The octaves [the expansion of quaternions, octonians] I think would be something that would fascinate Pythagoreans. There seems to be some odd connection between the way you remember the order of octaves and the musical scale and the fact that Shabat is the middle day of the week according to the Arizal and the recent result by Viazovska that the lattice is the most compact way of packing oranges in eight dimensions. I do not have any great results to report about this. But it seems to me that it must be more than a random coincidence. I think there must be some close connection, but I have not been able to figure out what it must be. Of course Hegel would have a field day with this. He would claim that this is what he was saying all along with his triads within triads.


I am sure lots of people would consider these to be totally unconnected, but maybe I was influenced by reading the Ari a few years back. In any case the Triads you do not need to go to Hegel for. Kant was the originator in his limited scheme. Only Hegel expanded the original idea of Kant. In any case you can see this same scheme --not in the Ari so much as in Shalom Sharabi [known as the Reshash].[That is the author of the Nahar Shalom.]

The curious thing which continues to be of interest to me is the connection with Kant . In Kelley Ross's modification of Kant he a theory of necessity, which  results in the description of eight modes of necessity. See his PhD thesis chapter 5 sec 1

Kant  does not want to expand the original triad. But I wonder.

Even  in Kant we get this triadic structure: