Relationship of amplifier sound to transformer quality?

Two tube power amp designers/builders who say yes are Tim de Paravicini (EAR-Yoshino) and Roger Modjeski (Music Reference). Roger will personally hand wind you one for $1,000. Tim says he can tell you how an amp reproduces bass by looking at the transformer.

Thanks bdp24. Do you know which transformer manufacturers are most highly regarded?

Jap manufacturer Hashimito transformers compete just as great as Audio Note hand-wounded transformers.

One of the reasons Jadis electronics sound so good is their hand-wound transformers. 

If the transformer is not wrapped with low frequency high permeability mu metal sound quality will suffer. No two ways about it. The toxic effects of the induced magnetic field extends to everything within what, about a foot of the transformer? At least.