converting to concert pitch

There's more to it. Philosophical pitch is derived from a seven-note major key scale, based on the perfect harmonies between the notes. In fact it's more generally known as "harmonic pitch."

I've never actually gone around the Circle of Keys twelve times, flattening or sharpening one note at a time to maintain the perfect harmonic intervals in each subsequent scale, but it's obvious that you won't return to C=256hz on the 12th iteration. It will have a fraction with a really large denominator and probably will be closer to D or Bb than to your original C.

Please feel free to do the math, I'd love to see the result.

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Anyway, in the harmonic C major scale the ratio of C to the A above it is 24:40 or 3:5. As you have surely already learned, the essence of "harmony" is ratios of small integers. A major sixth is 3:5 and the minor third it mirrors is 5:6. Those are nice small integers and a minor third is the smallest diad that sounds harmonious (to Western ears at least).

Oh wait, it's not always, is it?

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The major sixth between F and high D is 32:54, or 16:27 Those integers are not very small at all! On an ancient harmonically tuned instrument, you might actually be able to hear the subtle difference between the two major sixths. One might feel more "minor" than the other.

The scientific scale resolves this and many other issues. It is a chromatic scale with 12 tones. Each interval has been adjusted slightly to make it possible to play in all keys with the same 12 notes. The intervals are wrong, but somehow our ears forgive the dissonance and hear harmony. Don't ask me about that, I don't understand it either! But if 16:27 sounds just about as good to us as 5:6, there's clearly more going on in our perception of harmony than we can understand.

Anyway, the ratio of every two adjacent chromatic notes is the same. It's a "scientific half-tone" and it is the twelfth root of two. You can run around the Circle of Keys with it as many times as you want and you still come back to the original tuning of the note you started with. Because obviously the twelfth root of two raised to the twelfth power equals two.

So when you start with your scientific A=440hz, you do not calculate the pitch of your scientific C by multiplying 440 by 3/5! That would be mixing up the scientific scale with the harmonic scale.

You get it by dividing 440 by the twelfth root of two nine times.

Again, I'll let you do the math. Show us how different the two calculated values are.