/I as secondary chords:

1. Ab-Bbm-Eb7-Ab

2. Ab-Bbm-C7-F

3. F-Gm-A7-D

As we can see ii is reinterpreted as iv-V/ of the next tonic

Answer 5

As I mentioned in an earlier comment, check out Max Reger's book Modulation. In it you'l learn that you can travel just about any distance by modulating through the Neaopolitan chord. In your target key of D major, the Neapolitan chord is an E-flat major triad with G in the bass (first inversion). This chord would likely be followed by either an A major chord by itself, or preceded by a D major chord in second inversion. I'll leave it to you to decide how to get there, but E-flat is easily reached from A-flat major. You could try something like this;

1st statement: Stay in A-flat

2nd statement: Use the modulation to D described above

3rd statement: Confirm the modulation with a strong cadence in D.

If you keep the main melody line as similar to itself as you can in all three statements the somewhat surprising modulation will be less jarring to the singers and audience.

Answer 6

In A♭ major you have possible borrowed chords ♭III C♭ major and ♭VI F♭ major. The parallel minor key, Ab minor, source of the borrowed chords, is a lot simpler to read as enharmonic G♯ minor where the two borrowed chords would become B major and E major.

The target key is D major. Descending fifths to that key would be B E A D.

So your point of departure could be through borrowed chords in A♭ major reinterpreted as a series of secondary dominants leading to D major.