Turton has two different Geralds, one married to Gisele, and the second married (as her 1st husband) to Tutberge of Rheinfelden, daughter of Rudolph (who I have as an ancestor). Turton has Tetberge married 2ndly to Louis of Faucigny. AR has one Gerald m. to Gisele and 2ndly to Thietburga, widow of Louis of Faucigny (note the order of Tetberge/Thietburga's marriages is reversed). AR is considered, in general, to be "newer & more reliable" than Turton, but I am following Turton on this one.
AR has some possible "hickups" with its pedigree at this point: AR describes the first wife as "dead in 1060" and the birth year of Aimon, the son born by the 2nd wife, as "about 1050". I think it is strange (and probably significant) that AR can't find a way to describe the 1st wife's death to be "in sync" with the 2nd wife's child birth. Also AR's death date of "1061-1080" for Gerald is a pretty wide range. One could envision a Gerald I, d. 1061, and a Gerald II, d. 1080, which is what Turton has. In fact I think that Turton is right - at least in there being two Geralds. I will follow AR in other items, including the order of Tetberge/Thietburga's marriages..