Why do the Canadian and American ways of writing chords symbols differ?

Question

I've been studying music harmony (for the RCM program), and my teacher explained to me that the American way of writing chord symbols slightly differs from the Canadian way. For example, the chord progression [V⁶₄ - V⁷ - I] would be written as [I⁶₄ - V⁷ - I] in Canada. Also, the V⁴ chord is written as V⁴₂ in Canada. My question is, what caused these differences?

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Just for reference, the music textbook that I'm referring to right now introduces the I⁶₄ chord as the Appoggiatura ⁶₄ Decorating the Dominant (7th) Chord. To this, my teacher explained that the I⁶₄ chord is what is used in Canada (which makes more sense, as the notes are part of the I chord). I believe he also said the reason it is known as the "Appoggiatura ⁶₄" is because in second inversion, the I chord seems like a V chord with the other notes as an appoggiatura.

There are exercises in the book where you need to create a Bach Chorale (in SATB style), and the chord symbols are given. The end cadences sometimes like look this...

 8 ---- 7
 6 ---- 5
 4 ---- 3
V ------------ I

...with the numbers showing the movement of the voices. I⁶₄ makes a lot more sense to me, but I'm just curious if there is a localization-based reason for this.

Answer 1

As someone trained at an American music school, I would interpret those two chord progressions very differently, and contest that the system we use is any different from the one in Canada, or any part of Europe, for that matter.

What some people call "American Chords" might refer to one of the many disparate styles of notating jazz lead sheets. It is also introduced when harmony exceeds the capacity of roman numeral analysis. For example, there's no way to write C^7sus4b9 with figured bass notation. Also, though you gain context with something like C: I6/4 it's far easier to sightread C/G for most jazz musicians. Back to the examples you gave us, I would consider your "American" example to not be a "good" progression, since 6/4 chords are traditionally only used in bass arpeggiations or passing tones, and in pedal or cadential situations. The I6/4 chord is one such cadential situation, so it's possible that someone has confused the notation with the fact that both the I6/4 and the V7 after it take on a dominant function. It is probably worth noting that, were we actually realizing figured bass, there would be no roman numeral (just the bass note and the numbers), making this a non-issue.

That said, there are dozens of different ways that Music Theory professors like to teach harmony, so it's possible that your teacher heard of a situation where the V notation was being taught as dominant function cadential 6/4 chord. I would disagree, since it's confusing, and I certainly wouldn't call it an "American" style of writing roman numerals. My (American) theory textbook teaches the correct notation, and then brackets both chords with a V symbol to show dominant function.

Meanwhile, here is a pretty exhaustive source for figured bass realization that's about all I can say on the V4 vs. V4/2 question. Figured bass can be abbreviated, but almost always leaves out a 3, 5, or 8.

If any of this doesn't make sense, perhaps you could provide us with some notational examples?

Edit:

If I understand your edit correctly, they're NOT using the roman numerals that you mentioned there--they're showing you two different things. The figured bass is one thing (over whatever note is the root of the V chord) and then the V-I is the progression of dominant to tonic. In my answer I mention how some textbooks are fond of bracketing together [I6/4-V7] = [V]. That appears to be what you show here.
On http://www.harmony.org.uk/book/voice_leading/cadential_6_4.htm, they refer to this as a "Double Appoggiatura Cadential 6 4." I don't necessarily disagree with that notation, since they're not putting the 6/4 right next to the V, rather, they're showing the figured bass movement and then showing the progression as two different things. In contrast, you really can't interpret V6/4 itself any other way than what I notated above.

Answer 2

Different authors write the cadential 6-4 chord differently. Some books write I64 and others use V64; either is acceptable. There are other alternative notations I've seen used in similar cases. For example the pattern D7-G7-C is sometimes written II7-V7-I and sometimes as V7/V-V7-I. I often think of it the latter way when composing or analyzing but the former if playing. I also think of patterns with a cycle of fifths (C-E7-A7-D7-G7...) as a unit rather than as iterated secondary dominants.

I imagine that I would write what was most useful for the point I was trying to make when discussing the pattern.