This is something that I discussed fairly recently with a friend. I thought i'd put it out there for your perusal.
As I understand it, the socion sees quadra progression as going from alpha-beta-gamma-delta.
Although this may be true in some cases, It would seem that in terms of at least societal progression, societies do not really move through the four quadras, but really they cycle between three of them.
It would be that this cycle goes alpha-gamma-beta-alpha.
I see it that an aspect of social science, known as social progression law, is really a way to represent the quadras movement in a society.
Social progression goes from intellectual-aquisitor-warrior-intellectual. The fourth social type, the labourers occupy a special place within this.
The labourers do not have a social effect as such in society, they tend only to dominate if they do, for an extremely short period. However the labourers are essential to the society, as any society could not function without these people. So the labourers are sort of there but yet apart from it.
I think social progression law is representing the four quadras as thus:
Alphas - intellectuals
Beta - warriors
Gammas - aquisitors
Delta - labourers.
So therefore intellectual (alpha) - aquisitors (gamma)
- warriors (beta)
In times of plenty, the society has the age of the aquisitors (gammas), where commerce is plenty and there are many opportunities to create success and such financial endeavours. After a while this starts to run itself out, and in that new problems arise to the society. People become fearful of loosing what they have in such a scenario, the opportunities for such commerce are not what they were, therefore the warrior type/culture of society comes into play. Protecting what's already there, pushing into new territory by force to protect the wealth against other societies. After some time when this threat dies down, people look for a new change, away from both. It then becomes the age of the intellectuals-the alphas. Such things can perhaps be demonstrated when societies go through what is often called the Enlightenment period.
These conditions set by the Alphas, the intellectuals, sets the premise and leaves open opportunities for the gammas, the aquisitors come back into play, to make opportunities for trade, for aquiring things, for endeavour such as these, and therefore the cycle continues.
Each age of society requires the deltas, the labourers, in order to survive and provide the essential needs for each to exist and thrive, but they do not have an 'age' themselves.
I suppose this could represent political influences of the society also.
The law of social progression can be read here, if it's ok i'll post the link rather than copy or paste it, a summary from wikipedia:
Edit: I apologise in advance if there are any typing errors above: i'm typing from a cell phone on a long train journey home, I suppose looking for ways to keep myself occupied! Which isn't to say I don't think this could be interesting.