Do interest rates go up with each mortgage you get?

[Silvia Camponelli]

Hi all, quick q: I'm planning on getting 5-7 smaller multi-fam properties or 1-2 large ones.  If I were to get 5-7 separate mortgages, would the interest rate banks charge on each property likely go up?

If so, would it make sense then to get the largest loan first?

Goal is to minimize borrowing costs here over the entire portfolio.  Portfolio lenders seem to want 5.5-8pct, which is higher than rate I qualify for.