: Do C major and B-flat major have a deep connection? I decided to research where I can "easily" modulate from C major using dominant seventh chord (modulating to a major key we are). In this case I define the word "easily"

Let's take it bit by bit. Diatonic notes from key C - C D E F G A B.

C and D are in D7.

D and E are in E7.

F and G are in G7 (the dominant of C).

G and A are in A7.

A and B are in B7.

So already we can, using your criteria, modulate into G/Gm; A/Am; D/Dm; E/Em.

C and E are in C7.

D and F are in G7 (the dominant of C).

E and G are in C7.

F and A are in F7.

G and B are in G7 (dom of C again).

A and C are in D7.

B and D are in G7 (dom. again).

So keys F, B♭ and G are available.

That's a quick look at it!

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Do C major and B-flat major have a deep connection?

B flat major is the subdominant of C major's subdominant. That doesn't seem to be deep.

Question: Is it a 1) coincidence (overfit) 2) a bug in my calculation 3) a worthy observation worth a further meditation?

Given only those three choices I would say "coincidence." But really, it's just the outcome of the rule you created.

Requiring two common tones with the dominant seventh chord seems to categorically rule out using the subdominant of the new key. So we can't use these...

...where all the pivot chords are common to both keys.

Why categorically rule out those modulations to the modal regions? Other than to avoid the one to E minor with the E natural in the pivot that doesn't fit into B flat major.

The only thing that makes B flat major emerge is a arbitrary rule, not a real harmonic connection.

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I'll try to answer based on this comment from OP to Tim's attempt at an answer:

look. In order to use B7 I want to place a chord before it which is
either a minor or a major triad from c major scale, that will contain
at least two tones in common with B7. And there is no such

Okay, so the criteria seem to be two common tones on a major or minor triad from the C major scale, held in common the with dominant seventh of the new key. Is that correct?

If so, Tim has a large part of it, which I'll repeat here:

C and E are in C7.

D and F are in G7 (the dominant of C).

E and G are in C7.

F and A are in F7.

G and B are in G7 (dom of C again).

A and C are in D7.

B and D are in G7 (dom. again)

To Tim's list, I'd also add:

D and F are also in B-flat7.

E and G are also in A7.

A and C are also in F7.

B and D are also in E7.

We also need to consider:

C and G are in C7.

D and A are in D7.

E and B are in E7.

F and C are in F7.

G and D are in G7.

A and E are in A7.

B and F can't make a major/minor triad.

Overall, these allow modulations to F, G, A, B-flat, D, and E-flat. It's true that all of these destination tonic notes are contained in the B-flat major scale. (They are also contained in the G minor scale.)

As to whether this is a "coincidence," I'm not sure what the definition is for a "coincidence." These destination tonic notes all occur in particular scales because of the rules you set up. A dominant seventh chord is composed of three stacked thirds: a major third, and two minor thirds. As seen above, any of these thirds can be held as common tones from a major or minor triad, but the fact that there are two minor thirds allows for some ambiguity in possible destination keys. Since major and minor triads also contain a perfect fifth, and the dominant seventh contains a perfect fifth, that interval can also be a potential source of the two common tones.

It's not surprising that B-flat would emerge as a possible destination, as the production of the dominant seventh added to a pre-existing triad in a key means you get to stack a minor third on top of many intervals in the scale to form a dominant seventh. Thus, you can stack a minor third on top of C-E-G to get a C7. The addition of B-flat to the C major scale thus is simply created by the constraint that says you must start with triads and then allow dominant seventh chords. You can thus create a dominant seventh on any note of the C major scale using those criteria, except for B. But note why you exclude B: B7 is a perfectly fine secondary dominant (V/iii) and would lead to a destination key of E, which would not be in your B-flat major scale. However, the arbitrary constraint of holding two common tones between a triad disallows this potential modulatory chord. If instead the constraint was "build a dominant seventh on any note of the scale," then one might equivalently argue that F major is the scale with the "deep connection" to C major.

It's the rules you set up in your system that determine the outcome.

Similarly, E-flat occurs as a destination key due to the minor third ambiguity I mentioned above. You can use the D-F third in C major as both a 5th-7th in G7 or as a 3rd-5th in B-flat7. Note again that this particular case is special: the B-flat7 introduces two chromatic tones into the modulatory dominant seventh. No other chord does this. Well, B7 would have done it too, but you excluded that for arbitrary reasons above, so we're just left with B-flat7, which creates the pathway to E-flat. If you instead used criteria that said, "Keep two common tones, but also only allow one new chromatic pitch to be included in the dominant seventh," then both B-flat7 and B7 would be excluded, thereby excluding both E-flat and E as destination keys (and again allowing F major to have the "deep connection").

It all depends on the rules you set up. By varying them, you could easily allow modulation to many more keys or, alternatively, constrain possible modulations to only a few keys.

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