Two well-known simultaneous inference procedures for balanced one-way layout are Tukey's T-method and Scheffé's S-method. The T-method gives short simultaneous confidence intervals for all pair-wise comparisons while paying a high price for high-order comparisons. The S-method is appropriate when one has uniform interest in all contrasts, but gives long confidence intervals for pairwise and other low-order comparisons. In this paper we propose a family of procedures intermediate between the T- and S-methods. The resulting simultaneous intervals are shorter than Scheffé's for low-order comparisons, shorter than Tukey's for high-order comparisons and shorter than both Tukey's and Scheffé's for intermediate comparisons.