> The tension arises because the "Seventh" (A) is not a new addition, but a residue of the previous geometric state.
That residue in m.2 is common-tone voice leading. It's a technique that was used throughout the common-practice period to avoid tension, not introduce it. I'd bet the progression in m. 1-2 could be found in the figured bass at the beginning of a slow movement by Telemann or another Baroque composer.
Speaking of Baroque composers-- in the Coda of the 4th Ballade, Chopin has an exquisite passage of basso continuo plus accompaniment that would be right at home in a minor key aria by Handel. Except that:
1. There's no melody being accompanied.
2. It moves about 4x faster than it would have in the Baroque era.
I'd love to see a pianist play that passage by suddenly looking up and frantically nodding cues to an invisible, demonic singer.
You are absolutely right about the voice leading. Acoustically, it is the definition of smoothness.
When I wrote 'tension,' I was describing the Geometric Re-contextualization visible in the visualizer. The note (A) stays put, but the 'world' (the Surface) rotates underneath it.
To be clear: The Grammar is simply my attempt to describe the movements I observed in the 3D structure. I am not a theorist; I am a builder. I see the Structure itself (the lattice) as the real contribution.
My hope is that theorists will see this structure and develop their own grammars for it. I would love to see a 'Baroque Grammar' or a 'Jazz Grammar' that maps these geometric paths according to their specific stylistic rules. The geometry is neutral; the grammar is just the lens I used to read it.
That residue in m.2 is common-tone voice leading. It's a technique that was used throughout the common-practice period to avoid tension, not introduce it. I'd bet the progression in m. 1-2 could be found in the figured bass at the beginning of a slow movement by Telemann or another Baroque composer.
Speaking of Baroque composers-- in the Coda of the 4th Ballade, Chopin has an exquisite passage of basso continuo plus accompaniment that would be right at home in a minor key aria by Handel. Except that:
1. There's no melody being accompanied.
2. It moves about 4x faster than it would have in the Baroque era.
I'd love to see a pianist play that passage by suddenly looking up and frantically nodding cues to an invisible, demonic singer.