i wonder about scott connection [phase shift]
A wye-delta transformer can be configured so either side lags OR leads by 30 degrees.  It does not matter which side is wye or delta or which side is higher/lower voltage.  If it is an IEEE standard connected bank then the lower voltage side will lag the higher (C57.12.00-1987, 5.7.2).  

Here is an example of wye-delta bank and a delta-wye bank in an ABC phase rotation system, connected so both low-side lags by 30 (i use lower case letters for low-voltage side).  Notice that the 6 phase-neutral voltage phasors (the ones with only 1 subscript) have the exact same relationship in either case.  



I have never analyzed the Scott Transformer connection until tonight.  Per this Wikipedia article it was developed to convert 3-phase to 2-phase in a balanced fashion (evenly loads the 3-phase system) as Charles mentions above.  Below is my newly minted phasing derivation for this transformer arrangement.  The key is to use KVL to calculate the voltage across the primary of the teaser - then rest of it falls nicely into place.  Both transformers have N:1 ratio, but we are only using 86.6% of the top one.  



I hope this helps.  I believe this answers your question about how the turns ratio used can affect the resulting phase shift on secondary - it clearly does in this connection.  It is just all about scaling and adding/subtracting phasors.

Here is similar development for the variation of the Scott transformer to which you linked.  They added a neutral connection in the Teaser for use on 4-wire system.